T H E INTEGRAL SYMPLECTIC GROUPS A N D T H E EICHLER T R A C E OF Z ACTIONS OF R I E M A N N SURFACES P by QINGJIE Y A N G B.Sc. (Mathematics) Peking University, 1982 M.Sc. (Mathematics) Academia Sinica, 1985 A THESIS S U B M I T T E D IN P A R T I A L F U L F I L L M E N T O F T H E REQUIREMENTS FOR T H E D E G R E E OF D O C T O R OF PHILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES Department of Mathematics We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH C O L U M B I A March 1997 © Qingjie Yang, 1997 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Abstract Every conformal automorphism on a compact connected Riemann surface S of genus g gives rise to a matrix A in the integral symplectic group S P ^ Z ) by passing to the first homology group. If g > 2 then A has the same order as the automorphism. We consider the converse problem, namely which elements of finite order in SP2 (Z) are induced by some automorphism on some g Riemann surface S of genus g? A related problem is the determination of the conjugacy classes of torsion in S P ^ Z ) . To explain one of our main results let f(x) € Z[x] be an irreducible "palindromic" monic polynomial of degree 2g, that is one satisfying x 2g /(0) = 1, and let £ be a fixed root of f(x). — f(x) f(l/x) and Then there is a one-to-one correspondence between the conjugacy classes of integral symplectic matrices with characteristic polynomial f(x) and the classes of certain pairs (a, a), where o is an ideal of Z[£] and a is an element of Z[£] satisfying certain conditions. In the special case where f(x) = 1 + x + x -\ 2 h x ~, p l p is an odd prime, this result says that the number of conjugacy classes of elements of order p in S P _ i ( Z ) is p 2(P~ )/ /II, where h\ is the first factor of the class number of the cyclotomic extension. 1 2 If X e SP2 (1j) has a reducible characteristic polynomial of the form f(x)g(x), where g f(x) and g(x) are integral "palindromic" polynomials and coprime with coefficients in Z , then we prove that X is conjugate to a matrix of the form X\ * X2, where the star operation is an analogue of orthogonal direct sum. We determine completely those conjugacy classes of elements of order p in S P _ i ( Z ) which p can be induced by some automorphism on a Riemann surface with genus (p — l)/2. A complete list of the conjugacy classes of torsion in SP^{1.) is obtained. We give a complete set of realizable conjugacy classes in SP4(Z). We also study the Eichler trace of Z actions on Riemann surfaces. If A denotes the set of p all Eichler traces of all possible actions modulo integers and B = {x € Z [ £ ] | x + x G Z } / Z , we prove that the index of A in B is h\. There is group isomorphism between A and Q,, the group ii of equivariant cobordism classes of Z actions. Finally, we determine which dihedral subgroups p of GL (C) can be realized by an action on a Riemann surfaces of genus g. g iii Table of Contents Abstract ii Table of Contents iv List of Figures vi Acknowledgement Chapter 1. 1.1 1.2 ". The Conjugacy Classes of Type-I 14 15 18 20 27 27 29 32 34 37 42 The Symplectic Spaces Symplectic Transformations Symplectic Group Spaces 42 50 54 Order p elements in £ P _ i ( Z ) p A n Example Cyclotomic Units Realizable Elements of Order p Torsion in SP (Z) 75 4 i 61 61 63 68 Symplectic Complements Minimal Representatives The Case of f(x) = x + x + l Realizable Torsion Chapter 7. 7.1 14 Symplectic Spaces Chapter 6. 6.1 6.2 6.3 6.4 Preliminaries f Chapter 5. 5.1 5.2 5.3 1 4 Ideal Classes S-Pairs The Correspondence * Class Number of V The Rational Integer Case Chapter 4. 4.1 4.2 4.3 1 Direct Sum of Symplectic Matrices S-Polynomials Strictly Coprime Polynomials .. Group Actions on Riemann Surfaces Chapter 3. 3.1 3.2 3.3 3.4 3.5 Introduction Motivations Main Results Chapter 2. 2.1 2.2 2.3 2.4 vii 78 81 85 91 2 The Eichler Trace of Z p Actions on Riemann Surfaces The Eichler Trace 95 95 iv 7.2 Equivariant Cobordism 109 7.3 Dihedral Groups of Automorphisms of Riemann Surfaces 113 Bibliography 117 v L i s t of Figures 2.1 Fundamental Domain 23 5.1 Fundamental Domain (order p) 70 5.2 P~c<j< 5.3 1< j < p - c- 1 6.1 Fundamental Domain (order 6) . . . . 93 7.1 7.2 Cobordism of g = 0 Ill Cobordism with Canceling Pairs . 112 VI (p-l)/2 73 73 Acknowledgement It gives me great pleasure to thank my supervisor Dr. Denis Sjerve. Throughout my research, he carefully and patiently directed every step. He spent countless hours discussing the problem with me, and made numerous constructive suggestions to keep me in progress. I must also thank Professors K . Y . Lam and E . Luft for their valuable comments and advice. The financial support of U G F and Department of Mathematics is very much appreciated. Finally, I thank my wife Grace for her support and encouragement. This work is respectfully dedicated to Grace Gao. . vii Chapter 1 Introduction This thesis consists of two parts. The first part is the conjugacy classification of elements of the symplectic group over a principal ideal domain and the realizability of integer symplectic matrices by analytic automorphisms of compact connected Riemann surfaces. The second part is about the "Eichler trace" of group actions of Z , the cyclic group of odd prime order p, and p £>2p) the dihedral group of order 2p, on compact connected Riemann surfaces. 1.1 Motivations The first problem that we consider in this thesis is the determination of the conjugacy classes of matrices in the integral symplectic groups SP2 (T>), where V is a principal ideal domain, with a n given characteristic polynomial. Classification up to conjugacy plays an important role in group theory. The symplectic groups are of importance because they have numerous applications to number theory and the theory of modular functions of many variables, especially as developed by Siegel in [32] and in numerous other papers. But our original motivation for studying this problem came not from algebra but rather from Riemann surfaces. Let 5 be a connected compact Riemann surface of genus g (g > 2) without boundary. Let T € Aut (S), the group of analytic automorphisms of S. Then T induces an isomorphism of Hi(S) = H\(S,Z), the first homology group of 5, T* : ff (5) x Let {a, b} — { a i , . . . , a , b\,..., b } be a canonical basis of -Hi (5), that is the intersection matrix g g 1 for {a, b} is where I is the identity matrix of degree g. Let X be the matrix of T* with respect to the basis g {a, b}, i.e. T (ai,...,a ,bi,...,b ) t g = (ai,..., g a , & i , . . . , b )X. g g Since T» preserves intersection numbers, X'JX = J, where X' is the transpose of X. Hence X E SP2g{1'), the symplectic group of genus g over Z . If we fix a canonical basis of H\(S), there is a natural group monomorphism Aut(5)^SP (Z), 2 9 see [13]. Clearly, the matrices of T* with respect to different canonical basis are conjugate in SP (Z). 2g D e f i n i t i o n 1.1. A matrix X E SP2 (Z) is said to be realizable if there is T E Aut (5) for some 9 Riemann surface S such that X is the matrix of T* with respect to some canonical basis of ffi(S). Two questions naturally arise. 1: Can every X E SP {1) be realized? 2g 2: If the answer to Question 1 is no, which ones can be realized? Note that Aut (5) is finite, so we only consider torsion elements of SP2 (Z). g To answer these questions, we need some knowledge of the conjugacy classification of S P ^ Z ) . For example, consider elements of order p, where p is odd prime. Any action of Z on p S determines a representation p : Z -¥ GL (Y), where V is the vector space of holomorphic p g differentials on S. If T is a preferred generator of Z p(T) E GL (C). g p then this representation yields a matrix The trace of this matrix, x = t r ( T ) , is referred to as the Eichler trace. It 2 is an element of the ring of integers Z[(], where ( = e p . Suppose there are t fixed points Pi,..., Pt of T . The fixed point data is described as a set of integers modulo p, {ai,..., at}, one for each fixed point Pj, such that T i a acts on the tangent space at Pj by counterclockwise rotation through 2ir/p. The Eichler Trace Formula then determines the Eichler trace of T as where the kj are determined by the equations kja,j = 1 (mod p), 1 < j < t. See [13] for a proof of this result. Suppose we have two such automorphisms of order p, T\ : Si -» Si, T2 : S2 ->• 52, where S i and S2 have the same genus g. Let Xi, X 2 T 2 be the symplectic matrices induced by Ti, respectively. Then Xi and X2 are conjugate in SP2 (Z) if and only if their Eichler traces S x{Ti) and x ( T ) are the same, see A . Edmonds &: J . Ewing [5] and P. Symonds [35]. 2 The Riemann-Hurwitz formula for an order p element T € Aut (5) is 9 = P9o + *r(t-2) where (1.2) go = g {S/T), the genus of S/T, and t = Fix (T), the number of fixed points of T . We shall show that ai + • • • + at = 0 (mod p) is a necessary and sufficient condition that there be some T with order p and fixed point data {ai,..., at}. This implies there are only finitely many possibilities for the Eichler trace for fixed g. Therefore, there are only finitely many classes of order p matrices in SP2 (Z) which can be realized. The minimal polynomial of an element of 9 order p is x ~ p l + x~ p 2 H \-x + l, which is irreducible over integer ring Z . Hence the minimum g such that there is a element of order p in 5 P ( Z ) is g = 2s Y > 1. We consider this special PJ L case, only - ^ - classes of order p matrices in 5 P _ i ( Z ) can be realized. But we shall show that v p the number of conjugacy classes of order p matrices in S P _ i ( Z ) is 2 2 hi, where hi is the first p factor of the class number h of Z[£]. So in general most of the order p matrices in S P _ i ( Z ) is p not realizable. Furthermore, we shall answer Question 2 for this case. 3 The second problem we consider is to determine how much information about the action of Z is captured by the Eichler trace. We want to answer the following two questions. p Question 3: What element x € Z[(] can be realized as the trace of some action? Question 4: What is the relationship between two actions, not necessarily on the same surface, if they have the same trace? The primary motivation for these two questions are the papers of J . Ewing ([6], [7]). 1.2 M a i n Results In this section we will give main results of our thesis. A l l theorems in this section except for Theorem 8 and Theorem 9 are completely original. Proofs of the results in Theorem 8 and Theorem 9 have appeared previously (see [6], [7], [35]), but our approach is entirely new. To explain our results we need to develop some notation. Throughout this thesis T> will be a principal ideal domain with characteristic not 2, that means V is a commutative ring without zero divisors, containing 1, in which every ideal is a principal ideal. Let T denote the quotient field of V. Let M MnxmCD) For by n x m ( £ > ) be the set of nxm matrices over V. For sake of simplicity we denote M (V) when n = m, and let I be the identity matrix in n A e M (D), n B <E M„ (X>), we define the direct sum of A and B as ni 2 A+B = ^ D e f i n i t i o n 1.2. M {V). n e M (V). (1.3) ni+n2 The set of 2 n x 2 n unimodular matrices X in M2 (£>) such that n X'JX =J (1.4) is called the symplectic group of genus n over V and is denoted by SP2 (D)n Two symplectic matrices X, Y of SP2 ('D) are said to be conjugate or similar, denoted by X ~ Y, if there is a n matrix Q € S-F^n^) such that Y — Q~ XQ. l Let 4 (X) denote the conjugacy class of X. Remark. The definition is meaningful and clearly SP {V) is a subgroup of GL {T>), the general 2n 2n linear group with entries in V. It is well known that every symplectic matrix in SP (T>) has 2n determinant one [1]. It is readily verified that X belongs to SP {T>) if and only if X' belongs to SP (V). 2n A X = where A, B,C, D E M (V). If X E SP {V) n 2n 2n Let B the following conditions are satisfied: AB' = BA', CD' = DC' and AD' - BC' = I (1.5) A'B C'D = D'C and A'D - C'B = I. (1.6) as well as = B'A, Conversely, if one of (1.5) or (1.6) is true then X E SP {T>). 2n Given two matrices Xi=\ Ai B \ x \€M (V) and 2ni /'A\ 0 0 A Ci 0 = X 2 )(D) ni+n2 2j B 2 1 + (1.7) „ )(P). 2 0 x 2 (D), by e% E SP ( 2 0 D C 2 n 2 0 ^ x 2 \0 It is easy to check that X *X B |eM D 2 2 2 2 \C we define the symplectic direct sum of Xi and X Xi*X B 2 2 P>i) ,Ci A X = 0 DJ 2 if and only if Xi E SP {T>), for i = 1,2. 2ni Given two matrices Yi=\ C\i ^C i 2 C\ \ 2 C l 22 EM {V) 2nix2n2 and Y = 2 (D\\ D\ \D i D 2 2 22 | GM 2 t l 2 x 2 n i (D) where C y € M (T>), G M D niXn2 tj / Yi o y we define the quasi-direct sum by (V), n2Xni = 2 Cn 0 Dii 0 £> C21 0 \0 0 G M-2(ni+n2) (2>). C22 Z? 2 0 2 1 12 0 2 By an easy calculation we see that if m = n Yi,Y € Ci2^ 0 0 J = n, then F i o V 2 (1.8) 2 G SP\ (V) if and only if n SP (V). 2 2n Definition 1.3. A matrix X G SP (V) is said to be decomposable if it is conjugate to a 2n symplectic direct sum of two symplectic matrices which have smaller genera; otherwise, X is said to be indecomposable. When n is even, X is said to be quasi-decomposable if it is conjugate to XioX for some X\, 2 G SP {V). X n 2 Given a matrix X G M (T>), we denote the characteristic polynomial of X by 2n f {x) x = \xI-X\. If X G SP (V), then f (x) is "palindromic" and monic, that is 2n x x f(l) = f(x) 2n This is because X'JX = J, X' = /(0) = 1. (1.9) JX~ J~ , f (x) x and l l = \xI-X\ = \xI-X'\ = = \xI-X~ \ l x \X~ll\\X-'\ 2n and /(0) = det(X) = 1. Definition 1.4. A polynomial f(x) in V[x] of degree 2n (n > 1) is called an S-polynomial if it is a palindromic monic polynomial. A n S-polynomial f(x) G V[x] is said to be irreducible over T>, or is an irreducible S-polynomial in V[x], if it can not be expressed as the product of two S-polynomials (in V[x\) of positive degree. Otherwise, f(x) is termed reducible over V. A n S-polynomial of type-I is an irreducible S-polynomial which is also irreducible in the common sense, all other irreducible S-polynomials are said to be of type-II. Given a separable S-polynomial f(x) of degree 2n, let Mf be the set of all symplectic matrices, whose characteristic polynomials are f(x), M f We use Mf = {X G SP (V) 2n over V, that is | f (x) x = /(a;)}. (1.10) to denote the set of the conjugacy classes of Mf in SP {P )> 2n In Chapter 3 we deal with the case that f(x) be a fixed root of f(x). is a separable S-polynomial of type-I. Let £ Then 1/C is also a root of f(x). Let TZ = £>[C], S = Then S is the quotient field of TZ. A n ideal (fractional ideal) in S is a finitely generated 7^-submodule of S which is a free X>-module of rank 2n. A n integral ideal is an ideal which is contained in TZ. Two ideals o, b are equivalent if there are non-zero elements X,fi€.TZ such that Aa = fib. We denote the equivalence class of o by [o] and let C denote the collection of equivalence classes of ideals. C is an commutative monoid with respect to multiplication of ideals. The identity is in Let Pf be the set of pairs (o, a) consisting of an integral ideal a and an element a G 1Z such that a = a A a' and a = a, where the tilde denotes that conjugate such that C a = {a | a G a}, A = C 1 _ n /'(C) a n d a > 1S t n e = complementary ideal. Two such pairs (a,a) and (b,6) are said to be equivalent if there are non-zero elements A, /x G TZ such that Aa = fib and AAa = \x\xb. We denote by (a, a) the equivalence class of (a, a). Let Vf denote the set of all classes of Pf. Suppose X £ Mf. that is XC, = (a. There is an eigenvector a = ( a i , . . . G TZ 2n ,0:271)' Let a be the X>-module generated by ai,... ,a , 2n It is easy to check that a is an integral ideal in TZ and a — a. 7 corresponding to C, and let a = A~ a'Ja. 1 Furthermore we will prove that (a, a) 6 P and that the correspondence \& : M/ -> Vf, {X) —> (a, a), is well defined (cf. Section 3.3). Theorem 1. $ is bijection. Theorem 2. If f(x) is a separable S-polynomial, then Mj ^ 0. If TZ is integrally closed, then C is an abelian group. Also we have that Pf — {(a, a) | oo = (a) and a = a} and V turns out to be an abelian group where multiplication is given by (a, a)(b, b) = (ab, ab) (cf. Section 3.4). Let Co denote the subgroup of integral ideal classes defined by (1.11) Co = {o € C | oo = (a), a = a for some a € 7£} Let U = {u € U \ u = u} and C = {ml \ u Elf}, + C C U + where ?7 is the group of units in 71. Clearly, and they are subgroups of U. We shall show Theorem 3. There is a natural short exact sequence i _> c/+/c A A Co -> i (1.12) w/iere 0([u]) = (V[(],u) and i/>({a,a)) = [a]. Consequently, for the special case V = Z , we shall show Theorem 4. Let q m be the number of elements in Mf, where f(x) is the m-th cyclotomic polynomial. Then =< hi, m = 2 (mod 4), m ^ 2 (mod 4), and m is prime power, m ^ 2 (mod 4), and m is not prime power, where 4>{m) is the Euler totient function. 8 If m is an odd prime p, then <p(p) — p — 1. Hence we have Corollary 1.1. The number of conjugacy classes of order p elements in SP _i(Z) is 2^" hi. p In Chapter 4 we introduce symplectic spaces and symplectic group spaces. Let V be a symplectic space of rank 2n. In Section 4.1 we define (I, &)-normal sets of V and prove Theorem 5. Let the I + k elements a\,... , a;, there are 2n — I — k elements be an (I, k)-normal set of V. aj+i,..., « ,ftk+i-, n • • • >fin inV a i , . . . ,a ,Pi,... such that ,P n is a symplectic basis of Then n V. We relate symplectic matrices to symplectic transformations, and shall give a necessary and sufficient condition for decomposition. Let f(x) be a reducible S-polynomial in V[x], f{x) wherep\{x),... ,p m u\(x),..., u {x) m m E T>[x] are mutually coprime S-polynomials. Then there are m polynomials E !F[x] such that ui(x)qi(x) H where qi{x) = f(x)/pi(x), +u (x)q (x) m = 1, m for i = 1,... ,m. We shall show Theorem 6. Let X E Mf. only if Ui(X)qi(X) =pi{x)---p (x), Then X ~ X\ * • • • * X , m for some Xi E M , Pi i = 1 , . . . , m , if and E M2 (V), for i = 1 , . . . , m . n To every S-pair (a, a), defined in Section 3.2, we shall assign a symplectic structure and a G m action on a, where G m is the cyclic group on a fixed generator g of order m (cf. Section 4.3). Therefore a becomes a symplectic G -space, denoted by [a, a]. m Theorem 7. Two symplectic direct sums [oi,aj] * ••• * [a ,a ] and [bi,&i] * ••• * [b ,6 ] are r r isomorphic as symplectic G -spaces if and only if r = s, and there is an rxr m s s invertible matrix Q = {Qij)> Qij € •?"[(], satisfying the conditions q^aj C bj (/or = 1,... ,r) and 61 01 Q, (1.13) wAere Q = (g^). In Chapter 5 we consider order p matrices in S P _ i ( Z ) . The proof of Theorem 1 gives us p a way to find symplectic matrices of order p. First in this section we find a symplectic matrix 27T1 X of order p such that ^t(X) = {Z[(], 1), where ( = e ' . Then we give a complete answer to Question 2 for order p elements in S P _ i ( Z ) . Let p ^ sin — = -r-f, sin£ for(fc,p) = l» (1-14) be the cyclotomic units of Z[£]. By the Riemann-Hurwitz formula, an automorphism T : S —> S of order p, where S has genus Y~, has exactly 3 fixed points. Let the fixed point data of T be PJ {a, b, c}, where 1 < a, 6, c < p — 1, and a + 6 + c = 0 (mod p). We use M ( a , 6, c) to denote the symplectic matrix represented by T*. T h e o r e m 8. <b(M(a,b,c)) = {Z[(],u u u ) a b a+b This is similar to a result of P. Symonds[35] which was proved by using the G-signature. But we use an entirely different method to approach it. C o r o l l a r y 1.2. Let X € S P _ i ( Z ) be of order p. Then X is realizable if and only if p V(X) = (Z[(],u u u ). a b a+b for some integers a, b with 1 < a, b < p — 1 and a + b 7^ p. In Chapter 6 we shall give a complete set of conjugacy classes of torsion in 5Pi(Z). In addition, a list of realizable classes in SPi(Z) is obtained. 10 In Chapter 7 we shall answer Questions 3 and 4. Let A denote the set of all Eichler traces of all possible actions, that is A={ eZl(} X = tr(T)| X (1.15) where T is any automorphism of order p on any compact connected Riemann surface S. A simple calculation with the Eichler Trace Formula (1.1) shows that x + X — 2 — t for any x £ A where x denotes the complex conjugate of x- Thus A C B, where # = {x€Z[C] X+ * € Z J . (1.16) In Section 7.1 we shall show that B is a free abelian subgroup of Z [ £ ] of rank (p + l)/2 and determine a basis. Thus a reasonable first step in describing A is to determine the "index" of A in B. Unfortunately, it turns out that A is not a subgroup of B, so this does not make sense. O n the other hand, the quotient set A — A/Z, that is the elements of A modulo the integers, is a group, in fact a subgroup of B = B/Z. We prove that B is a free abelian group of rank (p — l)/2 and that the index of A in B is finite. Theorem 9. The index of A in B is h\. This theorem has appeared previously, see the two papers [6] and [7] of J . Ewing. The first paper is quite technical. It contains Theorem 9, but stated in terms of Witt classes and G-signatures. The second paper is an elegant exposition of the first. Theorem 9 gives a partial answer to Question 3. We shall find free generators of A, thereby answering completely Question 3. See Theorem 11. To an automorphism T: S —> S of order p we associate a "vector" [g;ki,... ,kt], where g is the genus of the orbit surface S/Z , p t is the number of fixed points, and the kj are the rotation numbers. The rotation numbers are unique modulo p, but their order is not determined. From the Eichler Trace Formula (1.1) it is clear that x — tr (T) does not depend on g or on the order of the kj. If a cancelling pair {k,p — rotation numbers {ki,--- ,kt}, where 1 < k < p — 1, appears amongst the set of then an easy calculation shows that their contribution to the 11 Eichler trace is 1 1 C - l fe + - l ~ (p~ k ' Thus we can replace the cancelling pair {k,p - k} by any other cancelling pair {l,p — 1} and not change the Eichler trace. Given two such automorphisms T i : 5 i —> S\, Ti: S —> S2 2 I we have two "vectors" [g; ki,... ,k ], [h;li,... t ,l ]. Let x i and X2 denote the respective Eichler u traces. T h e o r e m 10. x i = X2 if, o,nd only if, t = u and the set of rotation numbers {ki,... ,kt} agrees with {li,... ,l } up to permutations and replacements of cancelling pairs. u T h e o r e m 11. A is a free abelian group of rank {p — l)/2. It is freely generated by the mod Z representatives of the (p — l)/2 elements: Xr,s = j- + ^T~[ + f^s^i' w h e r e 1 <r <s < p - l and 1 + r + s = 0 (mod p). We shall give some geometric content to these theorems by relating equivariant cobordism of Z p actions on compact connected Riemann surfaces to A. To explain this let 0, denote the group of equivariant cobordism classes of Z actions. We show that the Eichler trace induces a p natural group homomorphism (f> : A -» T h e o r e m 12. <\> : A—> is a group isomorphism. Finally, in Section 7.3 we study the realizability problem for dihedral groups in GL (C). g This is a special case of a general problem. A group G of analytic automorphisms of a Riemann surface S of genus g > 1 can be represented as a subgroup R(S,G) of GL (C) g by passing to the induced action on the vector space V of holomorphic differentials. The problem is to determine those subgroups of GL (C) g which are conjugate to R(S,G) for some S and some G. In 1983, I. Kuribayashi proved that an element A of prime order in GL (C) g 12 is realizable if and only if A satisfies the "Eichler trace formula" [14]. In 1986 and 1990, I. Kuribayashi and A. Kuribayashi determined all realizable subgroups of GL (C) for g < 5 (see [15], [16], [17] and g [18]). We consider the dihedral group D . 2p Let D 2p B be generators with orders p and 2 respectively. are integers < 1. If D 2p be a subgroup of GL (C), g D 2p is called an IR-group if tr(A), D 2p tv(B) is an IR-group for some choice of A, B then it is an IR-group for all choices. We shall prove Theorem 13. and let A and is realizable if and only if it is an IR-group. 13 Chapter 2 Preliminaries In this chapter we collect some of the preliminaries needed for later chapters. 2.1 Direct Sum of Symplectic Matrices First we state some properties of symplectic direct sum and quasi-direct sum, (X *X )' = (y y )' = (xox)(y*y) = 1 2 l0 2 (X *X )(Y oY ) 1 2 l 1 = 2 2 1 2 {Xy*X ){Y *Y ) 2 1 2 (Xiox )(yioy ) 2 2 X[*X , (2.1) YJOY{, (2.2) 2 (XxYj o (X Y ), (2.3) (x{Y ) o (X Y ), (2.4) 2 2 2 = (XM) = {X Y )*{X Y ). 1 2 2 X * (X Y ), 2 2 2 1 We assume that all matrix multiplications are suitable. L e m m a 2.1. Let X\, X , 2 1. X\ * X ~ X 2 2. (Xi * X ) 2 3. If X x * 2 *X Y\, Y 3 be symplectic matrices. Then 2 X\. = X 3 X, * (X 2 x ~ Yi and X 2 * X ). ~ Y , then 2 3 X *X ~Yi*Y . In the following we assume X\ and X x 2 4. X\ o X 2 ~ X 2 o 2 2 have the same genus X\. 14 (2.5) (2.6) 5. X oX ~{-X )o(-X ). 1 6. If X\ ~ X , 2 1 2 then Io X\ 2 ~/o! . 2 Proof. (2) and (3) are easy. To prove (1) we let Q = I oI 2ni genus of X{, i = 1,2. Then Q~ (Xi *X )Q l = X *X\. 2 (5) by using Q = / * ( - / ) . For (6), HX 2 1 + n 2 ) ( Z ) , where nt is the Similarly we prove (4) by using Q = 7o7, 2 = Q' XiQ, then {Q- *Q- ){IoX ){Q*Q) l 2 e 5P („ 2n2 1 = IoX . x 1 2 • In general the converse of (3) in Lemma 2.1 is not true, but we have Lemma 2.2. Suppose X\, X , Y\ andY are symplectic matrices, f 2 i = 1,2. Suppose fi(x) and X 2 and f (x) 2 2 (x) = f .(x) x are coprime. Then X\ * X ~ Y\ * Y 2 = fi{x), for Y 2 if and only if X\ ~ Y\ ~ Y. 2 Proof. The sufficiency part has been proved. We consider the necessity. Note that any P € M ^ 2 ^(V) can be expressed in the form ni+ri2 P - Pi * P + P o P , 2 where P i € M (V), P 2ni e M (V), 2 P 2n2 3 GM (V), x 2 P 3 - P y , X P 2 2 2 X 3 = 0 since f {X\) 2 = P y 3 2 and X P 2 4 4 and P 2nix2n2 P . Let P be a symplectic matrix such that (X XP Z * X )P 2 = P{Y 4 2 2 is invertible. Similarly, we get P X 4 = 0. GM (V) are blocks of 2n2x2ni * Y ). We obtain X P X = P y . Then / ( X ) P 4 2 3 X 2.2 = PiY u = P / ( y i ) = 0, which yields 3 2 Hence P i , P 2 are symplectic, therefore X\ ~ y i and X ~Y . 2 X • 2 S-Polynomials Before we prove the following lemmas we make a Remark. Remark. Let f(x) = g(x)h(x), where f(x),g(x) and h(x) are polynomials over V. Then if two of them are S-polynomials so is the third. Lemma 2.3. Suppose that p(x) is an irreducible monic polynomial of degree n. 15 1 If x p(^) — p(x), then p(x) is S-polynomial of type-I or p{x) = x + 1. n 2 If x p{±) n = -p(x), then p(x) = x - 1. Proof. (1) If n is even thenp(x) is an S-polynomial of type-I. Assume n be odd. Thenp(—1) = 0, so x + 1 is a factor of p(x); but p(x) is irreducible, hence p(x) = x + 1. (2) Similar to the proof of (1) since = 0. • L e m m a 2.4. Lei /(a;) 6e an S-polynomial and assume / ( ± 1 ) = 0. Then f(x) = (xTl) g(x) 2 where g(x) is also a S-polynomial. Proof. Differentiate both sides of x f( 2n j ) = f(x) to see that 2nx - fC-)-x ~ f'( -) 2n l 2n 2 = f'(x). 1 But / ( ± 1 ) = 0, hence / ' ( ± 1 ) = 0, f(x) = (x^fl) g(x). 2 (2.7) It is obvious that g(x) is an S-polynomial by the above Remark. • L e m m a 2.5. Suppose f(x) is an S-polynomial of type-II of degree 2n. Then f(x)=p(0)x p(x)p(-) n X where p(x) is an irreducible monic polynomial with degree n. Proof. We will prove this by using the Unique Factorization Theorem. If / ( ± 1 ) = 0 then f(x) = {x +~ l ) , by Lemma 2.4. We can choose p(x) = 1 ^ 1 . 2 Now we consider the case / ( l ) ^ 0 and /(—1) # 0. Suppose that f(x) = p\{x) • • -p (x), m where p\ (x),..., p m (x) are irreducible monic polynomials of positive degrees n i , . . . , n . m 16 B y the Remark, none of pi(x). ..p (x) m Since f(x) is an S-polynomial since f(x) is an irreducible S-polynomial. is an S-polynomial, /(x)=x "/(I)=x> (i)-..^p (I). 2 1 Note that x pi(^) ni f{x). m is an irreducible polynomial, and neither x + 1 nor x - 1 are factors of There is k ^ 1, say k = 2, such that x^p^) Pi(x)p (x) = (0)p {x). Pl is an S-polynomial, and therefore f(x) 2 P2(x) =p(0)x p(±), and f(x) n = It is easy to verify that 2 = pi(x)p (x). Let p(x) 2 = pi(x). p(0)x p(^)p(x). • n P r o p o s i t i o n 2.1. Every S-polynomial f(x) Then is a product of irreducible S-polynomials. Apart from the order of the factors, this factorization is unique. Proof. Without loss of generality we assume that neither x + 1 nor x — 1 are factors of because of Lemma 2.4. We know that f(x) f(x), can be written as a product of irreducible monic polynomials, f(x) = pi{x)p {x) • • • Pk(x)qi(x)q (x) • • • q (x) 2 where the pi(x) 2 t (i = 1 , . . . , k) are S-polynomials of degree 2r; and qj(x) (j = I,...,I) are of degree Sj. Then x fil) = x ^ (l)x^p ^) n 2 Pl • 2 • • x *p (l)x i C-)x >q ( -) 2r s k s 1 qi • 2 •• x'lqtil) pi{x)p {x) • ••p (x)x^q (l)x^q (l) • • • £ <a,(±) s = 2 k 1 2 So we have qi{x)q {x) • • • (x) = x ' {l)x *q {D • • • s 2 Note that x iqj(\) s qi s qi 2 (j — 1,...,/) are irreducible polynomials. is Ij ^ j such that x iqj{\) s Then for each x iqj(^), s = qj(Q)qi.(x). It is easy to check that qj(0)qj(x)x Jqj(j) s there is an irreducible S-polynomial. By rearranging the order of qj{x) we get f(x) = pi(x)p (x) • • • p (x) 2 k qi{0)qi(x)x qi(±) Sl ••• q {o)qm(x)x q {±) Sm m m • The second part is simple. 17 2.3 Strictly C o p r i m e Polynomials We consider two polynomials f{x) = ax g(x) = bx n n m H l + b -ix m m ha , 0 + •• • + b, 1 m ^ 0, b n D e f i n i t i o n 2.1. f(x) n n m in V[x]. Assume m > 0, n > 0, and a + a -\x ~ 0 ^ 0. and ^(rr) are said to be strictly coprime over V if there are polynomials u(x) and v(x) in X>[x] such that u(x)f(x)+v{x)g{x) E x a m p l e . Let p (x) n —x~ n x + x~ n 2 = 1 + • • • + 1. Then p (x), m if and only if m, n are coprime. A n d p (x) m and p {x) n (2.8) p (x) n are strictly coprime over Z have a common factor of positive degree in Z[x] if and only if m, n have common factor great than 1. Recall that the resultant of f(x) and <?(a;) is > m rows Res (/, g) = det (2.9) > n rows bo) j The result we want to establish is P r o p o s i t i o n 2.2. Suppose either f(x) f(x) or g(x) is monic, that is either a n and g(x) are strictly coprime if, and only if Res (/,<?) «s a unit in V. 18 = 1 or b m = 1. Then Proof. Without loss of generality, let a — 1. n x u(x)f(x) + x v(x)g(x) k x v(x) k = x k = q (x)f(x) k Any x v(x) k can be written as € T>[x], and Vk{x) has degree less than n or k ^ ( x ) = 0. We set u (x) and g(x) are strictly coprime, then for any 0 < k < r n + n - l . k + v (x), where qk{x),Vk{x) k = x u(x) + qk(x)g(x) e V[x], then k Uk(x)f(x) and u (x) If f(x) + v (x)g(x) = x (2.10) k k has degree less than m or Uk{x) = 0. We may write k «(x) = d f V - + d f V + • • •+• 1 -2 fc If we equate the coefficients of x m + n _ 1 , x m + n ~ , 1 in Equations (2.10), we obtain the 2 following equations: E + E w f= i+j=l i+3=l 0<i<n 0<i<m 0<j<n-l 0<j'<m-l 1, l=k, (2.11) 0, Z ^ A:. Considering this as a system of m + n linear equations in the c( )'s and d^'s, fc (fc) c (fc) m - \ i ' ' ' (k) taken in the order (k) >o ) ^ n - u " " ' >^o ' c w e s e / (m+n-l) ( m-l c e t n a t & '^ " " ' c e s (/>9) (m+n-1) 0 = where the £> is the determinant ,(m+n-l) "n-1 " " " ,(m+n-l)\ "0 I £> = det V c (1) c (0) L m-1 'Tl-1 ^0 l(0) n-1 Z (0) / Since D € V, Res (/, 5) is a unit. Conversely, assume Res (f,g) is a unit in V. Then we can retrace the steps through (2.11) and (2.10) for k = 0 and conclude that there exist integral polynomials u (x)f(x) 0 + v (x)g(x) = 1. UQ(X), VQ(X) such that • 0 Remark. It is well known that f{x), g(x) have a common factor if and only if the Res(/,g) = 0. 19 We apply Proposition 2.2 to L , the (m + n - 2) x (m + n - 2) matrix defined by m<n rA m< 1 1 1 1 (2.12) / J where the entries are given by iij — s 1) i < i < j + m - 1,1 < j < n - 1 or j - n + 1 < i < j, n < j < m + n - 2, 0, otherwise. It is easy to see that ±1, d e t ( L , ) = Res(p m (2.13) 0, 2.4 (m, n) — 1, n (m, n) 7^ 1. G r o u p A c t i o n s on R i e m a n n Surfaces Throughout the thesis all Riemann surfaces S will be connected, orientable and without boundary. By the uniformization theorem the universal covering space U of S is one of three possibilities: the extended complex plane C , the complex plane C, or the upper half plane BL The letter U will always denote one of these three. If G is a finite group acting topologically on a surface 5 by orientation preserving homeomorphisms then the positive solution of the Nielsen Realization Problem guarantees that there exists a complex analytic structure on S for which the action of G is by analytic automorphisms (see [27], [11], [9] or [25]). Thus there is no loss of generality in assuming that the action of G is complex analytic to begin with, and we will tacitly do so. 20 The orbit space S = S/G of the action of G is also a Riemann surface and the orbit map 7r : S —>• S is a branched covering, with all branching occurring at fixed points of the action. If x 6 S is a branch point then each point in 7r (a;) has a non-trivial stabilizer subgroup in G. _1 To any action of G on S we associate a short exact sequence of groups u n - > r A G - > i , (2.14) with T being a discrete subgroup of Aut (U) and II a torsion free normal subgroup of T, as follows. Let 7r : U — > S denote the covering map. Then V is defined by r = { 7 e Aut (u) I 7T o 7 = g O 7T, (2.15) In other words T consists of all lifts 7 : U —> U of all automorphisms g: S —• 5, g € G . The subgroup T is unique up to conjugation in Aut (U). See the commutative diagram below. U -1> 5 U 5 The epimorphism 9: T —• G? is defined by # ( 7 ) = 3 , where 7 and g are as in (2.15). The kernel of 9: T —>• G is LT, the fundamental group of 5, and is therefore torsion free. The Riemann surface S = U/TI and the action of G on U/TI is given by g[z]u — [7(.z)]n, where z 6 U, g G G, and 7 € T is any element such that # ( 7 ) = g. Here the square brackets denote the orbits under the action of n . The orbit surface S = U / T , and the branched covering IT: S —>• S is just the natural map U/II -> U / T , [^] n •->• [2]p. Conversely, suppose 1—>n—>-T—> G —> I is & given short exact sequence of groups, where V is a discrete subgroup of Aut (U) and n is torsion free. Then this short exact sequence corresponds to the one arising from the action of G on the Riemann surface S = U/II defined above. Thus there is a one-to-one correspondence between analytic conjugacy classes of analytic actions by the finite group G on compact connected Riemann surfaces and short exact sequences 21 (2.14), where T is a discrete subgroup of Aut (U), unique only up to conjugation in Aut (U), and II is a torsion free subgroup of T. It is known that the signature of T must have form (g; mi,..., m;), where g is non-negative integer, each mj is an integer great than 1 and a factor of \G\, the order of G. As an abstract group r has a presentation of the following standard form (see [33] or [10]): (i) (") t + 2g generators A ,...,A , x X ,Y t X X ,Y . U g g t + 1 relations A™ = • •• = A™ = A x , . . . , A [X 1 Fx] • • • [X , X ] = 1. 1 t u g g For brevity, we refer to T by T(g;mi,.. .,m ). Moreover, consideration of non-Euclidean area t implies the Riemann-Hurwitz formula (2.16) where 7 is the genus of U / I L Now suppose G is the cyclic group Z and T € Z denotes a fixed generator. Actions of Z p p p on Riemann surfaces correspond to short exact sequences l - ) - i I - > r - > Z - > l . We see that t times p T must have the form T(g;p,... ,p), where g and t are non-negative integers. That is, as an abstract group T has the following presentation (i) t + 2g generators A\,... , A ,X\, Y ,... ,X , Y . (ii) t + 1 relations A\ = • • • = A\ = A • • • A [X , Y{\ • • • [X , Y ] = 1. t x g x t x g g g Any such group can be embedded in Aut (U) as a discrete subgroup in many different ways up to conjugation. In fact the set of conjugacy classes of embedding is a cell of dimension d(T) = Qg - 6 + 2t so long as 6g - 6 + 2t > 0. See [3] and [4]. The genus of the orbit surface S/Z p is g and the number of fixed points is t. Figure 2.1 illustrates a fundamental domain for a particular embedding when g = 0 and t — 3. It consists of a regular 3-gon P, all of whose angles are n/p, together with a copy of 22 A 2 Figure 2.1: Fundamental Domain P obtained by reflection in one of its sides. The generators A\, A 3 are the rotations by A, 2 2-K/P about consecutive vertices, ordered in the counterclockwise sense. In this case the cell dimension is d(F) = 6g — 6 + 2t = 0, in other words, up to conjugacy in Aut (IT) there is a unique subgroup of signature (0;p,p,p). In a similar manner, when g = 0 and t > 3, a fundamental domain for a particular Fuchsian t times group T of signature (0;p,. T. ,p) is given by P U R(P), where P is a regular t-gon all of whose angles are ir/p and R is a reflection in one of its sides. In this case F is the Fuchsian group generated by the rotations A\,... , A through 27r/p about consecutive vertices. The dimension t of the cell is d(T) = 6g — 6 + 2t = — 6 + 2t > 0. Thus the embedding is not unique up to conjugacy in Aut (U). t times Let T be any Fuchsian group of signature (g; p,. , p). Then an epimorphism 0: T — > Z is determined by the images of the generators. The relations in V must be preserved and the kernel of 6 must be torsion free, so 9 is determined by the equations 0(Aj) = T % 1 < j < t; 9(X ) = T \ 0{Y ) =T \l<k<g. 6 k c k The following restrictions must hold: (i) The a,j are integers such that 1 < a < p — 1 and X^=i j = 0 (mod p). a 3 (ii) The b , c k k are arbitrary integers mod p, except that at least one of them is non-zero if t = 0 (this guarantees that 6 is an epimorphism). 23 p It follows from the first restriction that the only possible values of t are t = 0 , 2 , 3 , . . . . Conversely, given integers a,j, b^, c satisfying conditions (i) and (ii), there is an epimork phism 6: F —> Z p S = with torsion free kernel II and a corresponding Z p action T : S —>• 5, where u/n. The integer t equals the number of fixed points of T: S —> S and g is the genus of the orbit surface S/Z . p A well known result of Nielsen [27] says that the topological conjugacy class of T: S —> S is completely determined by g and the unordered sequence ( a i , . . . , aj). the notation [g \ a\,... We use , at] to denote the topological conjugacy class of the homeomorphism T: S —> S determined by this data. If g = 0 we use the notation [a\,... , at], and usually order the aj so that 1 < a\ < ... < a < p — 1. t Of particular interest is the case g = 0. Then the orbit surface 5 / Z p is the extended complex plane C and T has the presentation (i) (ii) t generators Ai,... ,At- t + 1 relations A\ = • • • = A\ = A • • • A = 1. x t The epimorphism 6 is given by the equations 0(Aj)=T% (2.17) where a i , . . . , at satisfy the conditions 1 < ai < ... < a <pt P r o p o s i t i o n 2.3. 1, and t ^ a j = 0 (modp). (2.18) There is a one-to-one correspondence between the set of topological conju- gacy classes of automorphisms T: S -> S of order p and orbit genus 0, where S is an arbitrary compact connected Riemann surface, and sequences [a\,..., at] satisfying the conditions in (2.18). The integer t is the number of fixed points and the rotation numbers kj are determined by the equations kjaj = 1 (mod p), 1 < j <t. 24 Proof. It follows from the above that we can associate to an automorphism T: S -> S of order p, where S is any compact connected Riemann surface such that the genus of S/X p is 0, a sequence [a\,... , at] satisfying the conditions in (2.18). According to the results of Nielsen two such automorphisms are topologically conjugate if, and only if, the associated sequences are identical. Conversely, given any sequence [a\,... ,a<] satisfying (2.18) we can construct an automorphism T: S —> S of order p and orbit genus 0 as follows. Let T be any discrete subgroup of t times Aut (U) of signature (0;p^7~^p). Then Equation (2.17) defines an epimorphism 6: F -> Z with a torsion free kernel II, and this in turn determines an automorphism T of order p on S = U/II. The topological conjugacy class of T does not depend on the embedding of T, only on the signature and the sequence [a\,... , at]. Thus the correspondence is one-to-one on the level of topological conjugacy. A particular embedding of F in Aut (U) is the one indicated above; that is, F is the subgroup generated by A\, ... , A*, where the Aj are rotations by 27r/p about the vertices of a regular t-gon P , all of whose angles are ir/p. See Figure 2.1 for the case where t = 3. T h e fixed points of this action correspond to the orbits of the vertices, and thus there are t of them, P i , . . . , P*, where Pj is the orbit of the vertex of rotation for the generator Aj. The epimorphism 6 satisfies 0{Aj) = T i, a and therefore 0(A^) = T , where the kj satisfy kjaj = 1 (mod p), l < j < t . This k• implies that the automorphism T : S —»• 5 in a small neighborhood of Pj is represented by A • , J a rotation about Pj by an angle of Ikjixjp. In other words the rotation numbers are the kj for this particular embedding. This completes the proof since the number of fixed points and their rotation numbers are invariants of topological conjugacy. • We conclude this section by answering Question 3 in the introduction. This is just a matter of determining the possible sets of rotation numbers. Thus let {k\, • • • , kt) be any set of t numbers satisfying 1 < kj < p — 1, 1 < j < t, and let aj denote that number such that kjQj = 1 (mod p) and 1 < aj < p — 1. 25 p P r o p o s i t i o n 2.4. 1+ £ * = 1 ^ i - € A, if, Proof. First suppose that x = 1 + 2^7=1 and only if, £ j = 1 a j = 0 (mod p). — £ A . Thus there is an automorphism of order p, T: S —> S, on some compact, connected Riemann surface S, such that x(T) = X- ^ we can assume that the genus of S/Z p action of Z p f * ac is zero. According to the results of this chapter the on 5 corresponds to a short exact sequence 1—> II -> T —>• Z is abstractly isomorphic to the group presented by t generators A\,... p —>• 1. Here T ,At and t + 1 relations A\ = • • • = A\ = Ai • • • At = 1. The epimorphism 9 is determined by the equations 9(Aj) = 1 <fcj;< n T i, a p — 1. In order that 0 be well defined it is necessary that Y^j=i j — 0 (mod j>). a Next suppose that we are given a set {ki, • • • , satisfying the conditions of the proposi- tion. Then the short exact sequence above determines a Riemann surface S and an automorphism T: S —> S realizing x as an Eichler trace. 26 • Chapter 3 T h e Conjugacy Classes of T y p e - I It is well known that there is an one-to-one correspondence between the conjugacy classes of matrices of rational integers with a given irreducible characteristic polynomial f(x) classes of ideals in Z[x)/(f(x)) and the [22], [31], [36]. It is also known that under some conditions, the matrix class generated by the transpose of X corresponds to the inverse ideal class, [37]. E . Bender generalized this correspondence to matrices over an integral domain [2]. In this chapter we extend these methods and study symplectic matrices over V with a given separable characteristic polynomial of type-I. In particular, we give the the conjugacy class number of cyclic matrices with characteristic polynomial a cyclotomic polynomial in the integral symplectic groups. In Section 3.1 we shall review some results of ideal classes, most of them can be found in [19], [23] or any book on ideal theory. In Section 3.2 we introduce S-pairs. We prove Theorem 1 and Theorem 2 in Section 3.3. In Section 3.4 we shall prove Theorem 3. Finally, in Section 3.5 we shall consider the rational integer case and prove Theorem 4. 3.1 Ideal Classes Let f(x) G T> [x] be a monic irreducible and separable polynomial with degree n and ( be a n fixed root of f{x). Let T be the quotient field of V and /C be the splitting field over T of f(x). Let TZ = T>[C] and S = F[C]. Then S is the quotient field of TZ and TZ C S C fC. We also denote the set of non-zero elements of TZ by TZ*. The trace of an element a in S is defined as follows. Suppose the n different roots of f(x) are £ i , . . . , ( n G K with £i = (. Let a = a + a\( -I 0 27 + a„_i(" - 1 G S. The i-th conjugate of a is denned by a® = a + aiQ-\ 0 h a - Q~ . Then the trace of a is l n X n (3-1) It is clear that if a G TZ, then Tr (a) 6 V. Suppose a\,...,a G «5. Then the discriminant of a i , . . . , n (a? a? a is defined to be n (n)\ al (n) a. A ( a i , . . . , a „ ) = det \^ eg* (3.2) ah(«)J, A standard result is that A ( o ; i , . . . , a ) = det (Tr (a^ay)). 2 n L e m m a 3.1. ai,..., a are independent over T if, and only if A ( a i , . . . ,a ) n ^ 0. n For a proof see [19]. A n ideal (fractional ideal) in S is a non-zero finitely generated 7£-submodule of S which is a free P-module of rank n. A n integral ideal is an ideal which is contained in TZ. Assume that a and b are two ideals in S. The product ab is the collection of all possible finite sums of products ab, where a G a and 6 G b . With this definition ab indeed becomes an ideal in S. Let ot\,..., a r G S. Then a = { £ i a i -I ideal by ( a \ , a ) . r h£ a r r | & G TZ} is an ideal in S. We denote this It is clear that (OJI, .. .,a )(@i, ...,&) r = {aidi,.. .,aiP , • • -,a 0i, • • • ,a 3 ). s r r s (3-3) A n ideal a is called a principal ideal if there is an a in S such that a = (a). If a, (3 G S, then (a) = (P) if and only if a and /? are associates, i.e. they differ only by a unit factor. Two ideals a and b are said to be equivalent if there exist non-zero elements X,pETZ, such that Aa = pb. In fact the collection C of equivalence classes of integral ideals forms a monoid. 28 Let a be an ideal in S. The complementary ideal of o is a' = ^aeS Let a i , . . . , Tr(ao)c2?J. (3.4) a be a £>-basis of a. There is a dual basis a[,..., a' in <S", that is a basis such that n n Tr (a^Qj) = 6ij, where Sij is the Kronecker symbol. This is equivalent to either of the following equations or j ; a M = V XVi*^ = k (3.5) k We also have o' = Va[ + • • • + Va' (3.6) n because if (3 — ^ OiC^ with a» € J", then aj = Tr (/3aj). The following lemmas are given here without proof (for reference see [19]). Lemma 3.2. Let f'(x) be the derivative of f{x), the dual basis of 1, C , . . . , and = bo + b\X + • • • + 6 _ i x n n _ 1 . Then is bo b -\ n (3.7) Lemma 3.3. 71' = 71/ (/'(C)). Lemma 3.4. aa' C 71'. 3.2 S-Pairs In this section we assume that f(x) is a separable S-polynomial of type-I and degree 2n. If Q is a root of / ( x ) , then A- is also a root of /(x) and A G ^"(Ci)- Without loss of generality we Si Si assume that the 2n roots Ci = C> C2, • • • > (2n of /(x) satisfy C2i-iC2i = 1, for i = 1 , . . . , n. According to Galois Theory, there are 2n automorphisms 771 = 1, . . . , the Galois group of the extension field K/T, such that has the form a M = 771(a), for i = 1 , . . . , 2n. 29 771(C) of K in Gal (/C/JF), = Ci- Then the i-conjugate of a GS It is obvious that rj2 is an involution on the extension field S. We use a instead of 172(0) if a € S. It is easy to check that mi-i(oi) = r}2i{a) and mi(a) = V2i-i(a) (3.8) for a € S. Some notation is needed for the sake of convenience. We let A = (aij) and (A) = A™ = ( a £ ) (3.9) } Vk if A = (a^) is a matrix with entries in S, and 0 = {5 | a € a} for any ideal 0 in S. It is clear that a is also an ideal in S. The following lemmas are very useful. Lemma 3.5. Suppose M € M2 (F) and a,(3 G S 2n n 2n, there is 1 < k < 2n, where k depends on are two vectors. Then for any 1 <i,j such that a'^M(3^ Proof. Since V\, - • • ,V2n are permutations of the roots of f(x), is a root of f(x), Hence (a'Md^f ] = (a'Md^)^. for any 1 < i,j l ) l k = m k (a'Mn (3)) = ^ ( o ^ M r ^ / ? ) = a'^MB^. k n • and a = ( a i , . . . , a ) ' € «S™, w/iere a i , . . . , a independent over V, and a ' M a M = a'Na^ 2n 2 \ < k < 2 n such that a Ma^ ,(i) are < 2n, there is = ( a ' M a W ) W = 0. i.e. A'MB = 0, where A = ( o ^ ) P> = ^ 5 p ^ are 2n x 2n matrices. By Lemma 3.1, det A ^ O and det B Clearly A = - A by (2.7) and and 7^ 0, since « i , . . . , oj2n are independent over V, and therefore M = 0. C "/'(0- 2 n (for i = 1,..., 2n). TVien M = N. Proof. We only prove the special case N = 0. By Lemma 3.5, for any 1 < i,j 1_ < 2n, ri^ nj(( ) say ( . We have w (() = f]~ r]j(C,), therefore rjj(a) = 77;% (a), for any a € S. Lemma 3.6. Suppose M,N 6 M2 (F) Let A = < • /(J) = 0. Note that the pair (a, a) of an integral ideal 0 and an element a € 7c is an element of Pf if, and only if a = a A a' and a = a. From Lemma 3.3, we have 71' = 71/A and that is (71,1) € P / . Thus P / 7^ 0. 30 D e f i n i t i o n 3.1. A pair (a, a) consisting of an ideal a and an element a in S is said to be an S-pair, if there is a basis a\,..., a2n of a, such that ol J a W = S a A , for i = 1 , . . . , 2n, u where a = ( « i , . . . , a2 )'. n (3.10) The basis ct\,..., a^n is called a J-orthogonal basis of o with respect to a, and the vector a is called a J-vector with respect to the S-pair (a, a). Remark. By Lemma 3.5, we see that (3.10) is equivalent to a'WjaW) = < % a « A « . The bilinear form defined on column vectors a = (ai,..., {a,P) = a'jp a )' 2n is a non-degenerate skew-hermitian form. and /3 = (Pi,.. •, fan)' by In particular, if A. = a'Ja, then A = —A. Since A = —A it follows that if (a, a) is an S-pair, then a — a. L e m m a 3.7. A pair (a, a) is an S-pair if, and only if a = aAa' and a = a. (3.11) Proof. Suppose (a, a) is an S-pair. Let a = (a\,... , a n ) ' be a J-vector with respect to (a, a). 2 Let P = (Pi,..., fan)' = ^Ja. Then = % a' /?W W which implies Tr (aj/3,-) = so Pi, • • • > P2n is the dual basis of a\,..., a . Since det( J) = 1, we see that Pi,..., p n is also a 2 2 n basis of ^ o . Hence a = aAa'. For the converse, suppose (3.11). If A , . . . ,p2n Let 7 i , . . . , 72n be the dual basis of pi,..., p2n. where P = (Pi,... ,P2n)', 7 = (71, • • • ,j2n)'- is a basis of a then Pi,... ,P2n is a basis of a. Then Tr (PUJ) = and we have /j'^^O') = 6^, Since 0 = aAa', there is M € GL2n(V) such that Mp = a A 7 . Then P'MW = a ] W A /3' W W 7 = <J a A (3.12) U and / 3 ' M ' ^ ) = aAj' (P) Vi = -aAr (7')r/ r (/3) = -6 ?2 31 i ?2 U aA (3.13) For the last equality, we use Formula (3.8). Thus [3'M/? = -pVM'pP- w so M' = -M (for i = 1 , . . . , 2n), and (by Lemma 3.6). According to [26] there is Q G GL {T>) such that M = Q'JQ. 2n If a = Q(3, then a ' J S « = /3'M/3« = 5 l i a A. So a is a J-vector with respect to (a, a). • C o r o l l a r y 3.1. Suppose a is an integral ideal. Then (a, a) € Pj if and only of (a, a) is an S-pair. Proof. Suppose (a, a) is an S-pair. We need to show that a € 71. Since o C 71, then ^ = 71' C a'. But a A a ' = a, so aTZ c a, thus a € 11. The converse is clear. 3.3 • T h e Correspondence \& In this section we prove Theorem 1 and Theorem 2. Recall that Mj is the set of all the matrices in SP2n{T>) with characteristic polynomial f(x), Mf over SP (T>). 2n Suppose X € Mj. corresponding to £, that is Xa — (a. and A 4 / is the set of the similarity classes in There is an eigenvector a — (a\,..., Let a be the D-module generated by a.\,..., a = Va.\ -\ and a = A _ 1 o/J3. a.2n)' G a2 , i-e. n 1- Vain It is easy to check that a is an integral ideal in 71 and a = a. Thus a i , . . . , a2n are independent over V. Furthermore we have L e m m a 3.8. The pair (a, a) is an S-pair. Proof. We only need to prove that a'Ja^ = 0 (for i = 2 , . . . , 2n). Assume 2 < i < 2n. From Xa = (a we have X a ' ' = ( i a ^ and l a ' ' = ^rS' '. Hence 1 1 a'JS« = 1 ^a'X'JXS® = ^a'Ja®. The last equality follows from the fact that X € SP2 (V). n 32 (3.14) Since £ / 0 , we get a ' J a W = 0 . • Suppose Y is another element of Mf, and 3 = (3i,• • •, fan)' £ TZ 2n is an eigenvector corresponding to (, that is Y3 = (3- Let b be the integral ideal generated by 3\,..., 3in and 6 = A' 1 p'J/3. Lemma 3.9. X ~y if, and only if (a, a) = (b,b). Proof. Necessity. Suppose there is Q E SP {T>) such that Y = Q~ XQ. l 2n therefore XQfi = QY(3 = (Q3, that is Q3 is an eigenvector of X. Then QY = XQ and There are A, fi E TZ* such that Aa = nQ(3 = Qfi3- So Ao = fib, and AAa = A- \a'J\a l = A~\fiQ = A^fijlp'Q'jQp = 3)'J^QP A~ fi]i3'J3 l = IH&. Therefore (o, a) = (b,b). Sufficiency. Suppose A, fi E TZ* are such that Aa = fib and AAa = fifib. Then there is Q E GL2 (V) such that Aa = fiQ3, and thus n fiQY3 = fiQ(3 = (fiQ3 = ( A a = XXa = hence QYfi = XQ/3. Therefore QY = XQ, i.e. Y = It remains to prove that Q E SP (V). 2n 3'Q'JQ3 {i) = fiXQ3, Q~ XQ. l If i — 2 , . . . , In, then fifi^i JS«= 0 = 0 j/3«. If i = 1, then 3'Q'JQ3 = ^ a ' J 2 = -a'J5 = fifi a tfjp. Hence Q ' J Q = J (by Lemma 3.6). Let • denote the correspondence from Mf to Vf defined as above. Lemma 3.9 guarantees is well defined and injective. The proof of Theorem 1 is completed by following lemma. Lemma 3.10. $ is surjective. 33 Proof. Let (o, a) G Pf and a = (a\,..., C a i , . . . ,(a 2n ain)' be a J-vector with respect to (o, a). is another basis of o, and so there is X G GLiniP), Then such that Xa — C,a. It is clear that fx{x) = f{x). We only need to prove that X G SP {T>). We have 2n a'X'JXa^ = f o/J5 w = aA. Si Hence a'X'JXa® = a'Ja^ (for t = 1,..., 2n). By Lemma 3.6, X'JX = J. This completes the proof. • We now prove the Theorem 2. Proof of Theorem 2. B y Proposition 2.1, f(x) is a product of irreducible S-polynomials, f(x) = (x-l) (x 2k If Pi(x) is of type-I, then P pi + l) (x)---p (x). 2l Pl s ^ 0, thus there exists Xi G M . O n the other hand, if Pj(x) is Vi of type-II, then Pj{x) = q(0)x q(x)q(-^), where q(x) is an irreducible monic polynomial with ni degree (by Lemma 2.5). Let C be the companion matrix of q(x). Then Xj = C'q+Cq G M . 1 q Pj Hence hi * i-hk) * Xi * • • • X s G M. f That is Mf ^ 0. 3.4 • Class N u m b e r of Vf In this section we prove Theorem 3. Suppose 71 is integrally closed in S. Then aa = (a) if and only if o = a A a ' , see [19]. So C is a group, the identity is 71 and a - 1 = A a ' . We easily see that (a, a) G Pf if and only if aa = (a) and a = a. Then Vf is a group if we define multiplication in Vf by (a, a)(b,b) = (ab,a6). The identity is (7Z, 1) and the inverse of (a, a) is (a, a). For the proof of Theorem 3 we will need the following lemmas. 34 Lemma 3.11. Suppose (a, a) G Pf, A G TZ*. Then 1. (Aa, AAa) G Pf. 2. (a, Aa) G Pf if and only if Proof XeU . + For the first part we have AaAa = AAaa = ^AAa^ and AAa = AAa = AAa. Hence (Aa, AAa) G Pf. For the second part, if (a, Aa) G Pf then aa = (Aa) = (a); so A G U. We also have Aa — Aa = Aa, and so A = A. The converse is quite simple. • Lemma 3.12. Suppose (a, a), (a, 6) G Pf. Then (a, a) = (a, 6) if and only if | G C. Proof. Suppose (a, a) = (a, b). There are A, // G TV such that Aa = pa and AAa = ppb. If u — ^ , then u G U and | = uu, that is | G C. Conversely, suppose | = uu for some u G U. Then (a, a) = {a, mxb) = {ua, uub) = (a, b). Lemma 3.13. • Let (a, a), (b,b) G P / , and Aa = pb, for some X, p G 7£*. T/ien (a, a) = (b,ub) for some u G U . + Proof. If Aa = pb, then Aa = fib. Hence (AAa) = AaAa = pbpb = (pjib). Then there is a unit u G U, + such that AAa = uflub. Therefore (a, a) = (Aa, AAa) — (uh,pjiub) = (b,ub). • Now we can prove Theorem 3; namely there is a short exact sequence 1 -> U /C + Av f A Co - » 1 where <^([u]) = (TZ,u) and ^((a, a}) = [a]. Proo/ of Theorem 3. Clearly, (f> 1S w e u defined and a group monomorphism (by Lemma 3.12). i/> is also well defined and a group epimorphism (by Lemma 3.7). I/N£([U]) = tp({TZ,u)) = [TZ] (by definition) and Ker tp = Im <f> (by Lemma 3.13). This completes the proof. 35 • Remark. Lemma 3.11, Lemma 3.12 and Lemma 3.13 are also true even if 71 is not integrally closed in S. There is a bijective mapping between Vf and CQ X £ / / C . + C o r o l l a r y 3.2. IfV is the rational field Q, then there is an one-to-one correspondence between M f and 7Z /C, where 7l = {a + + G 71* \ a = a] and C = {aa\ae 71*}. P r o p o s i t i o n 3.1. If f(x) = x + x + l, then the number of conjugacy classes of Mf in SP (Q) 2 2 is infinity. Proof. Let 71 = Q[C], ( = e^. to show [p] ^ [q] in Let p, q be different primes with p = q = 2 (mod 3). We want 7l /C. + Suppose [p] = [q]. There are A = x\ + y\(, p, — x% + y ( € Z[C] such that AAp = pJJlq, that 2 is [x\ - xiyi + y\)p = [x\ - x y 2 2 + y )q. 2 Then there is an integer A; such that x\ - xiyi -\-y\ = kq x\ - x y 2 2 (3.15) +yl = kp This is impossible due to the fact that if the Diophantine equation x — xy + y = kp , where 2 2 r p = 2 (mod 3) and p \ k, has a solution, then r is even. By a theorem of Dirichlet, there are infinitely many primes of the form 3fc + 2, and so we have proved that 7l jC is an infinite group. • + In general we have C o n j e c t u r e . Let f(x) = x ~ v x + • • • + x + 1, p an odd prime. Then the number of conjugacy classes of Mf in SP _i(Q) is infinite. p 36 3.5 T h e R a t i o n a l Integer Case In this section, we assume V = Z and J = Q. Using the fact that the number of ideal classes 7 is finite, the unit group U is a finitely generated abelian group and U + +2 C C, we get P r o p o s i t i o n 3.2. M.j is finite. From now on we consider the m-th (m > 2) cyclotomic polynomial $ (x) m where C i , . . . , C<£(m) a r e t n e = (x-Ci)---(z-C^(m)) (3-16) primitive m-th roots of unity and <f>{m) is the Euler totient function. It is well known that the & (x) m has integral coefficients and is irreducible over Q. Also <& {x) m is an S-polynomial. We simply denote M<p and M$ m by M m m and M. m 2iri Let C = Cm = e~, TZ m = Z[( ]. Then the involution on TZ is just complex conjugation. m m We denote Cm by C . m P r o p o s i t i o n 3.3. For any X G M , m we have X / X - 1 . Proof. Let a G 7Zm ^ be an eigenvector of A corresponding to C Xa = (a. Then X~ a. = C"- Hence * ( A ) = (o, A - m l 5 _ 1 o / J a ) and * ( A _ 1 ) = {a,A' a'Ja). l If A were conjugate to X 1 we would have (o, A a ' J o 7 ) = (a, A a ' J a ) , that is we could find non-zero elements A,/x G TZ _1 _ 1 such that Aa = /xa and ^a'Ja = ^a'Ja. But this is impossible since a'Jo" = —a'Ja. • Let Ci be the set of integral ideal classes a such that aa is a principal ideal, Ci = {a G C | aa = (a) for some a G TZ }. m (3-17) Ci is a subgroup of C and by definition hi = \Ci\. It is easy to check that Co C C i . To show that Co = Ci we need L e m m a 3.14. Suppose ( is a primitive m-th root of unity. Then (1 — C) is a prime ideal ofTZ m if m is a prime power and I — ( is a unit of TZ if m has at least two distinct prime factors. m 37 See [39]. L e m m a 3.15. Co = C\. Proof. Suppose oo = (ao) where ao € Tl* . We need to find a unit u G U such that uao = ua~Q. m Let We see that uo is a unit because (ao) = (ao), and UQ — 1- According to [39] UQUO — UQ = ±C j for some integer k. If UQ — ( , for some integer I, then we can choose u — C, . Now fc 2L 1 we suppose UQ ^ C, , for any integer /. 21 Note that a = a (mod 1 - C ) (3.18) 2 for any a G 7Z . m Case 1. If m is odd, then UQ = —C , for some integer k. This is because if UQ = £ k uo = £ 2 f c ~ 1 + m 2 f c - 1 then , where 2k — 1 + m is even. By Lemma 3.14, either (1 — £) is a prime ideal in 7l m or 1 — £ is a unit in TZ . If 1 — £ is a unit, then = C m f c _ 1 = C '> f ° some integer /. We 2 r can choose u = (1 — Consider the case where (1 — () is a prime ideal in 7l - We want to show that UQ ^ — ( H m for any integer k. If a 0 G (1 - C)> then oo C (1 - () since ao = (ao). So either o C (1 — () or o C (1 — (). Both cases are the same and imply (ao) C (1 — 0 ( 1 — 0 - Let a\ = Then a\ G l¥ m and uo = fj-. Continuing this procedure, there is a G 71^ with a £ (1 — () such that UQ = | . Now suppose uo = ~ ( - Then, by (3.18), a = a = —C, a = —a (mod 1 — (), hence 2a = 0 k k (mod 1 — C)- Since (2) is a prime ideal different from (1 — C) we have a = 0 (mod 1 — Q, that is a G (1 — £)• Contradiction. Case 2. If m is even, then UQ = ( 2 K + L , for some integer k, since — 1 = ( . T Note that —C is also a primitive m-th root of unity, so either (1 + () is a prime ideal of 7Z or 1 + C is a unit in m 7Z . If 1 + £ is a unit in 7l , m m then we use u — (1 + QC k 38 In the case that (1 + Q is a prime ideal of TZ , we want to prove that UQ ^ ^ + for any 2fc 1 m integer k. For a similar reason as in Case 1, there is a 6 TZ* , a £ (1 + (), such that UQ = f. m Suppose u = 0 C - By (3.18) we have a = ( a 2k+1 (C-l)(C + --- + C + 2/ that ( 2l l)a = 0 (mod l - ( ) , thus 2 C~ (C 2i a {2l+1) = 2t+l (mod 1 - C ). This implies 2 + ••• + ( + l)a = 0 (mod 1 + C). We know + • • • + C + 1 £ (1 + C), hence a £ (1 + ()• Contradiction. Now we want compute the index [U : C] of C in U , + + • that is the order of U /C. Since + for m = 2 (mod 4), TZ = TZm, we assume that m ^ 2 (mod 4). First, we quote some results m of number theory (see [23] and [39]). Let W = {±( }, m a finite cyclic group consisting of the roots of 1 in TZ. Lemma 3.16 (Dirichlet). The unit group U ofTZ m a free abelian group of rank is the direct product W x V, where V is _ 1_ Lemma 3.17. [U : WU ] + 1, m prime power, 2, m not prime power. = < Lemma 3.18. If m is not a prime power, then 1 — Cm ^ WU + Proof. If there is an integer I such that C (l — Cm) € U , + m we only need to show that ~^ l m and (1 — Cm)(l Cm) ^ then (1 — ( ) ( l — Cm) € U . +2 m = —Cm ^ U . For this purpose we suppose —Cm £ U . 2 U . — 2 +2 So Then ~Cm = Cm f ° some /, which implies 41 — 2 = 0 (mod m) and m is even. Since m ^ 2 (mod 4), r we have m = 0 (mod 4). Thus 4/ — 2 = 0 (mod 4), which is impossible. This completes the proof. Lemma 3.19. Let k • = [U : C]. Then + m 2 2 , rn prime power, km — \ 4>(m) 2 2 . 1 , m not prime power. 39 is the direct product of Z2 and a free Proof. By Lemma 3.16 and Lemma 3.17, we see that U + abelian group with rank - 1, and then we get [U :U } + If m is a prime power, then C = U + U (1 - C)(l - 0U , 2^2^. (by Lemma 3.17 and Lemma 3.18). + which implies [C : U } +2 = m Um is not a prime power, then U = WU U(l-()WU 2 2 ^ . (Lemma 3.17), and we obtain k +2 We get C = U+ = +2 +2 = 2. Thus k = 2 * T - \ since i m [U+ : U+ } = [U+ :C][C: U }. 2 • +2 This completes the proof of Theorem 4 (by applying Theorem 3). E x a m p l e . Let m = 5. Then h\ = 1, 0(5) = 4, and hence g = 4. There are 4 classes of M 5 5 in SRi(Z). Here is a list of canonical matrices of M 5 , 0 -1 °1 0 -1 1 1 -1 / n 1i n 0 n \ X ( X = 3 0 0 x = 2 °J 0 0 1 -1 1 1 0 -1 1 0 -1 -1 -1 0 1 0 1° 1 - 1 - 1 0 - 1 0 f° / n 0 n 1 0 x= 4 0 - 1 - 1 0 0 Similarly a list of canonical matrices of M - 1i 0 0 0 - 1 0 0 -1 \o in 5P (Z) is -X, w n0 \ -X , 2 4 oy 1 -X , —X . 3 4 E x a m p l e . Let m = 8. Then hi = 1, 0(8) = 4, and hence g = 4. There are 4 classes in M . 8 8 A complete set of conjugacy classes of elements of order 8 in S P ( Z ) is 4 0 - 1 1 0 IoJ, Io(-J), - 1 0 11 - 1 1 0 0 -1 ^ A) 1 0 0 40 oy -1 0^ 1 0 - 1 - 1 1 - 1 0 0 \o 1 0 0 y E x a m p l e . Let m = 12. Then hi = 1, 0(12) = 4, and hence q® = 2. There are 2 classes of X G S P ( Z ) with characteristic polynomial /(x) = x - x + 1. Two non-conjugate matrices 4 2 4 are C / and Cy, where Cf is the companion matrix of 41 f(x). Chapter 4 S y m p l e c t i c Spaces If a symplectic matrix X is decomposable, then its characteristic polynomial f(x) is a reducible S-polynomial. In general, the converse is not true. In this section we want to find sufficient and necessary conditions for X to be decomposable. First, in Section 4.1 we introduce symplectic spaces and prove Theorem 5. In Section 4.2 we relate symplectic matrices to symplectic transformations and then prove Theorem 6. Finally, in Section 4.3 we shall discuss symplectic group spaces and prove Theorem 7. Some of the material in this chapter is known, see [12]. 4.1 T h e Symplectic Spaces We start with a definition: D e f i n i t i o n 4.1. Let V be a free V-module with rank 2n and suppose there is a skew symmetric inner product (, ) on it. V is called a symplectic space over V if there are 2n elements v\,..., v 2n of V such that their inner product matrix M(v ...,v ) u The ordered elements vi,...,v 2n 2n = ((vi,Vj)) 2nx2n = J. (4.1) form a symplectic basis of V. Two symplectic spaces are said to be isomorphic if there is a P-module isomorphism a which preserves their inner products, a is called a symplectic isomorphism. E x a m p l e . Let S be a Riemann surface with genus g > 1. Then Hi(S) form is a symplectic space over Z , with rank 2g. The following lemma says that a symplectic basis is a X>-basis. 42 with the intersection Lemma 4.1. Suppose V is a symplectic space over V with rank 2n. Then every symplectic basis is a D-basis ofV. Proof. Suppose vi,...,v 2n is a symplectic basis of V. Iiw\,..., vi = anwi H h ai w , v = a \Wi h a w , 2 2 H 2n is a X>-basis of V, then 2n 22n 2n (4.2) V2n = a iw\ H 2n where 2n w ha 2 n 2 n w 2 n € V (i, j = 1 , . . . , 2n). Let A = (a^) be the coefficient matrix. It is obvious that AM(w ...,w )A' u 2n = M{ ,...,v ) Vl = J. 2n Hence the determinant of A is a unit in V, therefore v\,..., v 2n Lemma 4.2. is a P-basis of V. • Two symplectic spaces over V are isomorphic if and only if they have the same V-ranks. Proof. The necessity is clear. For sufficiency, suppose v\,..., v 2n is a symplectic basis of V and w\,..., w 2n is a symplectic basis of W. If we define a : V -> W by a(vi) = Wi (for i = 1 , . . . , 2n), then cr is a symplectic isomorphism. • Lemma 4.3. Suppose two symplectic spaces V and W have the same V-ranks. Then a V-linear mapping a : V —> W which preserves inner products is a symplectic isomorphism. Proof. Let v\,..., v 2n is a symplectic basis of V. Then M(a(vi),...,a{v )) 2n = M{v u By Lemma 4.1, cr(vi),... ,cr(v ) is a basis of W. 2n therefore a symplectic isomorphism. ... ,v ) = J. 2n Hence a is a £>-module isomorphism and • 43 Consider V , 2n the X>-module of 2n-tuple over T>. For any two column vectors a, 8 € V , 2n we define a skew symmetric inner product on V 2n V 2n by (a, 8) = a'JB. It is easy to verify that with this inner product becomes a symplectic space, which we call the canonical symplectic space. Furthermore, if we put e; = (0,...,0,1,0,...,0)', then e i , . . . , e for i = 1,... ,2n, (4.3) is a symplectic basis of V , which we call the standard symplectic basis. 2n 2n In this section, we always assume that V is a symplectic space over V with rank 2n and ^l) • • • > V2n is a symplectic basis of V. Let v, w € V, and h a V2n v = aivi H We set a = (a\,..., and 2n w = b\Vi H h^n^n- (4-4) a )' and 3 — (bi,..., 62n)'> the coordinate vectors of v and w under the 2n basis v\,..., V2n- Clearly, we have (v, w) = a'J0. Suppose V i , V~2 are P-submodules of V. We use V\ © V2 to denote the module sum V\ + V2 if Vi n V2 = {0}. Vi and V2 are said to be orthogonal, written as Vi _L V2, if (^1,^2) = 0, for any elements v\ € V i , v € V2. Furthermore, suppose V i , V are symplectic subspaces of V. Then 2 2 Vi © V2 is called the symplectic direct sum of Vi and V2, denoted by V i * V 2 . Let a i , . . . , Ofc be elements of V. It is convenient to denote any greatest common divisor of a i , . . . , a by g.c.d (a\,..., k a )- We know that g.c.d ( a i , . . . , a ) = 1 if and only if there exist k k T*I, . . . , r € V such that r i a i H f- r a k k k = 1. In this case, we say that a\,..., a are relatively k prime. D e f i n i t i o n 4.2. A n element v (v ^ 0) of V is said to be primitive, if v = c to, where c £ T> and w E V, implies c is a unit in V. Let a\,..., a k € V. We say that a i , . . . , a k are coprimitive if for any relatively prime elements a-i,..., a € V, the linear combination a\a\ + • • • + a d k k primitive. A n ordered set of / + k (0 < k, I < n) coprimitive elements a\,..., is k a;, 81,..., 3 is k said to form an (I, Abnormal set if (a is 8 j ) = Sij, {a ) u aj 44 = (A, 8 j ) = 0, (4.5) for all possible i and j. If ati,..., a;, Pi,..., Qk form a (/, A;)-normal set, then their inner product matrix is (4.6) depending on whether I < k or I > k. Remark. If a\,..., Then a\,... , a 2 n are coprimitive, then every a; is primitive. Let a.\,..., a 2 n be a D-basis. are coprimitive. Thus an element of any £>-basis is primitive. A primitive element forms an (l,0)-normal set or a (0, l)-normal set. A n ordered set of 2n elements is a symplectic basis if, and only if, it forms an (n,n)-normal set. L e m m a 4.4. An element v — a\V\ H \-a2nV2n is primitive if and only if the greatest common divisor g.c.d (a\,... ,a2 ) = L n L e m m a 4.5. Let v E V be primitive, w E V and a, b be non-zero elements in V. If aw = bv, then a I b. L e m m a 4.6. Let a.\,..., a k be coprimitive. Then ot\,..., a k are independent and can be ex- tended to a V-basis of V. Proof. It is clear that a\,..., a k are independent. To complete the proof we need to show that V/W, where W is the subspace generated by ati,..., af~, is torsion free. Let v be a non-zero element in V and a be a non-zero element in T>. Suppose av is zero in V/W, that is av E W. Then av = a\cxi-\ ha^a*; for some a\,..., EV. Let g.c.d ( a i , . . . ,Ofc) = b. We have aj = bc\, where C{ E V and g.c.d (c\,... ,Ck) = 1. Then av — b(c\a\-\ \-CkCtk) and c\a\-\ \-Ckdk is primitive. Hence a \ b and therefore v E W. • L e m m a 4.7. An element v is primitive if, and only if there is an element w E V such that {v,w) = 1, that is v,w form an (1, l)-normal set. 45 Proof. By Lemma 4.4, if v is primitive then g.c.d ( a i , . . . , a ) 2n such that a i i c = 1- There are c i , . . . , c 2n € X> 1- Let w be an element of V such that the coefficient vector of w is = 6 = - J 7 , where 7 = ( c , . . . , C 2 ) ' - Then (v,w) = a'J(-Jj) = 0 / 7 = 1. n x The converse is clear. • L e m m a 4.8. If W is V-module summand ofV, then there is a primitive element w in W. Proof. This is because every D-basis of W can be extended to a P-basis of V. • P r o p o s i t i o n 4.1. If V = Vi + V and Vi ± V , then V = V i * V . 2 2 2 Proof. First, we prove that Vi D V = {0}. Let v 6 Vi fl V . Then for any w = wi + w , where 2 w\ € Vi and w 2 2 2 G V2, we have {v,w) = (v,wi) + (v,w ) = 0. Hence v = 0, that is V* = Vi © V . 2 2 Now we prove that Vi is a symplectic subspace of V by induction on rank(Vi), the rank of V i . If rank(Vi) = 1, then Vi _L V i , and so Vi _L V. Thus Vi = {0}, this is contrary to rank(Vi) = 1. Hence rank(Vi) = 1 is impossible. Suppose rank(Vi) > 2. Since Vi ± V , there 2 are two primitive elements wi, w of Vi such that (w\,w ) = 1 (by Lemma 4.8 and Lemma 4.7). 2 2 Let W be the symplectic subspace generated by w\ and w . If rank (Vi) = 2, we see that V\ = W 2 is a symplectic space. Suppose rank(Vi) > 2. We let U = {v e Vi | (v,w) = 0 for w € W}. If v 6 V i , then v — {v, w )w± 2 + {v, w±)w 2 E U. We see that V\ = U + W. By the same argument as above, Vi = W © U. Thus V = U © (W© V ). Also U ± (W + V ) and rank (U) = rank (Vi) - 2 2 2 by the definition of U. By induction, U is a symplectic subspace, and therefore Vi = W * U is a symplectic subspace too. By the same reasoning, V2 is a symplectic space. C o r o l l a r y 4.1. Suppose V i , . . . , V m are subspaces of V with 1. V = Vi + • • • + v , m 2. Vi ± Vj for i / j. 46 • Then V = V * • • • * V . x m L e m m a 4.9. Let a\,... ,otk, Pi,--- ,8k,Jo, 7 i , • • • ,7; be a V-basis ofV such that {ca,Pj) = &ij and (auatj) = (0^7,-) = 0. Then g.c.d ((70,71), • • • , (To, It)) = 1. Proof. Suppose there is a non-unit c G D such that c|(7o,7j), Let 7 = 70 — (jo,@i)a± 70, eti,...,ak for j = I,...,I. (4.7) — ••• — (70, Pk)etk- Then 7 is primitive since 70 is primitive and are independent over V. Any » e F can be expressed by k 1 v = ^2 {ai(Xi + bifii) + ^2 Cjjj i=l j=0 where a,i,bi,Cj G V. Hence k (7, v) = (70 - k ^2(lo, 8i)au i=l = ^2 J^o,8j) b j=i = Ci(7o,7l)H J2 l ( ii Q a + i&) + b i=l + J2 c j(7o,7j) - j=i i=i j=i L e m m a 4.10. Let ct\,... ,a>i be an (l,0)-normal h Pi,..., 8k, 7i, •••,7m ** a V-basis ofV 2. « i , . . • , dk, Pi, - • •, Pk form a (k,k)-normal set. Proof a We prove this lemma by induction on k. For k = 0 it is obvious (by Lemma 4.6). 47 • set of V. Then for any 0 < k < I, there are 8k, 7 1 , . . . , 7m> where m = 2n — k — /, in V such that 1. ai,...,Q J2 E M7o, &><*> r-q(7o,7i) which implies c| (7, v) by (4.7). This is contrary to Lemma 4.7. 81,..., ^2 j=0 Suppose it is true for k — 1. We have elements Pi,..., Pk-i, 71, • • •, 7m+i satisfying these two conditions. Set k-l k-l j=i k-l 3=1 k-l 72 = 72 - ^ ( J ' 7 2 ) / ? j + ^(/?j,72>ay, i=i j=i a fc-i fc-i 7m+i = 7m+i - J3<ai,7m+i)#; j=l + X^'^+iKj=l We have (a i l 7 ->=0 and (A,7^> = 0 (4.8) for i = 1,..., A; — 1 and j = 1,... , m + 1. Applying Lemma 4.9 to a\,... ,Q!fc-i, Pi, • • • ,Pk-i, ctfc,..., ai, 71,..., 7 ^ + D we see that there are c i , . . . , c i m+ in V such that ci<Q!fc,7i> + --- + Cm+i(afc.7m+i> = Note that here we use the fact (a ,aj) k ( - ) L 4 — 0 for j = 1,..., I. Now we can find a unit matrix A = (a(j) in GL +i(V) m with c i , . . . , c + i as itsfirstrow, m see [26]. Let Clearly, ai,... ,att, Pi,..., C1I1 + l- m+l7m+l> Pk = 7l = 02171 + " •+a2m+l7m+l, 7m = c «m+ll7l H ^ «m+lm+l7m+l- Pk, 7",. • • 1 7 forms a 2?-basis of V. Furthermore, let m P'l 9 = Pi-(c*k,Pi)Pk, 48 P'k-i = = Pk- 0k Then a\,...,ai, 8[,...,0' k (3[,..., B' , k 0k-i-(ak,Pk-i)Pk, 7",..., 7 is also a £>-basis of V. We shall verify that a\,..., M a, k form a (A;,A;)-normal set by using (4.8) and (4.9) Case 1. For i,j = l,...,k — l, (auftj) = ( i,8j - (a ,8j)0 ) a k = {ciuPj) - k {a ,8j)(ai,8k) k m+l = ( u0j) ~ ( k,8j) a a ^2 c ( a , , 7 ^ ) = (ai,8j) s=l = 6ij. s Case 2. For i = 1 , . . . , k, j = k, m+l iks=l Case 3. For i — k, j = 1 , . . . , k — 1, = <a,/3,-> - (a ,8 )(a ,8k) c f k 3 = 0. k Case 4. For jf = 1 , . . . , k — 1, m+l = (8j ~ (a ,8j)8k,8k) = E s(M) s=l = (faPk) k = °- c This completes the proof. • Proof of Theorem 5. Without loss of generality we can assume that k < I. symplectic subspace generated by ai,..., IfveV, a , 01,...,8k, k and V = 2 Let V\ be the V. r let k w = v- k ^2(v,0i)ai + Y2( i i)®ii=l i-1 v a It is easy to see that w € V . Hence V = V\ + V . B y Proposition 4.1, we see that V is a 2 2 symplectic subspace and V = V~i * V . 2 49 2 If k < I, then a i f c + i , . . . , « / form a (I — k, 0)-normal set of V . By Lemma 4.10 we can find 2 0k+i, • • •, 0 i in V such that a^+i, • • • , aj, 0 k + \ , • • •, 0 i form a (I — k, I — fc)-normal set. Then 2 a i , . . . , on, 0i,...,3i form an (/, /)-normal set. So we can suppose k — I. If k = I then a combination of a>i,..., a , k 0 i , . •. , 0 k and a symplectic basis of V is a 2 symplectic basis of V. • Remark. This theorem gives another way to prove that every normal array can be completed to a matrix in SP (V), see [29]. 2n 4.2 Symplectic Transformations D e f i n i t i o n 4.3. A linear transformation a of a symplectic space V is called a symplectic transformation if it preserves the inner product. A symplectic transformation a is reducible if there is a non-trivial cr-invariant subspace of V; otherwise it is called irreducible. A symplectic transformation a is decomposable if V can be decomposed as a symplectic direct sum of two non-zero symplectic cr-invariant subspaces; otherwise it is indecomposable. Remark. It is easy to see that every symplectic transformation maps a (k, /)-normal set to a (k, /)-normal set. Thus a symplectic transformation is a P-module isomorphism. Clearly, a decomposable symplectic transformation must be reducible. Now we shall see that the converse is also true. L e m m a 4.11. A symplectic transformation is decomposable if, and only if it is reducible. Proof. Suppose V\ is a non-trivial cr-invariant symplectic subspace. Then cr(Vi) = V\. Theorem 5, there is a non-trivial subspace V , such that V = V i * V . 2 2 (cr(Vi),Cr(V )) = ( V i , V ) = 0 , 2 V 2 By is cr-invariant since • 2 50 Let a be a linear transformation of V and X be the matrix of a with respect to a symplectic basis V2n-> i.e. a(vi,...,V ) = (cr(vi),...,CT(v2n)) = ( v i , . . . , U 2 n ) ^ - 2n We know that the inner product matrix of o(v\),..., a{v2n) is M{o(v\),..., (4.10) a ( v ) ) = X'JX. 2n Hence a is a symplectic transformation if and only if X G SP2 (V). Suppose a is a symplectic n transformation. Let v\,... ,V2 and w\,... ,w n 2n be two symplectic bases of V. Then there is a symplectic matrix Q G SP2 (T>) such that (w\,..., u; ) = (v%,..., v )Q. n 2n 2n Let X and Y be the symplectic matrices of a with respect to the bases v\,... ,V2 and w\,... ,W2nn calculation tells us Y = Q~ XQ, l that is X ~ Y. P r o p o s i t i o n 4.2. Suppose a is a symplectic transformation of V. Then a is decomposable if and only if X is decomposable. Furthermore, suppose V~i,...,V are a-invariant symplectic m subspaces of V, and V = V\ * • • • * V . Then X ~ X\ * • • • * X m matrices of a\V\,..., a\V m A simple m where X\,...,X m are the respectively. Proof. Let rank (Vj) = 2nj, and an,..., a.i , Sn,..., Qi ni ni be the matrix of a\V with respect to the basis an,..., i be a symplectic basis of V}. Let Xi ai , ni 0n, • • •, dim- We see that is a symplectic basis of V, and the matrix of a with respect to the basis (4.11) is X\ * • • • * X . m For the converse, we assume that X — X\ * • • • * X . Let V] be the subspace generated by m (i>i,... ,f2n)[0 * • • • * Xi * • • • * 0]. It is easy to see that V* is a a-invariant symplectic subspace of V and Vy + • • • + V = V. Thus V = V * • • • * V . m x m • L e m m a 4.12. Let a be a symplectic transformation of V, let p(x), q(x) G T>[x\ be mutually coprime polynomials, p(a)(a) = 0 and q(a){0) and let one of them be an S-polynomial. If a, 0 G V are such that = 0, then (a,0) = 0. Proof. Without lost of generality we assume that q(x) is an S-polynomial. There are two polynomials u(x),v(x) G 7J>[x] such that u(x)p(x) + v(x)q(x) = c, where c G V, c ^ 0. Then 51 c a = v(a)q(a)(a), and c since (a,0) q(a)(0) = (v(o)q(a)(a),0) = 0, and q{o~ ) = a~ l (v(a)(a),q{o~ ){0)) l = q(a), = (v(a)(a),0) = 0 where m is the degree of q(x). Here we use the fact 2m (a(a),0) = {a,a-H0)). • Let V be the canonical symplectic space V . Given any X £ SP2 {V), 2n n we can define a symplectic transformation a as follows, a{a) = Xa (for a € V ) 2n . It is well known that the matrix of a with respect to the standard basis e\,..., e-m is X. C o r o l l a r y 4.2. Lei K, be an extension field of J and X, u € JC with A ^ u and A/j ^ 1. 7/ 7 X € SP2n(K) and a, 0 G / C 2 n are suc/i that (X - \I) a = 0 and {X - til) 0 = 0, r s for some integers r, s, then a'J0 = 0. Proof. We apply Lemma 4.12 to X. Note that (x — A ) and (x - n) (x - j^) are mutually r s s coprime, and the latter is an S-polynomial. • Now we are ready to complete the proof of Theorem 6. Proof of Theorem 6. Suppose f(x) is a reducible S-polynomial and m fix) = Y[ (x) Pi 1=1 where p i ( x ) , . . . ,p (x) are mutually coprime S-polynomials. Let qi{x) = f(x)/pt(x). m m polynomials, U i ( x ) , . . . , u (x) m There are € T[x], such that ui(x)qi(x) + • • • + u (x)q (x) m 52 m = 1. (4.12) Suppose X ~ Xi*- • - * X , where X , G M m that X = g - ^ X i * • • -*X )Q. P i (for % = 1 , . . . , m ) . There is Q G SP (V) 2n Then c/(X) = Q~ \g(Xi) l m * • • • * g(X )]Q, such for any polynomial m g(x). By (4.12) and the fact that Pi(Xi) = 0 (for i = 1,... ,m), we obtain Hence Ui{X) {X) qi = Q [ 0 * • • • * J * • • • * 0]Q G - 1 7, i = j, 0, t^j. M {V). 2n For the converse, we regard X as the symplectic transformation a ->• symplectic space V . 2n of the canonical Let V = u (X)q (X)(V ) for i = l , . . . , m. 2n i i i (4.13) Then for each 1 < i < m, we have 1. Vi is submodule ofV , 2n because Ui(X)qi(X) 2. Vi is X-invariant, for X(V*) = X (ui(X) (X) qi 3. V 2n 4. Vi 1 = Vi + • • • + V , for •£Ui(X) (X) m qi € M2„(-D); [V )) 2n = (X)qi(X) (X (P )) = 2 n Ui = I; (i / j), by Lemma 4.12 and Pi(X)Vi = {0}. • Applying Proposition 4.2, we can complete the proof. C o r o l l a r y 4.3. Suppose f(x) and g(x) are strictly coprime S-polynomials, and X G Mf . g X is decomposable. E x a m p l e . Consider the case D = Z. Let 0 Ai 0 0 - 1 0 0 0 1 and 1 0 - 1 0 \0 -1 0 I) X 2 = 0 1 0 0 0 0 0 0 1 0 0 - 1 0 - 1 0 53 1 Then XXe u S P ( Z ) , and f (x) 2 4 Xl = f {x) %(x + l)(x 2 - 2 Clearly, X = (x + x + l)(x X2 x 2 + l)-±(x- - x + 1). We know that l)(x 2 + X + 1) = 1. is decomposable and \ (X +1) (X - X +I) £ M ( Z ) . But X 2 x since \{X x + I){Xl -X 2 + I)£ 2 4 2 is indecomposable, M (Z). 4 E x a m p l e . Let f(x) = (x + l)(x 2 x 2 ± x + 1). Any X € Mf is decomposable, since (x ± l)(a; + 1) - x{x ± x + 1) = ± 1 . 2 4.3 2 Symplectic G r o u p Spaces D e f i n i t i o n 4.4. Given a group G, a symplectic space V is called a symplectic G-space, or G-space, if G acts on V and every element of G preserves the inner product. Relative to a symplectic basis, V affords a symplectic representation of G. Let G be the cyclic group G , m generated by a fixed element g of order m, where m is a finite integer or infinity. To specify a G -space V , it suffices to give a symplectic matrix X. The characteristic m polynomial of X is independent of the representation, we call it the characteristic polynomial of the C7 -space. The set of all symplectic (7 -spaces with characteristic polynomial f(x) is m m denoted by Vf. D e f i n i t i o n 4.5. Two G-spaces V and W are equivalent, denoted by V = W, if there is a symplectic isomorphism a :V -> W such that the diagram G x V • V idxa GxW c > W is commutative, that is a(g o v) — g o (a(v)). Remark. Let Vf denote the set of equivalence classes in Vf. We have a natural one-to-one correspondence S, defined as above, between V / and Mf. 54 A G-space is decomposable if it is expressible as a symplectic direct sum of two non-zero G-subspaces; otherwise, it is indecomposable. A G-space is reducible if it contains a non-zero G-subspace of smaller rank. A non-zero G-space which is not reducible is called irreducible. A n analogue of Lemma 4.11 is P r o p o s i t i o n 4.3. V is decomposable if and only if it is reducible. E x a m p l e . If we have a group G acting on a Riemann surface S, then Hi(S) is a symplectic G-space by passing the action to homology. Suppose f(x) is an S-polynomial of type-I, and ( is a fixed root. Given any S-pair (o, a) G Pf (cf. Section 3.3), we know that o is a V-module since it is an ideal. We define a skew symmetric inner product as follows, Let m — order of £. We define the action of G m on a by g o x = x/(, that a = aAa'. Let a = ( a i , . . . , a.2n)', where ai,...,a for all x G o. Note is a J-orthogonal basis of a with 2n respect to a. Then the components of -^Jot form the dual basis of a±,... ,a.2n- This means the matrix Tr ( g - J ' ) is the identity matrix. On the other hand, Tr (fg. J') = Tr (ff-) J', Therefore we obtain a symplectic space, denoted by [a, a], and a i , . . . ,Qf2n is a symplectic basis. Also, it is easy to verify that g preserves the inner product and its characteristic polynomial is f(x). We have [a, a] G Vf. Before we prove the Theorem 7, we give the following lemmas, L e m m a 4.13. If Tr (ax) — Tr (bx) for all x G a, then a = b. Proof. T r is additive, so we only prove the special case where 6 = 0. P-basis of a. We obtain a system of 2n equations in the aW's, a (D W + . . . + x a 55 (2n) (2n) x = ^ Let xi,...,X2 n be a « 4 (1) ) + - which only has the 0 solution. Hence + a ( t e ) 4 a , ) =0, = • • • = o^ ™) = 0, so a = 0. • 2 L e m m a 4.14. Suppose a and b are ideals of TZ, and a : a —> b is a V-linear mapping with a(g o x) = g o a(x). Then there is a unique element q of S such that a(x) = qx for all x G o. (4-14) Proof. First note that a is 7£-linear. To prove this we write any element a of TZ as a P-linear combination of 1, 1/C, l / ( , . . . , l / ( 2 2 n _ 1 . It is easy to verify that a(ax) = aa(x). Let « o £ o- Then aoo(x) = a(aox) = a(cto)x. Set q = a(ao)/ao, we see that (4.14) is true. • Proof of Theorem 7. Suppose a is an symplectic isomorphism from the symplectic G -space m [ai, ai] * • • • * [a , a ] to [bi, &i] * • • • * [b , b ]. Thus there is an r x s matrix Q = (qij) with entries r r s s in 5 so that yi Q \x r for all ( x i , x ) ' r J G ai © • • • © o , and (y\,..., y )' G bi © • • • © b . Since a is an isomorphism, r s s Q has an inverse, and hence r = s. If we choose all x\,... ,x r qijXj G bi. Thus q^aj C bj for If a = ( a i , . . . ,a )' r i, j = to be zero except Xj, we obtain 1,... , r . and /? = (/?i,... ,/3 )' are in [oi,ai] * • • • * [a ,a ], then r r t fx ai r \ \ 0 i=l V 56 (4.15) and similarly 61 (a(a),a(0)) = Tr (4.16) V 1_ V Comparing each entry of (4.15) to (4.16), and using Lemma 4.13, we complete the proof of the first half. To prove the second half, we define a by = Q \Xr J \X r J a is a P-linear mapping from a\ © • • • © a to bi © • • • © b and preserves the inner product, r r hence a is isomorphism by Lemma 4.3. • C o r o l l a r y 4.4. If [01,01] * • • • * [a ,a ] = [bi, 61] * • • • * [b ,6 ], then r r r r (01 • • • a , a i • • • a ) = (bi • • • b , 61 • • • b ). r r r r Proof. For each generator a\ • • • a of 01 • • • a , the product (det Q)a\ • • • a can be expressed as r r r the determinant of the product matrix \ ai 0 0 a 0 0 Q 2 ••• a, whose i-th row consists completely of elements qijaj of bj. This proves that (det Q)ai • • • ar C bi • • • br. A similar argument shows that (detQ )bi • • • b C ai • • • a,. l r 57 Multiplying this last inclusion by detQ and comparing, it follows that bi • • • b (detQ)ai • • • a ; and it is easy to verify that 61 • • • b = (detQ)(detQ)a r r x r is equal to •••a . This completes r the proof. • Now we give some applications of Theorem 7. When r = 1, we have Corollary 4.5. [a,a] = [b,b] if, and only if (a,a) = (b,b). Proof. B y Theorem 7, [0, a] = [b, 6] if and only if there is A € S such that Aa C b and b = AAa, which is equivalent to (a, a) = (b,6). • Prom this corollary, we obtain a natural injective correspondence <]> : (a, a) —» [a, a] from Vf to Vf. The following lemma says $ is surjective. Lemma 4.15. For any V € Vf, there is an S-pair (a,a) € Vf such that V = [a,a]. Proof. Let v\,..., V2 be a symplectic basis of V. The action of g on V has a representative n X e SP {V). 2n We choose (a, a) € Vf such that * ( X ' X n ) = (a, a), suppose I- Xi \ \X2n J 7-1 \X2n where x \ , . . . , X 2 _ 1 J is a J-orthogonal basis with respect to (a,a). (f> : V —> a by 4>(VJ) = Xj. It follows that (x{,Xj) = We define the isomorphism Tr (-^XiXj) = preserves the inner product. and Vf and E between More precisely, we have Proposition 4.4. The correspondence That is, • Furthermore, we have one-to-one correspondences ^ between M/ Vf and Mf. 5ij = (vi,Vj). * o S o $ 58 is the identity ofVf. 4> Proof. Let (a,a) G Pf, and a = (ai,..., a\,..., ai2n a )' be a J-vector with respect to (o, a). 2n is symplectic basis of [a, a]. Let X be the matrix of g with respect to a\,..., a a We need to prove that ${X) = (a,a). S i n c e g ° { o t \ , . . . ,a ) = / _ 1 a = {a, we get = (a, a). Hence 2 n . \( ii • • • , 2n) = (ai, • • • , Q ; ) X , a a 2 n 2n and X Then = = (a, a). • The following proposition gives a method to compute n o S(V), for a symplectic G -space m V € Vf without needing to know a symplectic basis of V. Proposition 4.5. Suppose V £ Vf. Let a-i,... ,a 2n be a V-basis ofV, not necessarily symplec- tic. Let M be the inner product matrix of a\,..., a , and X be the matrix of g with respect 2n to a\,..., a . Let a — {a\,..., a )' € V be an eigenvector of X with respect to 2n 2n 2n ^ o S(V) = (a,a), where a is the ideal generated by ct\,... , a Proof. We choose a symplectic basis v\,..., v 2n to vi,... ,v2n- There is Q G GL (V) 2n such that 2 n and a = Then A~ a'Ma. l of V and let Y be the matrix of g with respect (01,... ,a ) = (v\,... ,v )Q. 2n 2n It follows that Y = QXQ- and M = Q'JQ. 1 If 0 = Qct, then Y0 = QXQ~ {Qa) l = QXa = Q(a - QQa = 0 . We see that 0 is an eigenvector of Y with respect to £• Now we need to show that 0 is a J-vector with respect to of (a, a). From the fact that Q is invertible, we see that the components of 0 form a £>-basis of a, and a = A~ a'Ma l = A~ a'Q'JQa l So * o E ( y ) = # ( Y ) = ( ,a). = A~ 0'J0. l • a For r = 2, we have Corollary 4.6. [a, a] * [b, b] = [R, 1] * [ab, ab] if and only if there are u € a and v G b such that ™ + l = l. a b V 59 (4.17) Proof. Suppose [o, a]*[b,b] = [71,1] * [ob, ab]. There is a 2 x 2 matrix Q = (c/y) with entries in S, so that q 7l C a, q i7Z C b and n 2 (4.18) Set u — qn, v = q \ and then compare the top left entries of both sides of Equation (4.18). 2 For the converse, suppose there are u G a, v G b such that (4.17) holds. Let Q = It follows that Q satisfies (4.18). Now we need to verify that - f a b C a and | a b C b. Since v G b, then —v G b, which implies -vb C bb = bAbb' C bTl, and thus - f b C 71. It follows that fab C a. Similarly, fab C b. Therefore [a, a] * [b,6] [71,1] * [ab, ab] by Theorem 7. This completes the proof. Example. Let 7l m • be as in Section 3.5. Then [7l , -1] * [7l , -1] ^ [7l , 1] * [7l , 1]. m 60 m m m Chapter 5 O r d e r p elements i n S P _ i ( Z ) p First, in Section 5.1 we will give examples of elements of order p in SP -i (Z). Then in Section 5.2 p 27T1 we will discuss the cyclotomic units of the cyclotomic field Q [ C ] , where ( = e p . A n d finally, in Section 5.3 we shall prove Theorem 8. 5.1 A n Example Theorem 1 gives us a way to find representatives for each cyclic matrix class in SP (V) with 2n characteristic polynomial f(x) irreducible and separable in T>. Suppose we have an S-pair (a, a) and a basis (3\,..., (3 of a, which is not necessarily J-orthogonal. The following steps will find a symplectic matrix X G SP (V) such that ^(X) = (a, a). 2n 2n 1. Find the dual basis 7 1 , . . . , j n of 0i,..., 2 (3 , that is solve the linear system 2n l'0 =6 (5.1) ii) li . . . , fan)' and 7 = ( 7 1 , . . . , 7 n ) ' ; where (3 = (f3 u 2 2. Find the integral matrix Y G GL {T>) such that Yd = C # 2n 3. Find the skew symmetric matrix M G GL (V) 2n such that M/3 = (2A7; 4. F i n d a matrix Q G GL <(D) such that M = Q'JQ; 2n 5. Let X = QYQ- . 1 Then X G SP {V) and * ( X ) = (a,a). 2n Let 71 = Z[(]. We shall apply this method to find X in fiP _i(Z) of order p and such that p V(X) = (71,1). We know that 1, C, • • • , C ~ p 2 is a basis of 71. 61 L e m m a 5.1. The dual basis of C p _ 2 is 7 1 , . . . , 7 _ i , where P (5.2) Jr Proo/. By Lemma 3.2, we need to verify where /(x) = xP' 1 + • • • + x + 1, and /'(() = P-2 p-2 Let 70 = 7 = 0. P p-2 7.-+1** = E 7 * + i * ' i=0 1=0 (x - 0 £ +1 p-1 p-2 - E 7.-+iC^ = E i=0 1=1 i=0 - E 7<+iC^ t=0 p i=0 • Thereby proving our assertion. Let 0 — ( l , C , . . . , (P ) ' and 7 = ( 7 1 , . . . , 7 - i ) ' . Then Y is the companion matrix 2 p 0 1 Cp_i -1 -1 ... - 1 and ( 0= 1 \ -1 and 7 = C-i -1 -1 P=^L ^p p p -1 -1 (5.3) - ... where L „ is the n x n matrix whose entries above the diagonal are 0 and the others are — 1. Since CP = C -iP p we have ( 0 = C'^0. Note that A = K ^ ( 62 ) / 2 w e see that Let M = Lp-iC .E+l f . By a long but routine computation, we see that 2 Af = 'L' -i V p 2 is a skew symmetric matrix, and M = Q' _ J -iQ -i, p l p p ' 0 1 where Q -\ p = I + L^-x G G L _ i ( Z ) . p Therefore we have shown P r o p o s i t i o n 5.1. Let Xp — (5.4) Qp-iGp-xQp^i -1 1 -1 1 1 where each block is a ^ x ^ m a i r * a ; - 1 Then X -1 G S P _ i ( Z ) with order p and $(X ) p p (0 - l \ \1 - U E x a m p l e . When p = 3, we see that X = p = (ft, 1). In Section 5.3 we shall see that all X is an element of order 3 in SP2(Z). p are realizable if p > 5, that is X T* with respect to some canonical basis of Hi(S), p is the matrix of for some analytic automorphism T of some compact connected Riemann surface S. 5.2 Cyclotomic Units The cyclotomic units in ft are Uk sin^r f, sm for(fc,p) = l - 63 (5.5) Since 1- C K = A i-C u, ft where A = -(^ l k (5.6) 1 c*^ and is a unit, we conclude that u E U . The following properties of the cyclotomic units + k are easy to verify: ui — 1 and u = —u mp+k u >0, m p _ = (-l) u m f c f c (5.7) l<k<p-l, k (5.8) u < 0, p + l < f c < 2 p - l . k L e m m a 5.2. Y!j=\ 2j+l = u u u k k + i + l . Proof. We use the trigonometric formulas, * * E ^ =E 0=1 s i i2i±^r n s i n E s i n 2 f P J = l * _J_ V 1 ' (2j + Z + l ) v T (2j + / - l ) 7 T cos cos - cos ^±m)E cos 2sin £ 2 s i n M s i n i ^ ± l l l V P r-2iF = u u i+i k k+ • From now on we let the i-th conjugate of ( be ( \ We have L e m m a 5.3. uf = {-lf - ^ ^ k l i+l u uj . l ik Proof. Using (5.6), we see that uf = ( - C ^ ) ^ 64 1 ^ l-C i-C* = ( _i)(^i)(Hi)^ i u r • Lemma 5.4. A ^ = ( - l ^ u ^ A . Proof. Since A = Lemma A « = We obtain ^ = ^ ( 5.5. Suppose X € S P _ i ( Z ) /ias order p, and * ( A ) = p f(I ) = k ^ ¥ L (o, ± (By Then a = A o ; ' Ja and _ 1 ^ ( '>,a*), where fc = Qkk' {k') a a • a € a|. Proof. Suppose a is a J-vector with respect to (a, a) and Xa — (a. a u ( / > , ( - l f ' \ / ) ) , fc k (k') = {-iy^ r\ a). T/ien where \ < k <p — \, k' is the inverse of k modulo p, and o^'^ = X n h = fc = A-V e ( f c , ) n c e Ja^ = = a ^(A-VJo)^) = (-1)*'" V^* 0 Lemma 5.4). This completes the proof. Lemma 5.6. u • C, for 2 < k < p — 2. k Proof. We only consider 2 < A; < - ^ . p Case I: k is even. For 4 < 2A; < p — 1, we get ujj ^ = —U2kU 2 1 2 < 0, and so ^ C. Case II: A; is odd. There i s l < i < p — 1 such that p + 1 < ki < 2p — 1. Then we have = u ki 7 u l Lemma 5.7. < 0) hence Uk^C. Ukuf , 1 • u ui ^ C , /or 1 < A;, / < and k ^ I. k 65 Proof. There is 2 < i < p - 2, such that il = k (mod p). Then Ukuf = ±u u~ 1 = ±uf\ 1 u ±uf^ But does not belong to C since if it did we would have ±ui € C by choosing the appropriate conjugate. This contradicts Lemma 5.6. Then u u = (u ul )uf £ C (since uf G C). l k t k • By Lemma 5.5, Lemma 5.6 and Lemma 5.7, the following corollary and Proposition 5.2 are easy to prove. C o r o l l a r y 5.1. The p—l elements [ ± l ] , [ ± i t 2 ] , . . . , [ i u ^ - i l are distinct in U /C. + 2 P r o p o s i t i o n 5.2. Let X p be the matrix given by Equation (5.4). Then X ,X ,..., p p X _ 1 p are not similar to each other. P r o p o s i t i o n 5.3. If ^- is odd, then there is an X G S P _ i ( Z ) of order p, such that there are Pj p just two different classes amongst X,... Proof. Let a = u • • • u^i. p l There is X G S P _ i ( Z ) of order p such that $ (X) = {71, a). Suppose 2 a G TV' ,X ~ . p (a ^ 0), Xa = C,a and a = A' a'Ja. 1 From Lemma 5.5 and the fact that 71^ = 71 l we get * ( X ) = (K,a ), f c k where a = ( - l ) * ' ^ , ^ * ' ) and k' is the inverse of k. Note that -1 1 k ffc'1 —l —l > = ±u 'U , •••u =i ,u , = ±m • • -Up-iu , — a 2 2k k E k k 2 ±au , v k k E+l hence a/a k = ±u ? k G C U ( - C ) . Therefore *P0, if <b(x ) = { a/a eC, k K ^(X- ), 1 i.e. X,..., X p ]fa/a €-C. k • are in two different classes. E x a m p l e . Let p — l. Let 0 0 0 1 - 1 0 X = 0 0 0 -1 °1 0 0 -1 0 0 -1 c +c-i 3 -c 6 a = 0 1 0 0 0 0 0 0 0 1 1 0 0 1 C + i 0 c1+c °J 66 2 I 6 J Then one can easily check that Xa = (,a, X G SPe(Z) and X = I. One can also check that 7 a = A o ; ' Ja = C + C = u^u^. _ 1 B y computing, we get 6 X ~ X ~ X 2 and 4 P r o p o s i t i o n 5.4. Suppose p = 1 (mod 3). X ~ X, X ~ X 3 5 T/iere is X ~ X . 6 G S P _ i ( Z ) o/ order p sitc/i £/ia£ p where k is the least positive solution of k + k + 1 = 0 (mod p). k 2 Proof. Since p = 1 (mod 3), then x + x + 1 = 0 (mod p) has a solution. Let k be the minimal 2 positive solution. There is an X G £ P _ i ( Z ) , of order p, with \I/(X) = (TZ,UkUk+i). Then by p applying Lemma 5.5 we get ty(X ) = (lZ,u), where k = (-^-^(-i^-^-^^,^ Note that k(p — k — 1) = mp + 1 and (fc + l)(p - A; - 1) = (m + l)p - A;. Hence X ~ X * . • To finish this section we give a proposition: P r o p o s i t i o n 5.5. There are integers ki,...,k , such that 2 < ki < • • • < k n < n Y~, and PJ Ukx • • • Uk G C if and only if hi, the second factor of the class number of TZ, is even. n Proof. Let C\ be the group generated by ± 1 , U 2 , • • • , zz±• u Suppose u kl • • -Uk = u G C and u G U . 2 n + Then [U + : C\] = /12, see [20]. We see that u £ C\ since u , • • • ) £ ^ i n a r e 2 generators. Let C2 be the group generated by ± 1 , u,U2,. • • , • Clearly, C\ C C C U + 2 f r e e and 2 [C : Ci] = 2, so 2\h . 2 2 If / i is even, there is u G U , + 2 all of rj are even. Thus u 2 v G Ci. It follows that = u kl u C\, but u •••u 2 v 2 kn G C i . Then u r -i = u^ • • - u ^ 2 P 2 1 where not for some distinct integers 2 < kj < ^- and some Pj • • • Uk G C. • n 67 Remark. In case that h is odd, the 2 ^ elements [ i u ^ u ^ • • • Uk ], where 2 < ki < • • • < k < 2 P j ^ , are all distinct. They are in fact the elements of U /C. + 5.3 n n Realizable Elements of Order p Theorem 8 is similar to a result of P. Symonds[35], but our approach is new. We consider short exact sequences of Fuchsian groups 1 -»• II ->• T(0;p,p,p) A Z ->• 1 p where T(0;p,p,p) — ( A i , A , A | A 1 A 2 A 3 = A\ = A\ = A\ = 1). If II is torsion free, then we 2 3 get an action of Z on S = U/II, with genus Now we indicate how to find all epimorphisms p 9 with torsion free kernel. The epimorphism 9 : Y —>• Z is determined by the images of the generators. The relations p in T must be preserved and the kernel of 9 must be torsion free, therefore 9 is determined by the equations A i -»• T ° , 9:1 A -> T , 6 2 A3 ^ T , c where T is a fixed generator of Z , l < a , 6, c < p — 1 and o + 6 + c = 0 (mod p). p We use M ( a , b, c) to denote the matrix class which is induced by T . Let V(a, b, c) denote the symplectic Z -space p H\(S) where the action of T on H\(S) is given byT*. T h e n S ( V ( a , 6 , c ) ) = M(a,b,c). The proof of Theorem 8 is based on Proposition 4.5. Hi(S), and M is the intersection matrix of a i , . . . , a _ i . p respect to a i , . . . , a _ i . Let a = ( a i , . . . , a _ i ) ' € p p ft p_1 Suppose a i , . . . , a _ i is a basis of p Let X be the matrix of T* with be an eigenvector of X with respect to C- It is easy to check that * ( M ( a , 6 , c ) ) = * o S(V(a,6,c)) = (a,A~ a'Ma), l where a is the ideal generated by a 1,..., cc _i. p Remark. If we prove the special case where a = 1 and 1 < b < tt(M(l,M) = 68 (n,u u i), b b+ , that is if we show that then Theorem 8 will follow. This is because M ( l , 6 , c ) = M ( l , c , 6) and M(a,b,c) is the a'-th power of M ( l , 6 i , c i ) , where aa' = 1 (modp), b\ = a'b (mod p), c\ = a'c (modp). Applying Lemma 5.5, we would get tt(M(a,6,c)) = (n,{-l) - u {u u )^) a l a bl bl+l and by Lemma 5.3, we could then have u= {-l) - u {u u )^ a l a bl = (-i) - «a(-i) f l a = ^ o ( 6 i - 1 ) ( * + 1 ) «6 aur (-i) 1 6 i ( a + 1 ) 1 t*( 6 l + 1 ) 0 «r 1 mp+b mp+a+b u = 1 bl+l u 1 ( - u 1 ) m u h ( - 1 ) m ' u a + 6 = U~ U U l b where m satisfies b\a = mp + b. We see that u/u u u a Thus we assume a = 1 and 1 < b < b a+b = u~ G C. 2 a+b Then ^ < c < p — 2. We choose a particular embedding of F in Aut (U), namely F is the subgroup generated by A i , A , A 3 , where A i , A , A 3 2 2 are rotations by 27r/p about the vertices vi,V2, ^3 of a regular triangle P, all of whose angles are -ir/p, see Figure 2.1. A fundamental domain of F consists of P together with a copy of P obtained by reflection in its side V1V3. Then a fundamental domain D of LT is the 2p-gon consisting of p copies of the fundamental domain of F obtained by the p rotations A\ (k = 0 , . . . ,p — 1), see Figure 5.1. Let e i , . . . , e 2 p be the 2p sides of D, and r\i = e j_i + e j (for i = 1 2 2 Then r / i , . . . , r) are closed paths on S and [r/i],..., [rj -i] forms a basis of H\(S), p p ,p). see [24]. T h e intersection matrix of [771],..., [r? _i] is somewhat complex, so we need to find another basis. p Since fl^+^A^A*-') = 1, then 7 = A[ ~ A^ +i l 1 A\~ G II is a boundary substitution { of D and so [e j_i]n = [—e2 +2i]n- In the interior of each side ej, we choose a point Ei such 2 C that [£?2i-i]n = [E2c+2i]u- Let /j denote the straight line segment from v\ to Ei in D. Let = J2i-\ — /2c+2i- Then ^ is a closed path on 5. It is clear that [&] = [iji] -\ 1- [rj +i] and [ni] H c 69 \-[r) ] = 0 in the homology group p Hi(S). Figure 5.1: Fundamental Domain (order p) Hence the transform matrix from [77]'s to [£]'s is the (p — 1) x (p — 1) matrix c+l< 1 -1 -1 }p-c-l -1 0 where the entries Xj,- are given by 1, l < J < p — c — 1 and j < i < j + c, — 1, p — c < j < p — 1 and j + c + l—p<i<j 0, otherwise. 70 — 1, By applying the Laplace expansion theorem to the last row we see that the determinant of this matrix is just the determinant of the (p — 2) x (p — 2) matrix L +i, - -i, c p c s e e Equation (2.9). Since p is an odd prime and 1 < c < p — 2, then c + 1, p — c — 1 are coprime, and therefore jdet L c + i ) P _ _ i | = 1 (See Section 2.3). Hence [ft],..., [ft-i] is a basis of c L e m m a 5.8. Hi(S). The matrix o/T* with respect to [ft],..., [ft>-i] is /o - l \ •l -l V Proof. Let f 2p+i = fi and ft+ fc = ft. Since 0(Ai) = T, we get T([/;] ) = [Ai(fi)] n n = [/i+2]n, f o r i = l , . . . , 2 p . Then ([€k]n) = T([f2k-i]n T — - [/2c+2Jfe]n) [/2fc+i]n — [/2c+2fc+2]n = [£fc+i]n for A: = 1,... ,p. Therefore T*([ft]) = [ft+i], for k = 1,... ,p - 1. But [ft] + • • • + [ft] = 0 and therefore = [6], r.fl&D = [6], r*(&- ]) = K -i], 2 P r.([e -i]) = -[ei]-K ] P 2 K -i]P • This proves the lemma. Now we compute the intersection matrix M of [ft],..., [ft-i]. Let m j j be the intersection number ft • ft of [ft] and [ft]. We have 71 L e m m a 5.9. For any 1 < i,j < p - 1, m^j = mj+ij+i and mij+i = - m i i P _j + 1 . Proof. T* preserves the intersection number of closed chains. By Lemma 5.8, i,j m = & • ij = p*di) • T*Hj) = ii+i • ij+i = m ij i. i+ + Iterating this formula we see that m i _ j + i = mj+i +i = rrij+i^ = —m\j+\. iP )P Let kj — m i j + i . Then m-ij+j = kj. Hence the intersection matrix is of the form M = fciMi + • • • + A ; _ M _ , P 2 P 2 where the Mj is the (p - 1) x {p - 1) matrix /o ... 0 1 A 0 0 -1 0 0 - 1 0 V The entries x$ of Mj are given by 1, x kl l - k = j, -1, 0, otherwise. By Lemma 5.9, we see that kj •- mij+i = — m i M = feiMi + k {M 2 2 k-l=j, i P + i _ j = —k -j, and therefore - M _ ) + • • • + kj^i ^ M H - I P 2 L e m m a 5.10. kj — p 1, 0, l<j<p-c-l, p - c < j < ^ i . 72 j . Proof. It is clear that the intersection of ft and ft (j = 1,..., ^) +1 is only one point, namely the vertex v . The verification of (5.9) is straightforward by referring to Figure 5.2 and 5.3. x r 2j+l Figure 5.2: p - c < j < (p - l)/2 Let 'l + < + ••• + c~^ l + c +• • • +C ~ a = p 2 p 3 i +C V i ) a is an eigenvector of C' _ with respect to the eigenvalue ft that is C _ a = (a. p x p 1 A~ a'Mia, L L e m m a 5.11. Let Vj A-'a'iMj-M^a, j = 2,...,^i. Then yj = u j. 2 Proof. Let 0 = (1 - ()a. We see that 0 = lk C~p k p—ip—i 0'MJ0 = E E A;=l ;=1 = p-i-j = E k=l E M i-fe=j - p-i - E A ft p-i-j p-i-i E k=j+l = 73 E fofo+j k=l - E fo+jPk k=l • p-l-3 , , P-1-3 = *=i E (i-c*- )(i-r*-'")- AE ;=i v fc ="E (I - c- p fc k=l (i-c- ^)(i-r ) fc r ' +c ) - ~E k 3 j P fc c^- k (i - r +v) k k=l (r - c - +c^- - r''-*)+(? -1 - i) (V - ?) fc ="if p fc fe = D(C*-?) + ( P - I - J ) (c-?) = fe=i E (c 2 fe -c ) + ( P + i - i ) (c-?). fc Hence for j = 1, /3'M{p = p For j = 2,..., pMjfl ((-(). we have - V'M^P = J2 2 (C - C") + (p ~ 1 - j) (V - ?) fc fc=i - '£ 2 1 (c - ?) fc k=j+l o > + 1 - P+i) (<r' - =p{t -?)- ~iz c ) j k (c - ?) j So we get =P , P ^ (ri 7*\ C ^ C - (C ~ 1) J 2j ,2±l\^ • Proof o/ Theorem 8. Let o be the ideal generated by the components of a. It is clear that a = TZ since 1 6 a. Now applying Lemma 5.2 and Lemma 5.11, we obtain A~ a'Ma l This completes the proof of Theorem 8. = u^u^i. • 74 Chapter 6 SP (Z) Torsion i n 4 We consider torsion elements of SPt(Z). The first question we consider is: for what positive integers d(d > 2), is there a matrix X G SP {Z) 2n its minimal polynomial m (x) having order d? If X has order d, then is a factor of x — 1, i.e. m (x) is a product of some different d x x cyclotomic polynomials, and its characteristic polynomial f (x) is a product of some cyclotomic x polynomials. Suppose d = pf • • -pp where pi,... ,pt are different primes. According to a result 1 of D . Sjerve [34], the degree of f (x) is not less then 4>{p i) H S x h 0(pj ) — 1, so ( 0(Pi ) + --- + 0 ( p f ) < 2 n + l . 1 We get If n = 1, then d must be 2, 3, 4, 6. If n = 2, then d must be 2, 3, 4, 5, 6, 8, 10, 12. (0 Let W x = -v - l \ and W = W i . Clearly, W _ = -W' V A x P r o p o s i t i o n 6.1. Suppose X € SP^CL) has order 3, 4, or 6. Ax + 1, and A ~ or and W = - J . 0 2 T/ien /x(^) = r n ( ) = x x + where A = 1 (resp. 0, —1) if the order is 3 (resp. 4, 6). This is an application of Theorem 1 or a corollary of Lemma 6.5. We denote by the set of elements of order d in SP^Z). I. Reiner gave a complete list of representatives of the conjugacy classes of involutions in all symplectic groups SP<2. {Z) [30]. n We state the special case for T here without proof. 2 75 P r o p o s i t i o n 6.2. Any X € T is conjugate to one of the three following matrices 2 -h, where U 1 0 1 -1, h*(-h) or U + U' (6.1) Now we suppose that d > 3. Let X € Tj. The possible minimal polynomials m x (x) and characteristic polynomials fx(x) are as follows: When d = 3, m(x) = (x + x + 1), /(x) = (x + x + l ) , (6.2) m(x) = (x - l)(x /(x) = ( x - l ) ( x + x + l ) . (6.3) 2 2 + x + 1), 2 2 2 2 When d = 4, m(x) = (x + 1), m(x) = (x- (6.4) /(x) = (x + l ) , 2 2 2 l)(x 2 + 1), /(x) = ( x - l ) ( x + l), (6.5) m(x) = (x + l)(x 2 + 1), /(x) = (x + l ) ( x + l). (6.6) 2 2 2 2 When d = 5, m(x) = f(x) = x + x + x + x + 1. 4 3 2 (6.7) 1 When d = 6, m(x) = (x - x + 1), = ( x - x + l) , 2 2 (6.8) 2 m(x) = ( x - l ) ( x - x + l), fix) = ( x - l ) ( x - x + 1), (6.9) m(x) = (x + l ) ( x - x + 1), fix) = (x + l ) ( x - x + l ) , (6.10) m(x) = (x + l ) ( x + fix) = (x + l ) ( x + x + l ) , (6.11) fix) = ( x - x + l ) ( x + x + l). (6.12) 2 2 2 2 X 2 + 1), 2 m(x) = ( x - x + l ) ( x + x + l), 2 2 2 2 76 2 2 2 When d = 8, m(x) = f(x) = x + 1. (6.13) 4 When d = 10, m(x) = /(x) = x - x 4 + x - x + 1, 3 (6.14) 2 When d = 12, m(x) = /(x) = (x - x + 1), (6.15) m(x) = /(x) = (x + l)(x + x + l), (6.16) m(x) = /(x) = (x + l ) ( x - x + l). (6.17) 4 2 2 2 2 2 Remark. The characteristic polynomials (6.7), (6.13), (6.14) and (6.15) are irreducible over Z. We have given a complete set of conjugacy classes for these cases (see Examples in Section 3.5). Remark. The characteristic polynomials (6.16) and (6.17) are products of two strictly coprime S-polynomials. According to Theorem 6, all matrices with characteristic polynomials (6.16) or (6.17) are decomposable (see Section 4.2). By Lemma 2.2 and Proposition 6.1, and the Remarks above, we obtain P r o p o s i t i o n 6.3. The number of conjugacy classes in T12 is 10. A complete set of non- conjugate classes is given by h o (-W), h o {-W); J *W, J *W, J *W, J *W; (6.19) J *(-W), J *(-W), J *(-W), J' *{-W')\ (6.20) 2 2 (6.18) 2 2 2 2 2 2 with respect to characteristic polynomials (6.15), (6.16), (6.17). For all other cases, we need to develop some new tools. In Section 6.1 we shall use symplectic complements to study the case where ± 1 is an eigenvalue of X. 77 In Section 6.2 we discuss the case of characteristic polynomials (6.2), (6.4) and (6.8). Then in Section 6.3 we consider the last case of (6.12). Finally, in Section 6.4 we shall give a list of conjugacy classes which are realizable. We use the program Maple V to calculate most of our results in this chapter. 6.1 Symplectic Complements A primitive integral 2n x (j + k) matrix j,k <n which satisfies the conditions A'JA = 0, B'JB = 0, and h A'JB 0 or (l, o) (depending on whether j > k or j < k) will be called a normal (j, A;)-array. According to Theorem 5 every normal (j,fc)-arraycan be completed to a symplectic matrix by placing n — j columns after the first j columns and n — k columns after the last k columns. Remark. Let a, 0 e Z 2 n . Clearly, a is (l,0)-array if and only if et is a primitive vector, and (a, (3) is a normal (1, l)-array if and only if a'Jf3 = 1. L e m m a 6.1. Suppose that X G SF"2n(Z) and / x ( l ) = 0. Then i a A 0 A & B 0 0 1 0 h X 1° where Y = A ,C C 0 B G SP (n-i)(Z), fx(x) = (x2 l) f {x), 2 Y a G Z , and a, f3, 7, 8 G Z ~ n l D, with £7, a AS - P C6 - D7, 7 Ca - A'0, 8 D'a - B'0. 78 (6.21) Furthermore, ifY~Yy Ai = 5i then ( \ 7i ai 0 Ai ai 0 0 X \0 5[\ Bi 1 0 Ci ft Proo/. Since 1 is an eigenvalue of X, there is a primitive vector 77 € Z 2 n such that Ar? = 77. By Theorem 5, we can find a integer symplectic matrix P with 77 as its first column. Then (l P~ XP l = Xi = i a S'\ 0 A a B 0 * b * C p DJ € \0 (A By computing we can see that the *'s are 0, b — 1, Y = x = {x - 2 B\ I G 5P2( _i)(Z), and a, P, 7, D) n \C 6 satisfy (6.21). Thus f (x) SP n(Z). l) g (x). 2 Y The second part is easy, merely conjugate by I*Q, where Q € 5P2(P) and Q~ YQ 1 L e m m a 6.2. Suppose X € SP4(Z), m (x) x = (x — l)(x 2 and X I*W' , A Moreover, these matrices are not conjugate. Proof. It is clear that I *W\ >* I *W' X (cf. Lemma 2.2). 79 • + Xx + 1) where A = 0, ± 1 . Then X is conjugate to one of I*W = Y\. By Lemma 6.1, we get X ~ X\ 0 A 0 0 B 1 0 C \0 (A di ei D) B\ where Y = € SP (Z) with / ( x ) = x + Xx + l. Then, from Proposition 6.1, Y ~ W 2 2 y A or Wj^. Without loss the generality we assume V ~ W\. Then X 1 a 2 6 0 0 a -1 0 0 1 0 ^0 1 ~x = 2 0 C2 2 2 Aa + c 2 -X) 2 /1 -a \ 2 where the last conjugacy is achieved by Q = I v° (A + 2)6 + c = 0 since m (x) = (x — l){x 2 2 x 0 0 0 0 \0 1 X, = . / 1 6 c 0 1 0 c -Ay 0\ / w7 +1 1 -1 ^ SRi(Z). We obtain + Xx + 1). This implies (A + 2) | c. Now we use Theorem 6 to see that X 3 is decomposable and use Proposition 6.1 to complete the proof. In fact let k (\ P = -1 k^ 4 1 V where k — € SP (Z) —k 7 k e Z . It is easy to check that P - -1 1 X P = I * W\. 3 • Similarly, we have L e m m a 6.3. Suppose X € S P ( Z ) , m ( x ) = (x + l)(x 2 4 x + Xx + 1) where X = 0, ± 1 . T/ien X is conjugate to one of (-1) * W x and and these matrices are not conjugate. 80 (-1) * W' , x Proof. Since m_ {x) = (x - l ) ( x x -W\ 6.2 2 - Xx + 1), we have -X ~ I * or 7 * W i . Note that A = W' . This complete the proof. • X M i n i m a l Representatives Let X G iSP2ri(Z) and 77 = ( x i , . . . , X2n)' € Z 2 n . If a = n'JXn then we say that X represents a. The set of values represented by X will be denoted by q(X). invariant, for if Y = Q~ XQ, where Q G SP {Z), X 2n o(Y) = q{Q~ XQ) then = {n'JQ- XQn l It is clear that q(X) is a conjugacy l | 77 G Z "} , 2 and so putting £ = Qrj gives (,'JXi Thus q(Y) = q(X). = n'Q'JXQn = n'JQ^XQn = n'JYr). Unfortunately, the converse is not necessarily true. The set q(X) is a set of integers, and consequently there is a non-zero 770 in Z 2 n such that |77 JXT7O| is least. If both n' JXr]o and —r] JXr)o = if^JXni occur, we resolve the ambiguity 0 0 0 by choosing the non-negative value. We write u(X) = r)' JXr]o- Clearly, if u(X) 0 7^ 0, the minimizing vector XQ must be primitive, and if u(X) = 0, we also can choose a primitive vector 770 such that rj' JXr)o = 0. 0 E x a m p l e . If X is quasi-decomposable, then u(X) = 0 since JX will have zero entries on the diagonal. L e m m a 6.4. Let f(x) = f (x) x be the characteristic polynomial of X. ,„ < (I)""* ! / W f c ! > ! ± . ml Then „ 2 Proof. Note that rj'JXrj is a quadratic form over Z . If M is a symmetric matrix belonging to M ( Z ) , and a — min {JT/MTJI 177 G z W , i ) ^ 0}, then n -J 2 81 |detM|». 2 , See [26]. Clearly, it is also true if M is a rational symmetric matrix. We know that rj'JXr) = \r,'{JX ric matrix. Because [JX)' + (JX)')r], where \(JX - X'J' \JX + (JX)'\ = \JX - JX' ] 1 = = -X'J = -JX' , 1 2 is a rational symmet- and \J\ = \X\ = 1, we see that 1 IJHX- !^ - + (JX)') I\ = / ( l ) / ( - l ) . Hence \t*(X)\ < • Remark. Note if X G SPi(Z) is a torsion element, then |/-*(A")| < 1 since is integer and the maximum of |/(l)/(—1)| is 16. L e m m a 6.5. Suppose X G S P ( Z ) , and 1 G q(X). Then 2n / n0 X n 0 _-1 1 n 0 \ 0 A a B 1 V a 6' C 0 DJ \0 , A B where \ I GSP 2 ( n _ ( Z ) , a G Z , and a, 0,J,8 G Z " " x ) Proof. Since there is a primitive vector rj G Z 2 n 1 aatfa/y (6.21). such that rj'JXri = 1, we see that {r},Xrj) is a normal (1, l)-array. Let P be the completion of the normal (1, l)-array (r),Xr]) to a symplectic matrix. Then P = and therefore rj * Xrj A) * b P- XP X = Xi = * *^ 0 A a B 1 i a 6' C 0 DJ \0 82 e 5P (Z). 2n • The remainder of the proof is similar to that of Lemma 6.1. C o r o l l a r y 6.1. Suppose X G S P ( Z ) , m (x) 2n where Y G S P ( - i ) ( ^ ) with m y (a;) = 2 n 2 0 = 0, and so 7= with 1 G q{X). 2 Then X ~ W *Y, X m (x). x we see that the entries of the matrix in Lemma 6.5 are: a = — A , Proof. Since X rj — -XXrj-rj, a = 0, = x +Xx+l, x 0, S = 0. • L e m m a 6.6. Suppose X G SP n(Z), and p(X) = 0. Th en 2 ^0 A a 1 7' a 6 0 C 0 D ^0 0 1 0J X A B ] G 5P -D < where I .C 2 ( n _ ( Z ) , a G Z , and a, 0,7,8 G Z n _ 1 1 ) sate/?/ (6.21). Proof. Note that we have a normal (2,0)-array (n,Xri), where 77 G Z L e m m a 6.7. Suppose X G SP4(Z), with m {x) x L 2 n • is primitive. — x + Xx + 1, w/iere A = 0 , ± 1 . T/ien 2 If u{X) = 1, Men X ~ W * W . A A 2. If p{X) = - 1 , Men X ~ W * W j . A 5. 7 / / x ( X ) = 0 and X = ±1, then X ~ W * W j . A 4. 7 / M ( X ) = 0, A = 0, and 1 G q(X), = (-J ) * J . then X~W *W{> 2 0 5. 7 / / x ( X ) = 0, A = 0, and 1 £ g ( X ) , Men X ~ W + W = (-^2) 0 Proof. (1) If / i ( X ) = 1, then by Corollary 6.1, X ~ W 0 x m 2 0 ^2, * Y , for some Y G 5 P ( Z ) , with 2 (x) = x + Xx + 1. From Proposition 6.1, Y ~ W or Wj[. Then X ~ W * W 2 Y A But ji(W * W' ) = 0, hence X ~ W * W . A x x x 83 x x or W * W' . A x (2) If n(X) -X = - 1 , then (i{-X) ~ V F _ * V7_ , and thus X A A = 1. It is clear that m_ (x) x = x 2 - Xx + 1, hence (W"_ * W _ ) = Wj[ * W' . A A x (3) -(5) In the following we assume that p,(X) = 0. By Lemma 6.6 we get y X ~ X\ 0 a 0 0 0 1 0 6 0 0 1 0 0 0 V ), a, b € Z . Let P 6 X(a) = x A b where Y = | W'~ Xb — a, 0 -1 a 0 1 -A 0 -a 0 0 -A -1 yo 0 1 0 /l Let Q - 0 0 0 Then P^XiP . Then Q~ X{a)Q l 0 \0 0 where 1 1 1 0 0 = X{a), = X(a - 2). So we 1 0 0 1 obtain X ~ X(0) or X ( l ) . It is clear that 1 e q(X(0)) if and only if A is odd, and always 1 € q(X(l)). For the case where 1 6 q(X), we get X ~ W\ * W{. This completes the proofs of (3), (4) and (5). • From Lemma 6.2, Lemma 6.3, and Lemma 6.7 we obtain the following two Propositions. P r o p o s i t i o n 6.4. The number of conjugacy classes in T3 is 5. A complete set of non-conjugate classes is given by W*W, W'*W\ h *W, h* W. W* W'; (6.23) (6.24) with respect to characteristic polynomials (6.2), (6.3). P r o p o s i t i o n 6.5. The number of conjugacy classes in T4 is 8. A complete set of non-conjugate classes is given by J2 * J2, J *J, 2 J2 * J2, 2 84 (-h) 0 h\ (6.25) h * J2, h * J\ (6-26) (-/2W2, (-h)*J - (6.27) 2 2 with respect to characteristic polynomials (6.4), (6.5), (6.6). 6.3 T h e Case of f(x) = x + x + 1 A 2 In this section we discuss the case that X £ SP (Z) has f {x) = x + x + 1. From Theorem 6 A 4 2 x it follows that: L e m m a 6.8. If X is decomposable, then X is conjugate to one of four non-conjugate matrices, W*{-W), Note that m (x) — x + x + 1, hence X 2 x2 (6.28) W*{-W'), W'*(-W), W'*{-W). is conjugate to one of three non-conjugate 2 matrices W * W, W *W , W* W . 2 Without loss of generality we assume that X 2 2 2 = X\ * X , where X\ and X 2 2 are either W or W . We can express X as 2 X = Pi * P + P o P 3 2 (6.29) 4 where the Pj's are 2 x 2 matrices. Then X = X(Xi * X ) = P1X1 * P X + P3X2 o P X X = {Xi * X )X = X1P1 * X P + X1P3 o X P . 3 2 3 2 2 2 2 4 2 2 U 4 Note that X has order 6. Then (JX )' = X' J' = -JX' = —JX . Therefore we have 3 Pi=aX , 2 3 P P P = -aXl 3 2 3 P4P3 = (1 - a )X , = (l-a )Xi, 2 4 3 2 2 (6.30) and det P3 = det P 4 = 1 — a for some a € Z. Also, since X £ SP^Z), we have 2 P JP[+P JP^ P[JP + P' JP = J, X A A P' JP + P' J P 2 2 z Pi'JP + P^JP 3 2 3 = J, X and = 0, 85 3 = J, ^ P2JP'2 + PiJP't = J , P\JP[ + P3JP2 = 0. (6.31) We state the following lemmas without proof. They are very easy to verify. Let P be a 2 x 2 matrix. Lemma 6.9. If PW = WP, then P has form P = al + bW. Lemma 6.10. If PW + WP = 0 then P = 0. Lemma 6.11. If PW = W P, ° V 6 then P = ( 2 Clearly, if P = al + bW, then det(P) = a - ab + b . 2 Now suppose that X From Equation (6.30), we see that P3 = bi + cW, where = W * W. 2 l 2 l b - be + c = 1 - a . Hence a = - 1 , 0 , 1 . 2 2 2 If a = ± 1 , then b = c = 0, thus X is decomposable. If a = 0, then P = P — 0, hence X = P3 o P x 2 4 is quasi-decomposable. We know that the Diophantine equation b — be + c = 1 has six integral solutions. 2 2 1. b = 1, c = 0, then P = 7, P = W ; l 3 4 2. 6 = 1, c = 1, then P = -W , P = -Iv^ ; 2 1 3 4 3. b = 0, c = 1, then P = W, P = W ' - ; 1 3 4 4. b = 0, c = - 1 , then P = - W , P = - W ' 3 - 1 4 5. 6 = - 1 , c = 0, then P = - 7 , P = 3 4 ; -W \ l 6. 6 = - 1 , c = - 1 , then P = W , P = V F . 2 m 3 4 By Lemma 2.1 and I o W ' ~ W ' o I F ' (use J * W ' as the conjugating matrix) we see that the 72 2 matrices P3 o P , in all 6 cases, are conjugate. So we obtain 4 Lemma 6.12. Suppose X 2 ~ W * W, l l I = 1,2. decomposable and conjugate to I o W . l 86 If X is indecomposable, then it is quasi- Now we consider the case that X = W * W. 2 Lemma 6.13. Suppose X 2 2 ~ W *W. Then X ~ X(a,b,c), where 2 (a X(a,b,c)= b —a c\ —c 0 b+ c —a a b+ c 0 -b a c -a) (6.32) for integers a, b, c satisfying a — 1 = b + be + c . 2 2 2 Proof. From (6.30), we see that X = {-aW ) * {aW) + P P , where P P 2 3 PW 3 = P3W . 2 4 3 = (1 - a )W 2 4 and Applying Lemma 6.11, we get P3 6 = ( c \ b+ c -b) K It is clear that det P = -(b 2 3 and P 4 I—c = \b b + c' c + be + c ) = 1 - a . 2 • 2 Remark. For any integral solution of a - 1 = b + be + c , X(a, b, c) 6 5P (Z), and its charac2 2 2 4 teristic polynomial is (6.12). Clearly, a ^ O . Remark. A n easy calculation proves that X (a, b, c) ~ X(—a, b, c). 5 Lemma 6.14. X(a, b, c) is decomposable if and only if a is odd. Proof. It is easy to check that 5 ( A - I) € M ( Z ) if and only if a is odd. 3 4 Lemma 6.15. • p(X(a, b, c)) /ms Me same sign as the non-zero number a. Proof. Let M = JX(a,b,c) + (JX(a,b,c))'. We want to prove that M is positive definite if 87 a > 0, and M is negative definite if a < 0. We see that M = 2a 2b+ c —a -b + c 2b + c 2a -b + c —a —a -b + c 2a -b + c —a -b -b -2c -2c 2a Its principal minors are: M i = 2a, M 2 2a 26 + c 26 + c 2a 2a 26 + c —a 26 + c 2a -6 + c —a -b + c = det 4a - 46 - 46c - c = 4 + 3c > 0, 2 / M 3 = det 1 2 2 2 j \ = 6(a - a6 - a6c - ac ) = 6a, 3 2 2 2a = 9. Hence M is positive or negative definite dependent according as a > 0 or a < 0, • C o r o l l a r y 6.2. X ( a , 6 , c) is quasi-indecomposable. C o r o l l a r y 6.3. X ( a i , 6 i , c i ) X(a ,62,c ) 2 «/a a 2 x 2 < 0. If a is even, then X ( a , 6, c) is also indecomposable. equation a — 1 = b +bc+c 2 2 2 It is known that the Diophantine has infinitely many solutions with a even. There are infinitely many X € SP4(Z), which are neither quasi-decomposable nor decomposable, of the form X(a, 6, c). In the following, we want to show that there are just two classes amongst X(a, 6, c), where a is even. For this purpose, we let V(x,y,z) = 2x 0 0 — 2x z x —x z y V x \-x —y x —x —z J where r a = — 3x + y + z, x = a — b — c, or b - -2x + z, y = 2a — 26 — c, c - —2x + y. ,z — 2a — b — 2c, Then V(x,y,z) = QX(a,b,c)Q~ , where 1 (l 1 -1 0 0 - 1 - 1 1 Q = 1 1 0-1 0 -v 1 v° \ 0 It is easy to see that a — 1 = 6 + 6c + c if and only if yz = 3x + 1, and a is even if and only 2 2 2 2 if x + y + z is even, and also a > 0 if and only if y > 0. Furthermore, we have L e m m a 6.16. Let x, y, z be integers satisfy yz — 3x + 1 and x + y + z is even. Then 2 1. Ify> 0, then V{x, y, z) ~ 2. Ify< 0, then V(x,y,z) V{0,1,1); ~ V(0,-1,-1). Proof. Suppose yz = 3x + 1, and x + y + z is even. If y is even, then y = 4k, where k is odd. 2 The reason for this is that x is odd, and then z is odd and 3x + 1 = 4/ where / is odd. If p is 2 an odd prime and y = 0 (mod p), then p = 1 (mod 3). This is because p ^ 3, and 3x + 1 = 0 2 (mod p). Thus we see that y has the form y= ±4rp i -P? r 1 where r = 0,1, rj > 0, and the pi are primes of the form 3/c + 1. Now suppose y > 0. First we want to prove there is a solution (u, v) of the Diophantine equation y = 3u + v 2 2 satisfying u + xv = 0 (mod y). If y = 1 then (0,1) is a such solution. 89 If y = 4, then x = ± 1 (mod 4). A solution is (1, +1). If y is an odd prime and y = 1 (mod 3) then it is well known that there are a,b G Z such that 3a + b — y, which implies (a — xb)(a + xb) = a — x 6 = a (3rc + 1) — yx = 0 (mod y). 2 2 2 Hence either a — xb = 0 (mod y) or a + 2 2 2 2 2 = 0 (mod y). So either (a, —b) or (a, 6) is a such solution. In general, we use induction on the factors of y. Suppose y = yiy , and (UJ, v{) are solutions 2 for yi (for i = 1, 2), that is ?/j = 3uf + v and U{ + xu; = 0 (mod y). Let 2 U = U\V 2 + ^2^1, u = V\V — "&U\U . 2 2 Then 3 u + v = y and 2 2 (u + xu)a; = (^1^2 + u vi)x + (v\V — 3 u i u ) x 2 2 = xv (u\ + xv{) + u v\X + U\u 2 2 2 2 (mod y) 2 = (u\ + xv\)(u + XW2) = 0 (mod y) 2 So u + xv = 0 (mod y) since (x,y) = 1. Therefore (u, v) is a solution for y. Now we can complete the proof. Suppose y = 3 u + v 2 2 and u + vx = 0 (mod y). Then v - 3xu = v + 3x v = (3x + l)v = 0 (mod y). Let 2 2 u P= u+xv v—Zxu y y u+xv 3m—v y y — v u \ —u V—diXU y rv u+xv y 2(u+xv) y 2 T h e n P e S P ( Z ) and P V ( 0 , 1 , 1 J P " = u 0 J That is V(0,1,1) 1 4 The second part is similar. ~v(x,y,z). • Remark. The u, v in the proof are coprime. We see that there is a primitive solution of the Diophantine equation 3u + v = m if m is a product of a power of 4 and odd primes of form 2 2 6k + I. 90 Putting all the results from Lemmas 6.2, 6.3, 6.7, 6.8, 6.12, 6.13 and 6.16 together, we have P r o p o s i t i o n 6.6. Any X € TQ, is conjugate to one of following matrices ~{W*W), -(W'*W') -{W*W')\ } h * (-W), h * (6.33) (-W); (6.34) - (7 * W), - (7 * w'y (6.35) {-h) ( - J ) * W; (6.36) 2 * W, 2 2 W*(-W), W*(-W'), W'*(-W), IoW, IoW', V{O l,l), W'*(-W); V(0,-1,-1). t (6.37) with respect to characteristic polynomials (6.8), (6.9), (6.10), (6.11), (6.12). 6.4 Realizable Torsion In this section we address the question of which classes of torsion in SP^Z) can be realized by a cyclic action on some Riemann surface. P r o p o s i t i o n 6.7. A complete list of realizable classes in SP (Z) is as follows 4 Order 2, -h, Order 3, W * W'; Order 4, (-J ) * J ; Order 5, Y, Order 6, - U + U'; 2 (6.39) Order 10, 1 Y\ Y\ no, i.i), ^(0,-1,-1); (W*W), (6.41) Y\ (6.42) (6.43) -Y, -Y\ A) 1 1 (6.40) 2 z, Order 8, where U (6.38) 0 -1. Y = 0 0\ 0 0 - 1 0 0 0 - 1 1 \i i -l -Y -Y\ (0 , and Z = - 1 0 -1 oy o 91 - 1 1 0 ^ 1 1 1 0 -l 0 o oy 4 (6.44) Consider the short exact sequence of groups 1 n - > where T A G-> 1 (6.45) F = T(g ;mi,.. . , m ) , G is a cyclic group and IT is torsion free. Recall the RiemannQ t Hurwitz formula where g is the genus of U/TI. For g = 2 the Riemann-Hurwitz formula becomes (6.46) Hence go must be 0 or 1. For go = 0 (resp. 1) we solve (6.46) for t and the mj. Then for each solution we find a Fuchsian group T and an epimorphism 9 : F —> G with torsion free kernel. To prove the realizability we choose a fundamental domain for F and use it to determine an intersection matrix. We illustrate this for the case of order 6; the other cases being similar. Suppose G = ZQ. If go = 1, then (6.46) has no solution. We assume that go = 0. We can find three solutions for (6.46). (i) t = 3, m i = 3, m = m = 6, (ii) (iii) 3 2 t = 4, mi = mi = 2, m = m = 3, 3 4 t = 4, m i = m = m$ — 2, m = 6. 2 4 If t = 4 and F = T(0; 2,2,2,6), then there is no epimorphism 9 such that n is torsion free. So we need only consider the first two cases. Case I, t = 3, m i = 3, m = 7713 = 6. That is 2 r = T(0; 3,6,6) = (A, B B \A 3 u 2 = B{ = B% = AB B X 2 There are two epimorphisms F —>• Z§. 01 : Bi -»> T , B^ 2 or T, 9 :< Bi -»• T , 5 2 £ -* 2 92 T , 5 = l) . where T is a fixed generator of Z . 6 Figure 6.1: Fundamental Domain (order 6) We first consider the case of the epimorphism 9\. A particular fundamental domain of II (see Figure 6.1) consists of 6 copies of the fundamental domain of T obtained by the 6 rotations B k (k = 0, . . . , 5 ) . The sides with the same label are identified in the Riemann surface S — U/II. It is easy to verify that [rji], [772], [vs], [%] is a canonical basis of Hi(S). induces a homomorphism T* : H\(S) ->• Hi(S) given by r)3 -> ~m + Vi, «4 ->• -V2+V3Hence the matrix of T* with respect to [771], [772], [773], [774] is V(0,1,1), and so V(0,1,1) is realizable. Similarly, consideration of 92 proves that V(0,-1,-1) is realizable. Case II, t = 4, m i = m = 2, m = m = 3. That is 2 3 4 r = r(0; 2,2,3,3) = ( A i , A , B B \Al 2 u = A\ = Bf = 5 2 93 3 2 = A AB B X 2 X 2 = 1> 9\ There are two epimorphisms 6 : T ->• Z 6 Ai -»• T , 3 0:1 A -»• T , 3 2 S i ->• T 2 (resp. T ) , £ 4 (resp. T ) . 2 -> T 4 2 Each # induces an action, denoted by T , on some Riemann surface S. Consider that epimorphism 6 such that 0(B\) = T . 2 basis of ti(X) Let X be the symplectic matrix of T* with respect to a canonical Hi(S). From a result of Macbeath[21], we see that T is fixed point free, and therefore = 2. Then X must be conjugate to one of the three matrices -(W*W), See Proposition 6.6. — (W * W) -(W'*W'), O n the other hand, X is realizable. 2 ~ W * W. -(W*W). Hence X ~ —(W * W), and so The other epimorphism leads to the same conjugate class. completes the proof of the case of order 6. 94 This Chapter 7 T h e E i c h l e r Trace of Z R i e m a n n Surfaces 7.1 Actions on p T h e Eichler Trace In this section we prove Theorem 9, 10 and 11. We begin by observing that the set A is not a subgroup of Z[£]. To see this suppose that x £ A , that is 1 4 is the Eichler trace of some automorphism T: S —> S. The possible values for the number of fixed points are t = 0, 2, 3 , . . . , and therefore the possible values of x + X = 2 — t are 2, 0, — 1, — 2 , . . . We also have x G A since * is the trace of T _ 1 1 : 5 —>• S. Therefore, if A were a subgroup we would have X + A 7 = 2 —i £ A, and hence Z would be a subgroup of A. But if n G A is an integer, n > 2, then n + n = 2n > 4 is not of the form 2 — t for an admissible t. Therefore A is not a subgroup. Recall that A is the set of realizable Eichler traces modulo Z . P r o p o s i t i o n 7.1. A is a subgroup ofZ[Q. Proof. Suppose x i and X2 are in A, say * X 1 = 1+ 1 ET*73I' j=i E77—T" X 2 = 1 + j=i 13 95 1 s Therefore x i + X2 = X, where X = 1 + E$=i sented by T i : S i — > S i and T2: S 2 E j U ^ijzp + — > S respectively, then x c a 2 I f X i and X2 are repre- n be represented by the equivariant connected sum of 7\ and T . Namely, for j = 1,2 find discs D j in Sj such that 2 Dj,Tj(Dj),... , T j ( D j ) are mutually disjoint. Excise all discs _ 1 T (Dj), k = 0 , 1 , . . . ,p - 1, k from S j , j = 1,2, and then take the connected sum by matching d(T {Di)) k to d(T {D )) for k 2 fc = 0 , 1 , . . . ,p — 1. T h e resulting surface S has p tubes joining S i and S . T h e automorphisms 2 T i , T can be extended to an automorphism T: S —»• S by permuting the tubes. The Eichler 2 trace of T is x- Thus A is closed under sums. If x € -A then also x £ A and x + X = 2 — t. Therefore x is the inverse of x once we reduce modulo the integers. The identity element of A is represented by any fixed point free action. • To determine the index of A in B we need a basis for B, but first we find a basis for B. Let m - (p- l ) / 2 . D e f i n i t i o n 7.1. Define elements 9\, 9 , ... ,9 2 Yl & j=m+l m in Z[£] by p-2 9i = C+ a n d °k = C - C , 2 < k k k < m. P r o p o s i t i o n 7.2. A basis of B is given by the m + 1 elements 1, 9\, 9 ,... , 9 . 2 m Proof. Suppose x = Ej=o j& £ ^[C]- Then a short calculation gives a p-2 X + X = 2ao - «i + Yl (J a + v-i ~ a a i) C ' j and therefore x £ -B if, and only if, a, + a _j = a i , 2 < j < p — 2. Solving for a + i , . . . , a _ p in terms of a i , . . . , a m m and substituting into x gives X = ao + ai#i + 02^2 -I Thus the elements 1, 81, 9 ,... , # 2 m r- a 0 . m m form a spanning set for 73. 96 p 2 Now suppose some linear combination is zero, say ao + a\9i + a 0 H 2 + a 0 2 m m = 0. It is easy to see that this is equivalent to a + C + • • • + a C + (a - a ) C m 0 a i m Thus we get ao = a\ = a = • • • = a 2 m + 1 m x + • • • + (ai - a )C " = 0. P 2 2 = 0, that is the elements are linearly independent. m Remark. Every integer n G B since 6\+ 0\ = —1. We also have £ — £ C - C" = i + 20i + 1 It follows that the elements 1, £ — 0 + • • •+ 2 £ —£~ ,... ,£ 2 2 m _ 1 • G B; in fact e. m — Q~ m form a basis for an index 2 subgroup of B. A n immediate corollary of Proposition 7.2 is C o r o l l a r y 7.1. is a free abelian group of rank (p — l)/2. .4 basis is aiuen bjy Me elements 01, 0 , • • • , 0m- 2 Before completing the calculation of the index of A in .B we first consider Question 4 from Chapter 1. Thus suppose two elements from A have the same Eichler trace, say 1+ * 1 1 u E _x j=i = 1+ s Ej=i£J,- _ ! • s This leads us into consideration of when certain linear combinations of the elements TJEZI zero, that is we want to solve the equation 23^=1 jd^x = 0 f° * r n e A R E integers x^. If s is any integer relatively prime to p then let R(s) denote that integer q such that 1 < Q< P — 1 and g = s (mod p), that is, s = [s/p]p + R(s). In what follows Y^jk=n denotes the sum over all ordered pairs (j, k) such that jk = n (mod p) and 1 < j < p — 1. L e m m a 7.1. fe=l s y jk=-l F n=l 97 \jk=n jk=-l J p-i , fc=i ^ y P JC* = I E?=i Proof. We use the identity ^ W_1) and get p-i j=i fc=i 3=2 k=l ^ = ^ n = l \jA:=n ^ 1 + ^ Now E[E« +J - - - + X ^ ) + HC 1 ^ n = l \jfc=n substitute (J?* = —1 — £ — —£ P 1 - 1 ^ S B +M E / V jk=-l F 1 j'fc=-l ~I + - E E ^* - E P = -- E 1 3x k jk=-l \jk=-l (T 1 J 1 u + v*k 1 I E ^ ) * - E O' + D ^ K ^ n = l \jfc=n - ; E + into the last term to see that 2 ETFTT = ^ i + - + v ) + f E k=l / / 2 n=l Yj'fc=n i? jk=—l • As a corollary we get C o r o l l a r y 7.2. Now ^ . = 0 i / , and on/y t/, Z = jx jk n = 0, / o r 1 < n < p - 1. k it is convenient to change the variables x\,... ,x -\ v to new variables y i , . . . , y - i p according to the equation yi = x , k where kl = 1 (modp). (7.1) Then Corollary 7.2 becomes C o r o l l a r y 7.3. Y%Z\ = 0 if, and only if, 98 £fc=l i?(nA;)y fc = 0, / o r 1 < n < p - 1. The coefficient matrix of this linear system is the (p—1) x (p—1) matrix M whose entry is M(jj) = R{ij). To solve this system of p—1 equations inp—1 unknowns y we apply a sequence k of row and column operations to the matrix M. We use the fact that R(ij) + R((p — i)j) = p. Recall that m = (p — l)/2. 1. Adding the i row to the (p - i) th th row, 1 < i < m, yields the matrix 2 m m + l m + 2 p-1 4 2m 1 3 p-2 * l?(2t) . R(mi) R{(m + l)i) R((m + 2)i) . R((p-l)i) m i?(2m) . R(m ) R({m + l)m) R((m + 2)m) . R((p-l)m) p p V P P P ^ p p P P P P Mi 2. Adding the j 2 column to the (p — j ) th ( , M 2 m p p P 4 2m p p P i R(2i) R(mi) p p P m R(2m) R{m ) p p P 2p \ = 2 P P p 2p 2p P P p 2p 2p st st column, 1 < j < m, yields the matrix 2 3. Subtracting the (m + l) (m + l) t h ... 2p J row from rows m + 2,... , p - 1 , and then subtracting the column from columns m + 2,... , p - 1 gives the new coefficient matrix 99 2 m p 0 0 4 2m p 0 0 R(2i) R(mi) p 0 i?(m ) p 0 0 / 1 M 3 = m R{2m) 2 P P p 2p 0 0 0 0 0 0 0 0 0 0 0 0 0 The variables z\. for this coefficient matrix are related to the y by the equations k Zk = Vk- Vp-k, 1 < k < m, z m + i =y i H m + Examination of the last m — 1 columns of M 3 h y _i, z +j = y +j, 2 < j < p - 1. p m reveals that m 2 M + 2 , • . . , 2 - i are completely p independent; whereas, z\,... , z +i must satisfy the matrix equation m I \ 1 2 2 4 i ii(2i) m il(2m) P P . / o \ m p 2m p Z2 . R(mi) p Zi . . R(m ) p 2 p 0 0 2p J \ Zm+l J V / 0 Now we apply another sequence of row and column operations to this last coefficient matrix. 1. Subtracting i times the first row from the i th 100 row, 2 < i < m, yields the coefficient matrix fl 2 . J m P 0 0 . 0 0 ~P 0 0 . • ~[3j/p]p • -[3m/p]p -2p 0 0 . • -[ij/p]p • —[im/p]p —(i — l)p 0 0 . • . — [m /p]p — (m — l)p Vp -[™j/p]p v • 2 p P 2. Subtracting j times the first column from the j column, 2 < j < m, yields the matrix th /1 J 2p 0 .. 0 0 P 0 0 .. 0 0 ~P 0 0 .. • ~[3j/p]p • -[2m/p]p -2p 0 0 .. • -[ij/p]p • • —[im/p]p -(i-l)p 0 0 .. • ~[mj/p]p •.. • \P -P — (m — l)p -[m / ]p 2 P — (m — l)p •• \ 2p The new variables Wj, after these last column operations, are related to the Zj by the equations w\ — z\ + 2z% + • • • + mz m It follows that w\ — w \ m+ = 0 and w , 2 W2 \ and Wj = Zj, 2 < j < m + 1. •• • , w m + 2wz -\ -[9/p]p . ~[3;/p]p -[Si/p]p . -[3m/p]p . are related by the equations h (m - l)w - 0, m ••• -[3m/p]p \ -[ij/ ]p P .. — [im/p]p [rnj/p}p .. -[m /p]p 2 101 ( w 3 \ ( 0 ^ 0 J \w m J \ 0J The coefficient matrix of this system can be row reduced to the matrix whose entry, 3 < i, j < m, is [ij/p]p — [(i — l)j/p]p, by first subtracting row m — 3 from row m — 2, then row m — 4 from row m — 3, etc., and then changing all signs. The resulting matrix is invertible, in fact its determinant equals ±p ~ hi, m Wj = where hi is the first factor of the class number [28]. Thus 2 0, 1 < j < m + 1. This proves that Yl =\ = k 0 ^> y m 2 M + o l u y ^ Vk ~ Vp-k for 1 < A; < p — 1, and y -u m P • • • , 2/>-i are completely arbitrary. Translating back to the x C o r o l l a r y 7.4. where x - = -y +2 m where y + , a n a variables we have: k 2^A-=1 2,--- , = 0 */> d tf> Xm X +2 an — k x p-k = f x ''' m 1 < k < p — 1, and or Xp— l , are completely arbitrary. We can now complete the proof of Theorem 10. Proof. Suppose x i = X2 are the Eichler traces of two actions, say * Xi = 1+ 1 £7fe—T 3=1 P = _ u 1 E7FTT' 1+ fc=l S S u p-1 ^ * = + E 77—T = + E 7FT7' 2 1 1 j=l where u k fc=l S S is the number of times k appears as a rotation number in x i , similarly. We immediately get t = u since x i + X i = 2 — t and X2 + X2 Xi — X2 = 0 gives the linear relation YX=i = k Corollary 7.4 that the vector x = (xi, • • • , x -i) p ij = 0' where x = u k = a n d v k is defined 2 — u. The equation — v. k It follows from is an integral linear combination of the vectors (• • • , 1, • • • , - 1 , - 1 , • • • , 1, • • •), 1 < j < m - 1, where the l's are in positions j, p — j; the — l's are in positions m, m + l ; and the other entries are zero. 102 For argument's sake suppose x = ej for some j. This means we can move from the vector of rotation numbers [u\,--- ,Up_i] to the vector [v\,--- , v _ i ] by replacing a canceling pair p {j\P — j} by the canceling pair {m,m + 1}. Taking linear combinations of the e} corresponds to a sequence of such moves. This completes the proof of Theorem 10. • The remainder of this section is concerned with the proof of Theorem 9. According to Proposition 2.3 and the Eichler Trace Formula (1.1) the set of Eichler traces is given by * A=<xe z[C] 1 .7 = 1 where the only restriction on the rotation numbers kj is that Yl)=\ R{kj ) l define x k to be the number of j , 1 <j A = <x € Z[C] <t, such that kj = k, then we can characterize A by X 1 1 p-i P-I = = 0 (mod p). If we +E T * ^ ' X fc=i^ *^ 0 A N D E^ _ 1 fc=i ) X * = ° (modp)L (7.2) J In the next lemma we show that by passing to A we can remove the restriction that the x k be non-negative integers. L e m m a 7.2. The set of Eichler traces modulo Z is given by p-1 A={x<ZZ[C] ^ = p-1 E7>rrT' fe=l ^ \ J2 ^ = ° (mod )[. R Xk P k=l J Proof. First note that by choosing all x = 1 in (7.2) we get an element x € A. In fact a short k calculation using Lemma 7.1 gives x — 1 — (p — l)/2, and thus this element represents 0 in A. By adding x sufficiently many times to an element in A we can ensure that all the coefficients x become positive, and this does not change its value in A. k • This description of A contains a lot of redundancy. In fact we have the following characterization of A. 103 L e m m a 7.3. The set of Eichler traces modulo Z is given by m m X U — 1 "a k=l 1 i k=l Proof. According to Lemma 7.2 a typical element X £ A can be represented by p—1 m E rk _ i X _ k ft=i where the x — m v Xp-k Z ^ / ^ f c _ i Z ^ /--k _ i ' + fc=i s Xk fc=i s s are integers satisfying J2k=\ R{k~ )x = 0 (mod p). Now we use the fact that l k k 1 1 r =- l + — to see that \ — where m E h Jfe=l ' The restriction on the integers Zfc becomes 2~Jfe=i -^(^ ) fc = 0 (mod p), since 1 p—1 ^Rik-^Xk m = k=l m ^Rik-^Xk + k=l m = 2; ^Rdp-ky^Xp-k k=l m ^Rik-^Xk + k=i ^ip-Rik-^Xp-k k=i = Y,R{k- )z m l k (modp) k=l and YX=\ R(k~ )x l k = 0 (mod p). • In Definition 7.1 we introduced elements #i, # , 2 ••• >#m and then in Corollary 7.1 we showed that the corresponding classes modulo Z , that is 9i, 62, ... ,9 , m formed a basis of B. To determine the index of A in B we want to express a typical element of A in terms of this basis. But first we need a definition. D e f i n i t i o n 7.2. For integers k, n define C(k,n) = R(k~ n) + R(k~ ) — p. 1 1 The following properties of the coefficients C(k,n) are easy to verify: 104 (i) C(k, n) + C(p - k, n) = 0 and C{k, n) + C(k,p - n) = 2 i ? ( £ r ) - p. 1 (ii) C(l,n) =n + l-p, C{k,l) = 2 R ( * T ) - p , C(p - 1, n) = p - n - 1, and C(k,p-1) = 0. 1 J The elements of A are those elements £ E Z[(] of the form L e m m a 7.4. ^ m m x = - E(E ^' K)^> c P n=l n fc=l where the only restriction on the integers z is E f c L i R{k~ )z = 0 (mod p). l k k Proof. By Lemma 7.3 a typical Eichler trace modulo Z is given by x, where x = E / c L i a n E f c L i (k- )z d R = 0 (mod p). Using Lemma 7.1 we have l k p-2 * = - \ E ^* + ^ E ( E ^ * - E jk=-l y The condition XlfcLi R{k~ )z n=l \jfc=n y cn jk=-\ / = 0 (mod p) can be written as E j f c = i J * * — 0 ( o d p), and so l m k — Y,jk=i (P ~ J) k = 0 (mod p). Therefore, modulo Z we have J2jk=-iJ k z z « - '*) < " - ; E f £ n=l yfc=n jfc=-l / - yfc=n n=l E «)<•• j'fc=-l / Note that the term corresponding to n = p — 1 contributes 0 to the sum. Z= jk n 3Z ~ E k i f c = - i )*k = £ £ = i and therefore = J x Also note that (EAU C(*> «)**) C n Next we break the sum up into two pieces, one piece for 1 < n < m, the other piece for the remaining values of then use\ properties m n,/ and m ^ m of /the m coefficients 1 C(k,n). \ X n=l \k=l m / m ) P » = 1 \jfc=l / . . m / m \ P 1 - 1 P P F \ E(E ^ « = 1 \*=1 P / m / m n + n=l \fc=l m / m / \ (\ \k=l J \ n=2 \ n = l \*=1 « - c ' m / m 1 E E ( ^ )2i ^ « P m k=l 105 \ = 1 \*=1 \ J _1 c _ 7 n + ^E^i)^)(c 1 m ^ n=l / m m + 1 + ---+c - ) p 1 \ \fe=l / The last equation follows from 9i = ( + £ m + 1 (- ( P -1 - 2 . • Any sequence [ai,... ,a-t], as in Proposition 2.3, determines uniquely up to topological conjugacy, a compact connected Riemann surface S and an analytical automorphism T: S -> 5 having order p, orbit genus 0, and whose Eichler trace is given by the equation * 1 X = 1+ X Tk~\' where kjdj = 1 (mod p), for 1 < j < t. (7.3) Let x [ o i , . . . , at] denote this Eichler trace. Then (i) x[ai,...,a ] t + x[bi,...,b ] u = x[ai, • • • ,a ,h,... ,b u]. t (ii) £ [ . . . , a , . . . , p - a , . . . ] = x [ . . . , a , . . . , p ^ a , . . . ] . If we define y to be the number of j, 1 < j < t, such that aj = k, then we obtain k 1 • • •, at] m / m \ =-E E ^ ' C P n=l _1 \A;=1 where z = y — Vp-k- This is because that y — X f t ( f c - i ) and k k k / ^ (-) 7 4 ky = 0 (mod p). k D e f i n i t i o n 7.3. Let K be the collection of m-tuples v — [z\,... , z ] satisfying the condition m m y] kzk = 0 (mod p). k=l Thus K is a free abelian group of rank m. A basis of K is given by the vectors ui = [ 2 , - 1 , 0 , . . . ,0], v = [1,... , 1 , - 1 , . . . ] , k v m = [1,0,... ,0,2], 106 2<k<m-l, where for 2 < k < m — 1, the 1 is in the first and the k entries, the —1 is in the (k + th l) st entry, and all other entries are zero. This is because the determinant of these m vectors is p. Now consider the group homomorphism L: K —> A defined by m m 71=1 fc=l 1 P Lemma 7.4 implies that L is an epimorphism. P r o p o s i t i o n 7.3. L is a group isomorphism. Proof. We first compute the images of the basis elements Vk, 1 < k < m, using properties of the coefficients C(k,n): * m / m \ Wi) = - E E ' * r c(fc U=l 71=1 P n) fe ; / t{ p m = E ~° ' n 71=1 - m / m 71=1 m P \ Vfc = l = -Y{C( M 1 l + C{k-\n)-C{{k + l)-\n))d n 71=1 P . m - V ((n + P 71 = £ 71 = * P '(k + l)n 1 m 7l=l ^ 1 - p) + R{kn) + R(k) -p- R({k + l)n) - 1 m = / . P . P. / 771 \ \fc=l / -lft m - £ ( C ( i , n ) + 2C(m-\n))ft; n=l 771 ^ = - E ( U , ") + C C r n " , n) - C((m + l)" , n)) ft^i C 1 1 71=1 = E 71=1 (m + l ) n mn P P l) 0 , n 107 + 1) + p) #n where we have used the equation kn = [y]p + R(kn). Now consider the m x m matrix M whose (k, n) entry is given by (k,n) M (m + l)n ran P . P . = -1 To complete the proof of the proposition we need only show that det(M) ^ 0. In fact we will show that the determinant of this matrix is ±h\, thereby completing the proof of Theorem 9. Note that all entries in the first row of M are - 1 . For each k, 2 < k < m, we subtract the first row of M from the k row. The resulting entries of the new k th th '{k + l)n . P row are kn .P . Clearly, the first column of these new entries is 0. This implies that det(M) = ± d e t ( V where 2 < k, n < m. where 3 < k, n < m. J The first column of this matrix is 0 , . . . , 0,1, hence det(M) = ± d e t ( if] ~ ^ ^ [ ^ 1 V J According to [28] the determinant of this matrix is ±h\. This proves the proposition since the determinant of M has only changed by a ± sign in the course of the above elementary row and column operations. • The proof of Theorem 9 follows from the fact that det(M) = ±h\ since the matrix M is the coefficient matrix for expressing the basis elements of A in the basis elements of B. Clearly, 2r,s = L(v ), r for 1 < r < m and 1 + r + s = p. Theorem 11. 108 This complete the proof of As mentioned in the introduction, J . Ewing proves our Theorem 9, but in a different setting. See Theorem (3.2) in [6]. To Explain how Ewing's results relate to ours we need some notation. Let W denote the Witt group of equivalence classes [V, 0, p], where V is a finitely generated free abelian group, 0 is a skew symmetric non-degenerate bilinear form on V, and p is a representation of Z p into the group of /3-isometries of V. To an automorphism of order p, T: S —>• S, we assign the Witt class [V,0,p], where V is the first cohomology group, 0 is the cup product form, and p is the induced representation on cohomology. This assignment is well defined up to cobordism and so defines a group homomorphism ab: Q, -> W, the so-called Atiyah-Bott map. The G-signature of Atiyah and Singer defines a group homomorphism from the group of Witt classes to the complex representation ring of Z , sig: W -¥ R(Z ). p P Let e: i l ( Z ) —> Z[£] p be the homomorphism that evaluates the character of a representation at the generator T € Z . p Let s: Q —» Z[£] denote the composite e o sig o ab:fi—> Z[£]. Ewing proves that s is a monomorphism whose image has index h\ in the subgroup R of Z[C] spanned by the elements (, — C~ , k = 1,... , m. From the Remark earlier in this section k k it follows that R has index 2 in B. If < g \ a\,... , at > denotes the cobordism class of T , see Section 7.2 for the notation, then The relationship between the G-signature a and the Eichler trace x is given by a = 2x +1 — 2, and from this it is an easy matter to translate Ewing's results into ours. 7.2 Equivariant C o b o r d i s m In this section we prove Theorem 12. To begin with suppose T i : S i —>• S i and T : S —> S are 2 2 2 automorphisms of order p on compact connected Riemann surfaces. We do not assume that the orbit genus of either S i or S is 0. We start with a standard definition. 2 109 D e f i n i t i o n 7.4. We say that T \ is equivariantly cobordant to T , written T\ ~ T , if there 2 2 exists a smooth, compact, connected 3-manifold W and a smooth Z action T : W —> W such p that (i) The boundary of W is the disjoint union of S\ and 5 , d(W) = S i U 5 . 2 (ii) T restricted to 2 agrees with T U T . x 2 The cobordism class of an automorphism T: S —> S depends only upon its topological conjugacy class [g \ a\,... ,a-t]. We denote this cobordism class by < g | a\,... the orbit genus g = 0, we denote it by < a\,... , at >, and if , a >. t The set of all cobordism classes of Z actions on compact connected Riemann surfaces p is denoted by Q. Addition of the cobordism classes of the automorphisms T\: S\ — > Si, T : S 2 —> S is defined by equivariant connected sum as follows. 2 2 that Dj, Tj{Dj),... ,T?~ (Dj) l Find discs Dj in Sj such are mutually disjoint for j = 1,2. Then excise all discs j = 1,2, k = 0 , 1 , . . . ,p — 1 from S\, 5 and take a connected sum by matching 2 T (Dj), k d(T {D\)) k to 9(T (/J )) for k = 0 , 1 , . . . ,p — 1. The resulting surface S has p tubes joining Si and 5 . fe 2 2 The automorphisms T\, T can be extended to an automorphism T: i f ? — > S by permuting the 2 tubes. The cobordism class of T does not depend on the choices made. Thus addition in Q, is given by the formula < g I 0 1 , . . . , a > + < h I 61,... , b >=< g + h | a . . . , a , h,... , b > . t u u t u (7.5) The next two lemmas show that O is an abelian group generated by the cobordism classes < a\,... , at >. The identity is represented by any fixed point free action, or by any cobordism class consisting entirely of canceling pairs, and the inverse of < g | a\,... , at > is represented by < 9 I P ~ ii • • • iP ~ t >• The proofs are not original, but are presented here to emphasize a a the relationship with A. L e m m a 7.5. < g | 0 1 , . . . , at >=< a i , . . . , at > • 110 Proof. Let T : S —> S represent the class < a i , . . . ,at >. First we take the product cobordism Wi = S x [0,1], where T is extended over W\ in the obvious way. Next we modify W\ on the top end S x {1} as follows. Take a disc D in S such that D , T(D),... ,T P _ 1 ( D ) are mutually disjoint, and then to each disc T (D) in S x {1}, k = 0 , 1 , . . . ,p — 1, attach a copy k of a handlebody H of genus o by identifying the disc T (D) with some disc D' C d(H). Let k W2 denote the resulting 3-manifold. See Figure 7.1. The action of Z can be extended to p W2 by permuting the handlebodies. The manifold W2 provides the cobordism showing that < g I a-i,... ,a >=< a-i,... ,a > . t • t Figure 7.1: Cobordism of g = 0 L e m m a 7.6. < a,p — a, 0 3 , . . . , a >=< t 1 | 03,... , a > = < 03,... , a > . t t Proof. The proof of this lemma is similar to the proof of the last one. Start with a product cobordism W\. Suppose PQ\ PI are the fixed points corresponding to the canceling pair {a,p — a}. Choose small invariant discs Do, D\ around PQ, P\ respectively, and then modify the cobordism at the top end by adding a solid tube D x [0,1] so that D x {0} = Do and D x {1} = D\. The automorphism T can be extended over this tube, and the resulting cobordism shows that < a,p - a,a ,... 3 ,a >=< 1 | a , . . . ,a > . t 3 Ill t • See Figure 7.2. Lemma 7.5 completes the proof. Figure 7.2: Cobordism with Canceling Pairs Define the isomorphism of Theorem 12, 4>: A -» Q,, by <f)(x[ai,... , a ]) =< « i , . . . , a > . t t The same relations hold for cobordism classes, see Equation (7.5) and Lemma 7.6, and therefore the mapping 4> is a well defined group homomorphism. Now we complete the proof of Theorem 12. The argument is analogous to one used in [8]. Proof. From the remarks above we know that c/>: A —> CI is a well defined group homomorphism. Lemma 7.5 implies that it is an epimorphism. It only remains to prove that (f> is a monomorphism. If there is an element in the kernel of 4> we can assume it is a generator, say x[ i > • • • j t]• a Suppose T : S —>• S represents [ a i , . . . ,at]. a Then there is a compact, connected, smooth 3- manifold W such that d(W) = S, and an extension of T to a smooth homeomorphism T: W —> W of order p, also denoted by T . The fixed point set of T: W —> W must consist of disjoint, properly embedded arcs joining fixed points in S to fixed points in S. The fixed points at the end of each arc will form a canceling pair {a,p — a}. In this way we see that [ a i , . . . , at] consists entirely of canceling pairs, and hence x[ i, a • • • > t] = 0 in A. a 112 • 7.3 D i h e d r a l G r o u p s of A u t o m o r p h i s m s of R i e m a n n Surfaces We conclude this thesis by proving Theorem 13. The essential nature of its proof is the relation between group actions on compact connected Riemann surfaces and Fuchsian groups, as well as the Lefschetz Fixed Point Formula. Let D be the dihedral group of 2p elements and T ,T 2p D p be two fixed generators of order p, 2 with the relations T 2p there is an embedding of D = T | = (T T ) = 1. Suppose 2 p p 2 in Aut (S). We have a faithful representation R : D 2p € 2 2p —• GL (C), g by passing to the space of holomorphic differentials on S, assuming g > 1. We want to characterize such groups R(D ). We denote by D (A, 2p GL (C) g 2p generated by A, B with the relations A p = B 2 B) any subgroup of = ( A B ) = I. Let G = D {A Bi) 2 { 2p (i = 1,2). Gi and G are said to be conjugate, denoted by G i ~ G , if there is Q € 2 such that Q~ GiQ = G , and strongly conjugate, denoted by G\ « G , if Q'^AiQ x Q" B\Q l 2 2 2 = B . A subgroup D (A, 2 2p g GL (C) g = A and 2 B) is said to be realizable if it is conjugate to some It is well known that the trace of an element of order 2 in GL (C) u R(D ). 2p is an integer, and the trace of an element of order p in GL (C) is an algebraic integer in the cyclotomic field Q(C)- A g subgroup G in GL (C) is called an I-group if all elements of G have integer traces. g Let X E D (A, 2p B) be of order p. Then X ~ X~ , and hence tr (X) = tr (X' ) l 1 = tr (X). Therefore tr (X) is a real number. Furthermore if tr (X) is rational, then tr (X) is an integer. Lemma 7.7. If some element X € D (A, 2p B) of order p has rational trace, then D (A,B) 2p an I-group and all elements of order p in D (A,B) 2p are conjugate. Proof. It is clear that tr (X) = k + ki{( + C ) + • • • + k {( m - 1 m negative integers k, k\, ..., k with k + 2(ki -\ m m m Lemma 7.8. +( _ m ) (m = ^ f ) , for some non- + k ) = g. But (,,... C over the rational field Q, so we have k\ — • • • — k , integer. is 1 p _ 1 are independent say /. Therefore tr (X) = k — I is an • Suppose Gi = D (Ai,Bi), 2p i = 1,2, are two I-groups. Then the following three conditions are equivalent. 113 1. G\ ~ 2. C?i « G; G2; 2 3. tr (A^ = tr ( A ) and tr (Bi) = tr ( 5 ) . 2 2 Proof. For a dihedral I-group we have the following canonical form G = D (Ai,B ), 2p \ Ai = \ (1 and where Xjy B x>y V where x + y + (p - 1)1 = g and tr (A ) = x + y - l . Since the number of blocks of J^'s in B , t even, tr(B XtV x y ) =x-y. is • If o- is an automorphism of S of finite order greater than 1, then we have the Lefschetz Fixed Point Formula, tr (a) + tr (a) = 2 - Fix (<T), where Fix (a) is the number of fixed points of a, see [38]. It is easy to deduce L e m m a 7.9. If D (A,B) 2p tx(B) is realizable, then D (A,B) 2p is an I-group with tr (A) < 1 and < 1. Thus we complete the proof of the necessity condition of Theorem 13. To any action of D 2p on S we can associate a short exact sequence of groups 1 -> II T ( g ; T V ^ T p , 2,.. 772) 4 D 0 2p -> 1 where V must has form T(g ;p,...,p,2,..7a) 0 = (X ..., u X , Y ..., Y , A go 114 u go u ..., A , B t u ..., B) s with relations A\ = ••• = A =B p = --- = B 2 2 t = [X Y ]---[X ,Y ]A ---A B ---B u l go go 1 t l = l s By the Riemann-Hurwitz formula (2.16) we see that s must be even. (7.6) From the results of Macbeath[21], we obtain that Fix (T ) — 2t and Fix (T ) = s. Hence if D2 {A, B) is realized by p 2 this action then tr(A) = 1 - t and ti(B) = P -^. 2 To prove the sufficiency condition of Theorem 13, we need the following lemma. Assume that D ( A , B) is an IR-group. 2 p L e m m a 7.10. Then ^ (g + (p — 1) tr (A) + ptv (B)) is a non-negative integer. Proof. This is an easy calculation. Let A, B be of forms A;, B , x>y as in the proof of Lemma 7.8. + (p-l)tr(A)-|-ptr(B) 5 = x + y + (p-l)l + (p-l)(x = p(x + y) + p{x - y) — 2px. + y-l)+p{x-y) Thus ^ (g + (p — 1) tr (A) +ptv (B)) = x is a non-negative integer. • Now we can complete the proof of Theorem 13. Proof of Theorem 13. Let t = 1 - tr (A), s = 2 - 2tr (B), and go = ^-(g + (p-I) 2p t We define an epimorphism 0 : T(go; p,. ^p, tr (A)+ptr(B)). s 2,. T. ,2) —> D as follows: 2 p Case 1: If tr (A) = 1 and tr (B) = 1, then t = 0, s = 0, and o > 2. We set 0 0(X )=0(Yi)=T 1 p and 0(*i) = 0(YO = T 115 2 (for i = 2,... )- ff0 Case 2: If tr (A) = 1, tr (73) = 0, then t = 0 and s = 2, and go > 1. We define 9(B ) = 9{B )=T l 2 and 2 0(X*) = 0 ^ ) = T . p Case 3: If tr (A) = 1 and tr (73) < - 1 , then t = 0 and s > 4. We define 9{B )=T T and bi i p 2 0(X,-) = 0(Yj) = 1, where bi are integers (not all the same) with 0 < bi < p — 1 and E i = i ( l ) ° i = 0 (mod p). — l Since s is even, 9 preserves the group relations, and hence is an epimorphism. Case 4: If tr (A) < 0 and tr (73) = 1, then t > 1, s = 0, and g > 1. We define 0 0(Ai)=T*, 6(Xj)=T£> where a^, C j are integers with 1 < aj < and 0(1$) = T , 2 p — 1 and 5Z' a^ + 2 E j l j =1 Cj = 0 (mod p). Case 5: If tr (A) < 0 and tr (73) < 0, then t > 1 and s > 2. We define 0(Ai)=T % p where a u e(B ) = T} T i j 2 and bj are integers with 1 < a < p - 1 and £ { = 1 0{X ) = 9{Y ) = 1 k k OJ + E j = i ( - ! ) s + 1 O j = 0 (mod p). Let IT = Ker(0). We get a short exact sequence of Fuchsian groups 1 -> IT -> r{g ;p^^~p, 2 , . . . , 2) A 7 J -»• 1. 0 2p It is easy to check that II is torsion free. By Lemma 7.8, we get an action of 7 J which realizes D {A;B). 2p on S = U/II • 2p C o r o l l a r y 7.5. The minimal genus of D 2p is p — 1. 116 Bibliography [1] E . Artin, Geometric Algebra, Interscience tracts in pure and applied mathematics, vol. 3, Interscience Publishers, New York, 1957. [2] E . Bender, Classes of Matrices over an Integral Domain, Illinois J . Math. 11 (1957), 697702. [3] L . Bers, Universal Teichmiiller Space, Conference of Complex Analysis Methods in Physics, University of Indiana, June 1968. [4] C . J . Earle, Reduced Teichmiiller Space, Trans. Amer. Math. Soc. 129 (1967), 54-63. [5] A . L . Edmonds & J . H. Ewing, Surface Symmetry and Homology, Math. Proc. Camb. Phil. Soc. 99 (1986), 73-77. [6] J . Ewing, The Image of the Atiyah-Bott map, Math. Z. 165 (1979), 53-71. [7] , Automorphisms of Surfaces and Class Number: An Illustration of the G-Index Theorem, Topological Topics (I. M . James, ed.), London Math. Soc. Lecture Notes Series, vol. 86, Cambridge University Press, 1983. [8] L . Edmonds &; J . H. Ewing, Remarks on the Cobordism Group of Surface Diffeomorphisms, Math. A n n . 259 (1982), 497-504. [9] D . Gabai, Convergence Groups are Fuchsian Groups, Ann. Math. 136 (1992), 447-510. [10] W . Harvey, Discrete Groups and Automorphic Functions, Academic Press, New York, 1977. [11] S. P. Kerckhoff, The Nielsen Realization Problem, Ann. Math. 117 (1983), 235-265. [12] M . A . Knus, Quadratic and Hermitian Forms over Rings, Grundlehre der Mathematischen Wissenschaften, vol. 294, Springer Verlag, New York, 1991. [13] H . M . Farkas & I. Kra, Riemann Surfaces, second edition ed., Graduate Texts in Mathematics, vol. 71, Springer Verlag, New York, 1992. [14] I. Kuribayashi, On Automorphisms of Prime Order of a Riemann Surface as Matrices, Manuscripta Math. 44 (1983), 103-108. [15] , On an Algebraization of the Riemann-Hurwitz Relation, Kodai Math. J . 7 (1984), 222-237. [16] , Classification of Automorphism Groups of Compact Riemann Surfaces of Genus Two, Tsukuba (1986), 25-39. [17] I. Kuribayashi & A . Kuribayashi, Automorphism Groups of Compact Riemann Surfaces of Genera Three and Four, J . of Pure and Applied Algebra 65 (1990), 277-292. 117 [18] A . Kuriyabashi, Automorphism Groups of Compact Riemann Surfaces of Genus Five, J . of Algebra 134 (1990), 80-103. [19] S. Lang, Algebraic Numbers, Addison-Wesky, Reading, Mass., 1964. [20] , Cyclotomic Fields I and II, Springer-Verlag, New York, 1990. [21] A . M . Macbeath, Action of Automorphisms of a Compact Riemann Surface on the First Homology Group, Bull. London Math. Soc. 5 (1973), 103-108. [22] C . G . Latimer & C . C . MacDuffee, A Correspondence between Classes of Ideals and Classes of Matrices, Ann. of Math. 34 (1933), 313-316. [23] D . A . Marcus, Number Fields, Springer-Verlag, New York, 1977. [24] W . S. Massey, Singular Homology Theory, Graduate Texts in Mathematics, vol. 70, SpringVerlag New York Inc., New York, 1980. [25] B . Eckmann & H. Miiller, Plane Motion Groups and Virtual Poincare Duality of dimension two, Invent. Math. 69 (1982), 293-310.[26] M . Newman, Integral Matrices, Academic Press, New York, 1972. [27] J . Nielsen, Abbildungklassen Endliche Ordung, Acta Math. 75 (1943), 23-115. [28] L . Carlitz & F . R. Olson, Maillet's Determinant, Proc. Amer. Math. Soc. 6 (1955), 265269. [29] I. Reiner, Symplectic Modular Complements, Trans. Amer. Math. Soc. 77 (1954), 498-505. [30] , Automorphisms of the Symplectic Modular Group, Trans. Amer. Math. Soc. 80 (1955), 35-50. [31] , Integral Representations of Cyclic Groups of Prime Order, Proc. Amer. Math. Soc. 8 (1957), 142-146. [32] C . L . Siegel, Symplectic Geometry, Amer. J . Math. 65 (1943), 1-86. [33] G . Jones &: D . Singerman, Complex Functions, Cambridge Univ. Press, Cambridge, 1987. [34] D . Sjerve, Canonical Forms for Torsion Matrices, Journal of Pure and Applied Algebra 22 (1981), 103-111. [35] P. Symonds, The Cohomology Representation of an Action C Math. Soc. 306 (1988), 389-400. p on a Surface, Trans. Amer. [36] O. Taussky, On a Theorem of Latimer and Macduffee, Canadian J . Math. 1 (1949), 300302. [37] , On Matrix Classes Corresponding to an Ideal and its Inverse, Illinois J . Math. 1 (1957), 108-113. [38] J . W . Vick, Homology Theory, Academic Press, 1973. [39] L . C . Washington, Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982. 118
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The carbon and nitrogen isotopic compositions of particulate organic matter in the subarctic northeast… Wu, Jinping 1997
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Title | The carbon and nitrogen isotopic compositions of particulate organic matter in the subarctic northeast Pacific Ocean |
Creator |
Wu, Jinping |
Date Issued | 1997 |
Description | The aim of this study was to determine the variations of δ¹³C and δ¹⁵N of particulate organic matter (POM) and the factors that control the isotopic fractionation from inorganic substrates to zooplankton and lead to variations in the isotopic composition of suspended and sinking POM in the subarctic northeast Pacific Ocean. Along a transect from Station L (48°39'N, 126°40'W) to Station P (50°N, 145°W), surface δ¹⁵NO₃⁻ decreased from 11.2‰ to 7.6‰, while [NO₃⁻] increased from 3 to 12 pM. The δ¹⁵N trends for bulk POM, < 5 pm POM, 50-253 pm POM, < 253 pm POM and zooplankton were similar to that for nitrate. The fractionation factor (e) for the nitrogen isotopes was 5‰. Trophic enrichment of ¹⁵N was observed with mesozooplankton being isotopically heavier than suspended POM by 3.9‰ at Station P and 2.2‰ at Station L. The range of δ¹⁵N among six zooplankton groups was 3.7‰. At Station P, sinking δ¹³C[sub POM] and δ¹⁵N[sub POM] at 3800 m between 1982 and 1990 show significant annual changes. The δ¹³C[sub POM] ranges from -25.3‰ to -22.0‰ and the δ¹⁵N[sub POM] ranges from 10.2‰ to 7.7‰, isotopically depleted values occurring in summer and heavier values occurring in winter. The change in trophic length may be the principal controlling factor of the variations. The δ¹³C[sub POM] at 200,1000 and 3800 m show a ¹³C enrichment with depth. Suspended POM δ¹³C and δ¹⁵N increased with depth and were higher than those of sinking POM, suggesting that the two types of POM did not interact in deep water. At Station L, high frequency variations of δ¹³C and δ¹⁵N in summer are due to the periodic injection of subsurface nitrate by upwelling, which leads to diatom blooms, resulting in a heavier δ¹³C and a lighter δ¹⁵N in sinking POM. The sinking δ¹⁵N[sub POM] value (8%c) is much heavier than that of nitrate (4%c) at 500 m, suggesting that the lateral input of lower nitrate δ¹⁵N from the open ocean (3‰) was important. This study develop our understanding of the isotopic biogeochemistry of carbon and nitrogen in the ocean. |
Extent | 4287937 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-04-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0053274 |
URI | http://hdl.handle.net/2429/6846 |
Degree |
Doctor of Philosophy - PhD |
Program |
Oceanography |
Affiliation |
Science, Faculty of Earth, Ocean and Atmospheric Sciences, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1997-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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