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The carbon and nitrogen isotopic compositions of particulate organic matter in the subarctic northeast.. 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 Created | 2009-04-06T21:01:06Z |
Date Issued | 2009-04-06T21:01:06Z |
Date | 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 |
Collection |
Retrospective Theses and Dissertations, 1919-2007 |
Series | UBC Retrospective Theses Digitization Project [http://www.library.ubc.ca/archives/retro_theses/] |
Date Available | 2009-04-06T21:01:06Z |
DOI | 10.14288/1.0053274 |
Degree |
Doctor of Philosophy - PhD |
Program |
Oceanography |
Affiliation |
Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 1997-05 |
Campus |
UBCV |
Scholarly Level | Graduate |
URI | http://hdl.handle.net/2429/6846 |
Aggregated Source Repository | DSpace |
Digital Resource Original Record | https://open.library.ubc.ca/collections/831/items/1.0053274/source |
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T H E I N T E G R A L S Y M P L E C T I C G R O U P S A N D T H E E I C H L E R T R A C E OF Z P A C T I O N S OF R I E M A N N S U R F A C E S 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 R E Q U I R E M E N T S F O R T H E D E G R E E O F D O C T O R O F P H I L O S O P H Y 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 U N I V E R S I T Y 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 refer- ence 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 A b s t r a c t 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 SP2g(Z) are induced by some automorphism on some 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 x2g f(l/x) — f(x) and /(0) = 1, and let £ be a fixed root of f(x). 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 + x2 -\ h xp~l, p is an odd prime, this result says that the number of conjugacy classes of elements of order p in SP p _i(Z) is 2(P~ 1)/ 2/II, where h\ is the first factor of the class number of the cyclotomic extension. If X e SP2g(1j) has a reducible characteristic polynomial of the form f(x)g(x), where 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 SP p _i(Z) which 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 p actions on Riemann surfaces. If A denotes the set of 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 Zp actions. Finally, we determine which dihedral subgroups of GLg(C) can be realized by an action on a Riemann surfaces of genus g. iii Table of Contents Abstract ii Table of Contents iv List of Figures vi Acknowledgement vii Chapter 1. Introduction 1 1.1 Motivations 1 1.2 Main Results 4 Chapter 2. Preliminaries 14 2.1 Direct Sum of Symplectic Matrices 14 2.2 S-Polynomials 15 2.3 Strictly Coprime Polynomials . . ". 18 2.4 Group Actions on Riemann Surfaces 20 Chapter 3. The Conjugacy Classes of Type-I 27 3.1 Ideal Classes 27 3.2 S-Pairs 29 3.3 The Correspondence * 32 3.4 Class Number of Vf 34 3.5 The Rational Integer Case 37 Chapter 4. Symplectic Spaces 42 4.1 The Symplectic Spaces 42 4.2 Symplectic Transformations 50 4.3 Symplectic Group Spaces 54 Chapter 5. Order p elements in £ P p _ i ( Z ) 61 5.1 A n Example 61 5.2 Cyclotomic Units 63 5.3 Realizable Elements of Order p 68 Chapter 6. Torsion in SP 4(Z) 75 6.1 Symplectic Complements 78 6.2 Minimal Representatives 81 6.3 The Case of f(x) = xi + x2 + l 85 6.4 Realizable Torsion 91 Chapter 7. The Eichler Trace of Z p Actions on Riemann Surfaces 95 7.1 The Eichler Trace 95 iv 7.2 Equivariant Cobordism 109 7.3 Dihedral Groups of Automorphisms of Riemann Surfaces 113 Bibliography 117 v Lis t of Figures 2.1 Fundamental Domain 23 5.1 Fundamental Domain (order p) 70 5.2 P~c<j< ( p - l ) / 2 73 5.3 1 < j < p - c - 1 73 6.1 Fundamental Domain (order 6) . . . . 93 7.1 Cobordism of g = 0 I l l 7.2 Cobordism with Canceling Pairs . 112 VI 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 In t roduct ion 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 p , the cyclic group of odd prime order p, and £>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 SP2n (T>), where V is a principal ideal domain, with a 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* : ffx(5) Let {a, b} — {a i , . . . , ag, b\,..., bg} be a canonical basis of -Hi (5), that is the intersection matrix 1 for {a, b} is where Ig is the identity matrix of degree g. Let X be the matrix of T* with respect to the basis {a, b}, i.e. Tt(ai,...,ag,bi,...,bg) = (ai,..., ag, & i , . . . , bg)X. 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 A u t ( 5 ) ^ S P 2 9 ( Z ) , see [13]. Clearly, the matrices of T* with respect to different canonical basis are conjugate in SP2g(Z). Definition 1.1. A matrix X E SP29(Z) is said to be realizable if there is T E Aut (5) for some 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 SP2g{1) be realized? 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 SP2g(Z). 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 Zp on S determines a representation p : Z p -¥ GLg(Y), where V is the vector space of holomorphic differentials on S. If T is a preferred generator of Z p then this representation yields a matrix p(T) E GLg(C). The trace of this matrix, x = tr(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 P i , . . . , 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 Tai 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 Si and S2 have the same genus g. Let Xi, X2 be the symplectic matrices induced by Ti, T2 respectively. Then Xi and X2 are conjugate in SP2S(Z) if and only if their Eichler traces x{Ti) and x(T 2 ) are the same, see A. Edmonds &: J . Ewing [5] and P. Symonds [35]. The Riemann-Hurwitz formula for an order p element T € Aut (5) is 9 = P9o + *r(t-2) (1.2) where 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 SP29(Z) which can be realized. The minimal polynomial of an element of order p is xp~l + xp~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 5P 2 s (Z) is g = PJYL > 1. We consider this special case, only v - ^ - classes of order p matrices in 5P p _ i(Z) can be realized. But we shall show that the number of conjugacy classes of order p matrices in SP p _i(Z) is 2 2 hi, where hi is the first factor of the class number h of Z[£]. So in general most of the order p matrices in SP p _ i(Z) is 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 p is captured by the Eichler trace. We want to answer the following two questions. 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. Al 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 n x m ( £ > ) be the set of nxm matrices over V. For sake of simplicity we denote MnxmCD) by Mn(V) when n = m, and let In be the identity matrix in Mn{V). For A e Mni(D), B <E M„ 2(X>), we define the direct sum of A and B as A + B = ^ e Mni+n2(V). (1.3) Definit ion 1.2. The set of 2nx2n unimodular matrices X in M2 n (£>) such that X'JX = J (1.4) is called the symplectic group of genus n over V and is denoted by SP2n(D)- Two symplectic matrices X, Y of SP2n('D) are said to be conjugate or similar, denoted by X ~ Y, if there is a matrix Q € S-F^n^) such that Y — Q~lXQ. Let (X) denote the conjugacy class of X. 4 Remark. The definition is meaningful and clearly SP2n{V) is a subgroup of GL2n{T>), the general linear group with entries in V. It is well known that every symplectic matrix in SP2n(T>) has determinant one [1]. It is readily verified that X belongs to SP2n{T>) if and only if X' belongs to SP2n(V). Let X = A B where A, B,C, D E Mn(V). If X E SP2n{V) the following conditions are satisfied: AB' = BA', CD' = DC' and AD' - BC' = I (1.5) as well as A'B = B'A, C'D = D'C and A'D - C'B = I. (1.6) Conversely, if one of (1.5) or (1.6) is true then X E SP2n{T>). Given two matrices Ai B x \ A2 B2 Xi=\ \€M2ni(V) and X2 = | e M 2 n 2 ( D ) , , C i P>i) \C2 D2j we define the symplectic direct sum of Xi and X2 by Xi*X2 = e % 1 + „ 2 ) ( P ) . /'A\ 0 Bx 0 ^ 0 A2 0 B2 Ci 0 Dx 0 \0 C2 0 D2J It is easy to check that XX*X2 E SP2(ni+n2)(D) if and only if Xi E SP2ni{T>), for i = 1,2. (1.7) Given two matrices C\i C\2\ (D\\ D\2 Yi=\ E M2nix2n2{V) and Y2 = | G M 2 t l 2 x 2 n i ( D ) ^C2i C22l \D2i D22 where Cy € MniXn2(T>), Dtj G Mn2Xni(V), we define the quasi-direct sum by Yi o y 2 = G M-2(ni+n2) (2>). (1.8) / 0 C n 0 Ci2^ D i i 0 £> 1 2 0 0 C21 0 C22 \ 0 2 1 0 Z?2 0 J By an easy calculation we see that if m = n 2 = n, then F i o V 2 G SP\n(V) if and only if Yi,Y2€ SP2n(V). Definition 1.3. A matrix X G SP2n(V) is said to be decomposable if it is conjugate to a 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 XioX2 for some X\, X2 G SPn{V). Given a matrix X G M2n(T>), we denote the characteristic polynomial of X by fx{x) = \xI-X\. If X G SP2n(V), then fx(x) is "palindromic" and monic, that is x2nf(l) = f(x) and /(0) = 1. (1.9) This is because X'JX = J, X' = JX~lJ~l, and /(0) = det(X) = 1. fx(x) = \xI-X\ = \xI-X'\ = \xI-X~l\ = x2n\X~ll\\X-'\ 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), over V, that is Mf = {X G SP2n(V) | fx(x) = /(a;)}. (1.10) We use Mf to denote the set of the conjugacy classes of Mf in SP2n{P>)- In Chapter 3 we deal with the case that f(x) is a separable S-polynomial of type-I. Let £ be a fixed root of f(x). 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. There is an eigenvector a = ( a i , . . . ,0:271)' G TZ2n corresponding to C, that is XC, = (a. Let a be the X>-module generated by ai,... ,a2n, and let a = A~1a'Ja. It is easy to check that a is an integral ideal in TZ and a — a. Furthermore we will prove 7 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 Co = {o € C | oo = (a), a = a for some a € 7£} (1.11) Let U+ = {u € U \ u = u} and C = {ml \ u Elf}, where ?7 is the group of units in 71. Clearly, C C U+ 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 qm be the number of elements in Mf, where f(x) is the m-th cyclotomic polynomial. Then m = 2 (mod 4), = < m ^ 2 (mod 4), and m is prime power, hi, 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 SPp_i(Z) is 2^" hi. 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;, be an (I, k)-normal set of V. Then there are 2n — I — k elements aj+i,..., «n,ftk+i-, • • • >fin inV such that a i , . . . ,an,Pi,... ,Pn is a symplectic basis of 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) =pi{x)---pm(x), wherep\{x),... ,pm E T>[x] are mutually coprime S-polynomials. Then there are m polynomials u\(x),..., um{x) E !F[x] such that ui(x)qi(x) H +um(x)qm(x) = 1, where qi{x) = f(x)/pi(x), for i = 1,... ,m. We shall show Theorem 6. Let X E Mf. Then X ~ X\ * • • • * Xm, for some Xi E MPi, i = 1,...,m, if and only if Ui(X)qi(X) E M2n(V), for i = 1 , . . . ,m. To every S-pair (a, a), defined in Section 3.2, we shall assign a symplectic structure and a Gm action on a, where Gm is the cyclic group on a fixed generator g of order m (cf. Section 4.3). Therefore a becomes a symplectic Gm-space, denoted by [a, a]. Theorem 7. Two symplectic direct sums [oi,aj] * ••• * [ar,ar] and [bi,&i] * ••• * [bs,6s] are isomorphic as symplectic Gm-spaces if and only if r = s, and there is an rxr 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 SP p _i(Z) . The proof of Theorem 1 gives us 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 SP p _i(Z). Let sin — ^ = - r - f , for(fc,p) = l» (1-14) s i n £ be the cyclotomic units of Z[£]. By the Riemann-Hurwitz formula, an automorphism T : S —> S of order p, where S has genus PJY~, has exactly 3 fixed points. Let the fixed point data of T be {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[(],uaubua+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. Coro l lary 1.2. Let X € SP p _i(Z) be of order p. Then X is realizable if and only if V(X) = (Z[(],uaubua+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={XeZl(} X = t r ( T ) | (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. On 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/Zp, 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 — where 1 < k < p — 1, appears amongst the set of rotation numbers {ki,--- ,kt}, then an easy calculation shows that their contribution to the 11 Eichler trace is 1 1 C f e - l + (p~k - l ~ ' 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 : 5i —> S\, Ti: S2 —> S2 I we have two "vectors" [g; ki,... ,kt], [h;li,... ,lu]. Let x i and X2 denote the respective Eichler 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,... ,lu} up to permutations and replacements of cancelling pairs. 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 Zp actions. We show that the Eichler trace induces a 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 GLg(C). 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 GLg(C) by passing to the induced action on the vector space V of holomorphic differentials. The problem is to determine those subgroups of GLg(C) 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 GLg(C) is realizable if 12 and only if A satisfies the "Eichler trace formula" [14]. In 1986 and 1990, I. Kuribayashi and A. Kuribayashi determined all realizable subgroups of GLg(C) for g < 5 (see [15], [16], [17] and [18]). We consider the dihedral group D2p. Let D2p be a subgroup of GLg(C), and let A and B be generators with orders p and 2 respectively. D2p is called an IR-group if tr(A), tv(B) are integers < 1. If D2p is an IR-group for some choice of A, B then it is an IR-group for all choices. We shall prove Theorem 13. D2p is realizable if and only if it is an IR-group. 13 Chapter 2 Prel iminar ies 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, (X1*X2)' = X[*X2, (2.1) (yl0y2)' = YJOY{, (2.2) (X1*X2)(YloY2) = (XxYj o (X2Y2), (2.3) (x1ox2)(y1*y2) = (x{Y2) o (X2YX), (2.4) {Xy*X2){Y1*Y2) = (XM) * (X2Y2), ( 2 . 5 ) ( X i o x 2 ) ( y i o y 2 ) = {X1Y2)*{X2Y1). (2.6) We assume that all matrix multiplications are suitable. L e m m a 2.1. Let X\, X2, X3, Y\, Y2 be symplectic matrices. Then 1. X\ * X2 ~ X2 * X\. 2. (Xi * X2) * X3 = Xx * (X2 * X3). 3. If Xx ~ Yi and X2 ~ Y2, then Xx*X2~Yi*Y2. In the following we assume X\ and X2 have the same genus 4. X\ o X2 ~ X2 o X\. 14 5. X1oX2~{-X1)o(-X2). 6. If X\ ~ X2, then Io X\ ~ / o ! 2 . Proof. (2) and (3) are easy. To prove (1) we let Q = I2ni oI2n2 e 5 P 2 ( „ 1 + n 2 ) ( Z ) , where nt is the genus of X{, i = 1,2. Then Q~l(Xi *X2)Q = X2*X\. Similarly we prove (4) by using Q = 7o7, (5) by using Q = / * ( - / ) . For (6), HX2 = Q'lXiQ, then {Q-1*Q-x){IoX1){Q*Q) = IoX2. • In general the converse of (3) in Lemma 2.1 is not true, but we have Lemma 2.2. Suppose X\, X2, Y\ andY2 are symplectic matrices, fx (x) = fY.(x) = fi{x), for i = 1,2. Suppose fi(x) and f2(x) are coprime. Then X\ * X2 ~ Y\ * Y2 if and only if X\ ~ Y\ and X2 ~ Y2. Proof. The sufficiency part has been proved. We consider the necessity. Note that any P € M2^ni+ri2^(V) can be expressed in the form P - Pi * P2 + PZ o P 4 , where Pi € M2ni(V), P2 e M2n2(V), P 3 G M2nix2n2(V), and P 4 G M2n2x2ni(V) are blocks of P . Let P be a symplectic matrix such that (Xx * X2)P = P{YX * Y2). We obtain XXPX = PiYu X2P2 - P 2 y 2 , X X P 3 = P 3 y 2 and X 2 P 4 = P 4 y 2 . Then / 2 ( X X ) P 3 = P 3 / 2 ( y i ) = 0, which yields P 3 = 0 since f2{X\) is invertible. Similarly, we get P 4 = 0. Hence P i , P 2 are symplectic, therefore X\ ~ y i and X2~Y2. • 2.2 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 xnp(^) — p(x), then p(x) is S-polynomial of type-I or p{x) = x + 1. 2 If xnp{±) = -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)2g(x) where g(x) is also a S-polynomial. Proof. Differentiate both sides of x2nf( j ) = f(x) to see that 2nx2n-lfC-)-x2n~2f'(1-) = f'(x). (2.7) But / ( ± 1 ) = 0, hence / ' ( ± 1 ) = 0, f(x) = (x^fl)2g(x). 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)xnp(x)p(-) 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 ) 2 , by Lemma 2.4. We can choose p(x) = 1^1. Now we consider the case / ( l ) ^ 0 and /(—1) # 0. Suppose that f(x) = p\{x) • • -pm(x), where p\ (x),..., pm (x) are irreducible monic polynomials of positive degrees n i , . . . , nm. By the 16 Remark, none of pi(x). ..pm(x) is an S-polynomial since f(x) is an irreducible S-polynomial. Since f(x) is an S-polynomial, Note that xnipi(^) is an irreducible polynomial, and neither x + 1 nor x - 1 are factors of f{x). There is k ^ 1, say k = 2, such that x^p^) = Pl(0)p2{x). It is easy to verify that Pi(x)p2(x) is an S-polynomial, and therefore f(x) = pi(x)p2(x). Let p(x) = pi(x). Then Proposi t ion 2.1. Every S-polynomial f(x) 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 f(x), because of Lemma 2.4. We know that f(x) can be written as a product of irreducible monic polynomials, where the pi(x) (i = 1,.. . , k) are S-polynomials of degree 2r; and qj(x) (j = I,...,I) are of degree Sj. Then / ( x ) = x 2 " / ( I ) = x > 1 ( i ) - . . ^ p m ( I ) . P2(x) =p(0)xnp(±), and f(x) = p(0)xnp(^)p(x). • f(x) = pi{x)p2{x) • • • Pk(x)qi(x)q2(x) • • • qt(x) xnfil) = x2^Pl(l)x^p2^) • • • x2r*pk(l)xsiqiC-)xs>q2(1-) • • • x'lqtil) = pi{x)p2{x) • ••pk(x)x^q1(l)x^q2(l) • • • £ s<a,(±) So we have qi{x)q2{x) • • • qi(x) = xs'qi{l)xs*q2{D • • • Note that xsiqj(\) (j — 1,...,/) are irreducible polynomials. Then for each xsiqj(^), there is Ij ^ j such that xsiqj{\) = qj(Q)qi.(x). It is easy to check that qj(0)qj(x)xsJqj(j) is an irreducible S-polynomial. By rearranging the order of qj{x) we get f(x) = pi(x)p2(x) • • • pk(x) qi{0)qi(x)xSlqi(±) ••• qm{o)qm(x)xSmqm{±) The second part is simple. • 17 2.3 Strictly Coprime Polynomials We consider two polynomials f{x) = anxn + an-\xn~l H ha 0 , g(x) = bmxm + bm-ixm-1 + • • • + b0, in V[x]. Assume m > 0, n > 0, and an ^ 0, bm ^ 0. Definit ion 2.1. f(x) 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) = 1 (2.8) Example . Let pn(x) — xn~x + xn~2 + • • • + 1. Then pm(x), pn(x) are strictly coprime over Z if and only if m, n are coprime. And pm(x) and pn{x) 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 Res (/, g) = det The result we want to establish is > m rows > n rows bo) j (2.9) Proposi t ion 2.2. Suppose either f(x) or g(x) is monic, that is either an = 1 or bm = 1. Then f(x) and g(x) are strictly coprime if, and only if Res (/,<?) «s a unit in V. 18 Proof. Without loss of generality, let an — 1. If f(x) and g(x) are strictly coprime, then xku(x)f(x) + xkv(x)g(x) = xk for any 0 < k < r n + n - l . Any xkv(x) can be written as xkv(x) = qk(x)f(x) + vk(x), where qk{x),Vk{x) € T>[x], and Vk{x) has degree less than n or ^(x) = 0. We set uk(x) = xku(x) + qk(x)g(x) e V[x], then Uk(x)f(x) + vk(x)g(x) = xk (2.10) and uk(x) has degree less than m or Uk{x) = 0. We may write «fc(x) = dfV- 1 +dfV - 2 + • • •+• If we equate the coefficients of x m + n _ 1 , x m + n ~ 2 , 1 in Equations (2.10), we obtain the following equations: E + E w f = i+j=l 0<i<n 0<j'<m-l i+3=l 0<i<m 0<j<n-l 1, l=k, 0, Z ^ A:. (2.11) Considering this as a system of m + n linear equations in the c(fc)'s and d^'s, taken in the order (fc) (fc) (k) (k) c m - \ i ' ' ' >co ) ^ n - u " " ' >̂ o ' w e s e e t n a t & ' ̂ e s (/>9) = where the £> is the determinant / (m+n- l ) (m+n-1) , (m+n-l) , ( m + n - l ) \ ( c m - l " " ' c 0 "n-1 " " " "0 I £> = det V c ( 1 ) c ( 0 ) L m - 1 'T l -1 l(0) n-1 0̂ 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 UQ(X), VQ(X) such that u0(x)f(x) + v0(x)g(x) = 1. • 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 Lm<n, the (m + n - 2) x (m + n - 2) matrix defined by r A m< 1 1 1 1 (2.12) / J where the entries are given by i i j — 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 det(L m , n ) = Res(p ± 1 , (m, n) — 1, 0, (m, n) 7̂ 1. (2.13) 2.4 Group Actions on Riemann Surfaces Throughout the thesis all Riemann surfaces S will be connected, orientable and without bound- ary. By the uniformization theorem the universal covering space U of S is one of three possi- bilities: 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 homeo- morphisms 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_1(a;) has a non-trivial stabilizer subgroup in G. 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> U 5 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 Ax,...,At, XX,YU Xg,Yg. For brevity, we refer to T by T(g;mi,.. .,mt). Moreover, consideration of non-Euclidean area implies the Riemann-Hurwitz formula where 7 is the genus of U/IL Now suppose G is the cyclic group Zp and T € Zp denotes a fixed generator. Actions of Zp on Riemann surfaces correspond to short exact sequences l - ) - i I ->r ->Z p -> l . We see that 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\,... , At,X\, Yx,... ,Xg, Yg. (ii) t + 1 relations A\ = • • • = A\ = Ax • • • At[Xx, Y{\ • • • [Xg, Yg] = 1. 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 See [3] and [4]. The genus of the orbit surface S/Zp 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 (") t + 1 relations A™1 = • •• = A™1 = A x , . . . , At[Xu Fx] • • • [Xg, Xg] = 1. (2.16) t times d(T) = Qg - 6 + 2t so long as 6g - 6 + 2t > 0. 22 A 2 Figure 2.1: Fundamental Domain P obtained by reflection in one of its sides. The generators A\, A2, A 3 are the rotations by 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\,... , At through 27r/p about consecutive vertices. The dimension 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 p 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(Xk) = T 6 \ 0{Yk) =Tc\l<k<g. The following restrictions must hold: (i) The a,j are integers such that 1 < a3 < p — 1 and X^=i aj = 0 (mod p). (ii) The bk, ck 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 It follows from the first restriction that the only possible values of t are t = 0,2,3,.. . . Conversely, given integers a,j, b^, ck satisfying conditions (i) and (ii), there is an epimor- phism 6: F —> Zp with torsion free kernel II and a corresponding Zp action T : S —>• 5, where S = 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/Zp. 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). We use the notation [g \ a\,... , 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\ < ... < at < p — 1. 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) t generators Ai,... ,At- (ii) t + 1 relations A\ = • • • = A\ = Ax • • • At = 1. The epimorphism 6 is given by the equations 0 ( A j ) = T % (2.17) where a i , . . . , at satisfy the conditions t 1 < ai < ... < at <p- 1, and ^ a j = 0 (modp). (2.18) Proposi t ion 2.3. 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 arbi- trary 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/Xp 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 automor- phism T: S —> S of order p and orbit genus 0 as follows. Let T be any discrete subgroup of Aut (U) of signature (0;p^7~^p). Then Equation (2.17) defines an epimorphism 6: F -> Z p 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. The 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) = Tai, 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. t times 25 Proposi t ion 2.4. 1 + £ * = 1 ^ i - € A, if, and only if, £ j = 1 a j = 0 (mod p). Proof. First suppose that x = 1 + 2^7=1 — £ A. Thus there is an automorphism of order p, T: S —> S, on some compact, connected Riemann surface S, such that x(T) = X- ^ n f a c* we can assume that the genus of S/Zp is zero. According to the results of this chapter the action of Z p on 5 corresponds to a short exact sequence 1—> II -> T —>• Z p —>• 1. Here T is abstractly isomorphic to the group presented by t generators A\,... ,At and t + 1 relations A\ = • • • = A\ = Ai • • • At = 1. The epimorphism 9 is determined by the equations 9(Aj) = Tai, 1 < fcj; < p — 1. In order that 0 be well defined it is necessary that Y^j=i aj — 0 (mod j>). 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 automor- phism T: S —> S realizing x as an Eichler trace. • 26 Chapter 3 The Conjugacy Classes of Type- 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) and the classes of ideals in Z[x)/(f(x)) [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>n[x] be a monic irreducible and separable polynomial with degree n and ( be a 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 = a0 + a\( -I + a „ _ i ( " - 1 G S. The i-th conjugate of 27 a is denned by a® = a0 + aiQ-\ h an-XQ~l. Then the trace of a is n It is clear that if a G TZ, then Tr (a) 6 V. Suppose a\,...,an G «5. Then the discriminant of a i , . . . , an is defined to be A ( a i , . . . , a „ ) = det (a? a? \^ eg* a a. (n)\ l (n) («) , ah J A standard result is that A 2 ( o ; i , . . . , a n ) = det (Tr (a^ay)). (3-1) (3.2) L e m m a 3.1. a i , . . . , an are independent over T if, and only if A ( a i , . . . ,an) ^ 0. 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\,..., ar G S. Then a = { £ i a i -I h £ r a r | & G TZ} is an ideal in S. We denote this ideal by ( a \ , a r ) . It is clear that (OJI, .. .,ar)(@i, ...,&) = {aidi,.. .,aiPs, • • -,ar0i, • • • ,ar3s). (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 Tr(ao)c2?J. (3.4) Let a i , . . . , an be a £>-basis of a. There is a dual basis a[,..., a'n in <S", that is a basis such that Tr (a^Qj) = 6ij, where Sij is the Kronecker symbol. This is equivalent to either of the following equations X V i * ^ = or j ; a M = V (3.5) k k We also have o' = Va[ + • • • + Va'n (3.6) 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), and = bo + b\X + • • • + 6 n _ i x n _ 1 . Then the dual basis of 1, C , . . . , is bo bn-\ Lemma 3.3. 71' = 71/ (/'(C)). Lemma 3.4. aa' C 71'. (3.7) 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, . . . , of K in Gal (/C/JF), the Galois group of the extension field K/T, such that 771(C) = Ci- Then the i-conjugate of a G S has the form a M = 771(a), for i = 1,. . . , 2n. 29 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 Vk(A) = A™ = ( a £ } ) (3.9) 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 € M2n(F) and a,(3 G S2n are two vectors. Then for any 1 <i,j < 2n, there is 1 < k < 2n, where k depends on such that a'^M(3^ = (a'Md^)^. Proof. Since V\, - • • ,V2n are permutations of the roots of f(x), for any 1 < i,j < 2n, ri^lnj(()) is a root of f(x), say ( k. We have wk(() = f]~lr]j(C,), therefore rjj(a) = 77;% (a), for any a € S. Hence (a'Md^f] = m (a'Mnk(3)) = ^ ( o ^ M r ^ / ? ) = a'^MB^. • Lemma 3.6. Suppose M,N 6 M2n(F) and a = ( a i , . . . , a 2 n ) ' € «S2™, w/iere a i , . . . , a 2 n are independent over V, and a ' M a M = a'Na^ (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 < 2n, there is \ < k < 2 n such that a,(i)Ma^ = (a 'MaW)W = 0. i.e. A'MB = 0, where A = ( o ^ ) and P> = ^5p^ are 2n x 2n matrices. By Lemma 3.1, det A ^ O and det B 7̂ 0, since « i , . . . , oj2n are independent over V, and therefore M = 0. • Let A = C 1 _ " / ' (0 - Clearly A = - A by (2.7) and /(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 / 7̂ 0. 30 Definition 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 JaW = Su a A , for i = 1,.. . , 2n, (3.10) where a = ( « i , . . . , a 2 n ) ' . 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,..., a2n)' and /3 = (Pi,.. •, fan)' by {a,P) = a'jp is a non-degenerate skew-hermitian form. 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 2 n ) ' be a J-vector with respect to (a, a). Let P = (Pi,..., fan)' = ^Ja. Then a ' W / ? W = % which implies Tr (aj/3,-) = so Pi, • • • > P2n is the dual basis of a\,..., a 2 n . Since det( J) = 1, we see that Pi,..., p2n is also a basis of ^ o . Hence a = aAa'. For the converse, suppose (3.11). If A , . . . ,p2n is a basis of a then Pi,... ,P2n is a basis of a. Let 7 i , . . . , 72n be the dual basis of pi,..., p2n. Then Tr (PUJ) = and we have /j'^^O') = 6^, where P = (Pi,... ,P2n)', 7 = (71, • • • ,j2n)'- Since 0 = aAa', there is M € GL2n(V) such that Mp = aA7 . Then P'MW] = a W A W / 3 ' 7 W = <JU aA (3.12) and /3 'M'^) = aAj'Vi(P) = -aAr ? 2(7')r/ ir ? 2(/3) = -6U aA (3.13) 31 For the last equality, we use Formula (3.8). Thus [3'M/?w = -pVM'pP- (for i = 1,. . . , 2n), and so M' = -M (by Lemma 3.6). According to [26] there is Q G GL2n{T>) such that M = Q'JQ. 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). • Coro l lary 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 aAa' = a, so aTZ c a, thus a € 11. The converse is clear. • 3.3 The 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), and A4/ is the set of the similarity classes in Mf over SP2n(T>). Suppose X € Mj. There is an eigenvector a — (a\,..., a.2n)' G corresponding to £, that is Xa — (a. Let a be the D-module generated by a.\,..., a2n, i-e. a = Va.\ -\ 1- Vain and a = A _ 1 o / J 3 . 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 ' 1 ' = ( i a ^ and l a ' 1 ' = ^rS'1'. Hence a ' J S « = ^a'X'JXS® = ^a'Ja®. (3.14) The last equality follows from the fact that X € SP2n(V). Since £ / 0, we get a'JaW =0. • 32 Suppose Y is another element of Mf, and 3 = (3i,• • •, fan)' £ TZ2n 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 SP2n{T>) such that Y = Q~lXQ. Then QY = XQ and therefore XQfi = QY(3 = (Q3, that is Q3 is an eigenvector of X. There are A, fi E TZ* such that Aa = nQ(3 = Qfi3- So Ao = fib, and AAa = A-l\a'J\a = A~\fiQ 3)'J^QP = A^fijlp'Q'jQp = A~lfi]i3'J3 = 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 GL2n(V) such that Aa = fiQ3, and thus fiQY3 = fiQ(3 = (fiQ3 = (Aa = XXa = fiXQ3, hence QYfi = XQ/3. Therefore QY = XQ, i.e. Y = Q~lXQ. It remains to prove that Q E SP2n(V). If i — 2 , . . . , In, then 3'Q'JQ3{i) = J S « = 0 = 0 j / 3 « . fifi^i If i = 1, then 3'Q'JQ3 = ^ a ' J 2 = - a ' J 5 = tfjp. fifi a 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\,..., ain)' be a J-vector with respect to (o, a). Then C a i , . . . ,(a2n is another basis of o, and so there is X G GLiniP), such that Xa — C,a. It is clear that fx{x) = f{x). We only need to prove that X G SP2n{T>). We have a'X'JXa^ = f o / J 5 w = aA. S i 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. By Proposition 2.1, f(x) is a product of irreducible S-polynomials, f(x) = (x-l)2k(x + l)2lPl(x)---ps(x). If Pi(x) is of type-I, then Ppi ^ 0, thus there exists Xi G MVi. On the other hand, if Pj(x) is of type-II, then Pj{x) = q(0)xniq(x)q(-^), where q(x) is an irreducible monic polynomial with degree (by Lemma 2.5). Let Cq be the companion matrix of q(x). Then Xj = C'q+Cq1 G MPj. Hence hi * i-hk) * Xi * • • • Xs G Mf. That is Mf ^ 0. • 3.4 Class Number of Vf In this section we prove Theorem 3. Suppose 71 is integrally closed in S. Then aa = (a) if and only if o = aAa', see [19]. So C is a group, the identity is 71 and a - 1 = Aa' . 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 XeU+. Proof 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 Avf 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 . Coro l lary 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*}. Proposi t ion 3.1. If f(x) = x2 + x + l, then the number of conjugacy classes of Mf in SP2(Q) is infinity. Proof. Let 71 = Q[C], ( = e^. Let p, q be different primes with p = q = 2 (mod 3). We want to show [p] ̂ [q] in 7l+/C. Suppose [p] = [q]. There are A = x\ + y\(, p, — x% + y2( € Z[C] such that AAp = pJJlq, that is [x\ - xiyi + y\)p = [x\ - x2y2 + y2)q. Then there is an integer A; such that x\ - xiyi -\-y\ = kq (3.15) x\ - x2y2 +yl = kp This is impossible due to the fact that if the Diophantine equation x 2 — xy + y2 = kpr, where 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 Conjecture. Let f(x) = xv~x + • • • + x + 1, p an odd prime. Then the number of conjugacy classes of Mf in SP p_i(Q) is infinite. 36 3.5 The Rational Integer Case In this section, we assume V = Z and J7 = Q. Using the fact that the number of ideal classes is finite, the unit group U+ is a finitely generated abelian group and U+2 C C, we get Proposi t ion 3.2. M.j is finite. From now on we consider the m-th (m > 2) cyclotomic polynomial $m(x) = (x -Ci ) - - - ( z -C^(m)) (3-16) where C i , . . . , C<£(m) a r e t n e primitive m-th roots of unity and <f>{m) is the Euler totient function. It is well known that the &m(x) has integral coefficients and is irreducible over Q. Also <&m{x) is an S-polynomial. We simply denote M<pm and M$m by Mm and Mm. 2iri Let C = Cm = e~, TZm = Z[(m]. Then the involution on TZm is just complex conjugation. We denote Cm by C m . Proposi t ion 3.3. For any X G Mm, we have X / X - 1 . Proof. Let a G 7Zmm^ be an eigenvector of A corresponding to C5 Xa = (a. Then X~la. = C"- Hence * ( A ) = (o, A _ 1 o / J a ) and * ( A _ 1 ) = {a,A'la'Ja). If A were conjugate to X - 1 we would have (o, A _ 1 a'Jo7) = (a, A _ 1 a ' J a ) , that is we could find non-zero elements A,/x G TZ 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 TZm}. (3-17) Ci is a subgroup of C and by definition hi = \Ci\. It is easy to check that Co C Ci . 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 ofTZm if m is a prime power and I — ( is a unit of TZm if m has at least two distinct prime factors. 37 See [39]. L e m m a 3.15. Co = C\. Proof. Suppose oo = (ao) where ao € Tl*m. We need to find a unit u G U such that uao = ua~Q. Let UQ — We see that uo is a unit because (ao) = (ao), and UQUO — 1- According to [39] UQ = ±C f c j for some integer k. If UQ — (2L, for some integer I, then we can choose u — C,1. Now we suppose UQ ^ C,21, for any integer /. Note that a = a (mod 1 - C 2) (3.18) for any a G 7Zm. Case 1. If m is odd, then UQ = —Ck, for some integer k. This is because if UQ = £ 2 f c - 1 then uo = £ 2 f c ~ 1 + m , where 2k — 1 + m is even. By Lemma 3.14, either (1 — £) is a prime ideal in 7lm or 1 — £ is a unit in TZm. If 1 — £ is a unit, then = C f c _ 1 = C2'> f ° r some integer /. We can choose u = (1 — Consider the case where (1 — () is a prime ideal in 7lm- We want to show that UQ ̂ — (H 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 = ~( k - Then, by (3.18), a = a = —C,ka = —a (mod 1 — (), hence 2a = 0 (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 7Zm or 1 + C is a unit in 7Zm. If 1 + £ is a unit in 7lm, then we use u — (1 + QCk- 38 In the case that (1 + Q is a prime ideal of TZm, we want to prove that UQ ^ ^ 2 f c+ 1 for any integer k. For a similar reason as in Case 1, there is a 6 TZ*m, a £ (1 + (), such that UQ = f. Suppose u0 = C2k+1- By (3.18) we have a = (2t+la = C~{2l+1)a (mod 1 - C2). This implies (C-l)(C2/ + --- + C + l)a = 0 (mod l - ( 2 ) , thus (C2i + ••• + ( + l)a = 0 (mod 1 + C). We know that (2l + • • • + 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), TZm = TZm, we assume that m ^ 2 (mod 4). First, we quote some results 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 ofTZm is the direct product W x V, where V is a free abelian group of rank _ 1 _ Lemma 3.17. 1, m prime power, [U : WU+] = < 2, m not prime power. Lemma 3.18. If m is not a prime power, then 1 — Cm ^ WU+ and (1 — Cm)(l — Cm) ^ U+2. Proof. If there is an integer I such that Cm(l — Cm) € U+, then (1 — ( m ) ( l — Cm) € U+2. So we only need to show that l~^m = —Cm ^ U2. For this purpose we suppose —Cm £ U2. Then ~Cm = Cm f ° r some /, which implies 41 — 2 = 0 (mod m) and m is even. Since m ^ 2 (mod 4), we have m = 0 (mod 4). Thus 4/ — 2 = 0 (mod 4), which is impossible. This completes the proof. • Lemma 3.19. Let km = [U+ : C]. Then km — \ 2 2 , rn prime power, 4>(m) . 2 2 1 , m not prime power. 39 Proof. By Lemma 3.16 and Lemma 3.17, we see that U+ is the direct product of Z2 and a free abelian group with rank - 1, and then we get [U+ : U+2} = 2 ^ . If m is a prime power, then C = U+2 (Lemma 3.17), and we obtain km = 2^2^. Um is not a prime power, then U = WU+U(l-()WU+ (by Lemma 3.17 and Lemma 3.18). We get C = U+2 U (1 - C)(l - 0U+2, which implies [C : U+2} = 2. Thus km = 2 * T i - \ since [U+ : U+2} = [U+ :C][C: U+2}. • This completes the proof of Theorem 4 (by applying Theorem 3). Example . Let m = 5. Then h\ = 1, 0(5) = 4, and hence g 5 = 4. There are 4 classes of M 5 in SRi(Z). Here is a list of canonical matrices of M 5 , / n i n n \ / n n i n \ X ( X3 = 1 0 °1 f° 0 -1 0 0 -1 0 x2 = 0 0 1 -1 0 -1 1 1 1 0 -1 1 -1 °J 1° 1 0 -1 0 1 -1 -1 0 - 1 - 1 0 1 - 1 0 0 0 - 1 - 1 0 0 x4 = 1 0 0 0 0 - 1 0 0 \o -1 1 oy Similarly a list of canonical matrices of Mw in 5P 4(Z) is -X, -X2, -X3, —X4. Example . Let m = 8. Then hi = 1, 0(8) = 4, and hence g 8 = 4. There are 4 classes in M 8 . A complete set of conjugacy classes of elements of order 8 in SP 4 (Z) is IoJ, Io(-J), 0 - 1 1 0 - 1 0 1 1 - 1 1 0 0 0 -1 0 oy ^ A) 1 -1 0 ^ 1 0 - 1 - 1 1 - 1 0 0 \o 1 0 0 y 40 Example . Let m = 12. Then hi = 1, 0(12) = 4, and hence q® = 2. There are 2 classes of X G SP 4 (Z) with characteristic polynomial /(x) = x 4 - x2 + 1. Two non-conjugate matrices are C / and Cy, where Cf is the companion matrix of f(x). 41 Chapter 4 Symplect ic 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 trans- formations 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 The Symplectic Spaces We start with a definition: Definit ion 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\,..., v2n of V such that their inner product matrix M(vu...,v2n) = ((vi,Vj))2nx2n = J. (4.1) The ordered elements vi,...,v2n 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. Example . Let S be a Riemann surface with genus g > 1. Then Hi(S) with the intersection form is a symplectic space over Z , with rank 2g. The following lemma says that a symplectic basis is a X>-basis. 42 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,...,v2n is a symplectic basis of V. Iiw\,..., w2n is a X>-basis of V, then vi = anwi H h ai2nw2n, v2 = a2\Wi H h a22nw2n, (4.2) V2n = a2n iw\ H h a 2 n 2 n w 2 n where € V (i, j = 1,.. . , 2n). Let A = (a^) be the coefficient matrix. It is obvious that AM(wu...,w2n)A' = M{Vl,...,v2n) = J. Hence the determinant of A is a unit in V, therefore v\,..., v2n is a P-basis of V. • Lemma 4.2. 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\,..., v2n is a symplectic basis of V and w\,..., w2n 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\,..., v2n is a symplectic basis of V. Then M(a(vi),...,a{v2n)) = M{vu ... ,v2n) = J. By Lemma 4.1, cr(vi),... ,cr(v2n) is a basis of W. Hence a is a £>-module isomorphism and therefore a symplectic isomorphism. • 43 Consider V2n, the X>-module of 2n-tuple over T>. For any two column vectors a, 8 € V2n, we define a skew symmetric inner product on V2n by (a, 8) = a'JB. It is easy to verify that V2n 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)', for i = 1,... ,2n, (4.3) then e i , . . . , e2n is a symplectic basis of V2n, which we call the standard symplectic basis. 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 v = aivi H h a2nV2n and w = b\Vi H h ^ n ^ n - (4-4) We set a = (a\,..., a2n)' and 3 — (bi,..., 62n)'> the coordinate vectors of v and w under the basis v\,..., V2n- Clearly, we have (v, w) = a'J0. Suppose Vi , 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\ € Vi , v2 € V2. Furthermore, suppose Vi , V2 are symplectic subspaces of V. Then Vi © V2 is called the symplectic direct sum of Vi and V2, denoted by Vi*V2. Let a i , . . . , Ofc be elements of V. It is convenient to denote any greatest common divisor of a i , . . . , ak by g.c.d (a\,..., ak)- We know that g.c.d ( a i , . . . , ak) = 1 if and only if there exist T*I, . . . , rk € V such that r ia i H f- rkak = 1. In this case, we say that a\,..., ak are relatively prime. Definition 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\,..., ak € V. We say that a i , . . . , ak are coprimitive if for any relatively prime elements a-i,..., ak € V, the linear combination a\a\ + • • • + akdk is primitive. A n ordered set of / + k (0 < k, I < n) coprimitive elements a\,..., a;, 81,..., 3k is said to form an (I, Abnormal set if (a i s 8 j ) = Sij, {au aj) = (A, 8 j ) = 0, (4.5) 44 for all possible i and j. If ati,..., a;, Pi,..., Qk form a (/, A;)-normal set, then their inner product matrix is Remark. If a\,..., are coprimitive, then every a; is primitive. Let a.\,..., a 2 n be a D-basis. Then a\,... , a 2 n 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\,... ,a2n) = L 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.\,..., ak be coprimitive. Then ot\,..., ak are independent and can be ex- tended to a V-basis of V. Proof. It is clear that a\,..., ak 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. (4.6) depending on whether I < k or I > k. 45 Proof. By Lemma 4.4, if v is primitive then g.c.d ( a i , . . . , a2n) = 1- There are c i , . . . , c2n € X> such that a i c i = 1- Let w be an element of V such that the coefficient vector of w is 6 = - J 7 , where 7 = (c x , . . . , C2 n ) ' - Then (v,w) = a'J(-Jj) = 0 / 7 = 1. 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. • Proposi t ion 4.1. If V = Vi + V2 and Vi ± V2, then V = V i * V 2 . Proof. First, we prove that Vi D V2 = {0}. Let v 6 Vi fl V2. Then for any w = wi + w2, where w\ € Vi and w2 G V2, we have {v,w) = (v,wi) + (v,w2) = 0. Hence v = 0, that is V* = Vi © V2. Now we prove that Vi is a symplectic subspace of V by induction on rank(Vi), the rank of Vi . If rank(Vi) = 1, then Vi _L Vi , 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 ± V2, there are two primitive elements wi, w2 of Vi such that (w\,w2) = 1 (by Lemma 4.8 and Lemma 4.7). Let W be the symplectic subspace generated by w\ and w2. If rank (Vi) = 2, we see that V\ = W is a symplectic space. Suppose rank(Vi) > 2. We let U = {v e Vi | (v,w) = 0 for w € W}. If v 6 Vi , then v — {v, w2)w± + {v, w±)w2 E U. We see that V\ = U + W. By the same argument as above, Vi = W © U. Thus V = U © (W© V2). Also U ± (W + V2) and rank (U) = rank (Vi) - 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. • Corol lary 4.1. Suppose V i , . . . , Vm are subspaces of V with 1. V = Vi + • • • + vm, 2. Vi ± Vj for i / j. 46 Then V = Vx * • • • * Vm. 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 ,7 j ) , for j = I,...,I. (4.7) Let 7 = 70 — (jo,@i)a± — ••• — (70, Pk)etk- Then 7 is primitive since 70 is primitive and 70, eti,...,ak 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 k l (7, v) = (70 - ^2(lo, 8i)au J2 (Qiai + bi&) + ̂ 2 i=l i=l j=0 = ̂ 2 bJ^o,8j) + J2 c j ( 7 o , 7 j ) - J2 E M7o, &><a*> j = i j = i i = i j = i = C i ( 7 o , 7 l ) H r - q ( 7 o , 7 i ) which implies c| (7, v) by (4.7). This is contrary to Lemma 4.7. • L e m m a 4.10. Let ct\,... ,a>i be an (l,0)-normal set of V. Then for any 0 < k < I, there are 81,..., 8k, 7 1 , . . . , 7m> where m = 2n — k — /, in V such that 1. ai,...,Qh Pi,..., 8k, 7i, •••,7m ** a V-basis ofV 2. « i , . . • , dk, Pi, - • •, Pk form a (k,k)-normal set. Proof We prove this lemma by induction on k. For k = 0 it is obvious (by Lemma 4.6). 47 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 3=1 k-l k-l 72 = 72 - ̂ ( aJ '72)/?j + ̂ (/?j,72>ay, i=i j=i fc-i fc-i 7m+i = 7m+i - J3<ai,7m+i)#; + X ^ ' ^ + i K - j=l j=l We have ( a i l 7 - > = 0 and (A,7̂ > = 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 , . . . , cm+i in V such that ci<Q!fc,7i> + --- + Cm+i(afc.7m+i> = L ( 4- 9) Note that here we use the fact (ak,aj) — 0 for j = 1,..., I. Now we can find a unit matrix A = (a(j) in GLm+i(V) with c i , . . . , c m +i as its first row, see [26]. Let Pk = C1I1 + l-cm+l7m+l> 7l = 02171 + " •+a2m+l7m+l, 7m = «m+ll7l H ^ «m+lm+l7m+l- Clearly, ai,... ,att, Pi,..., Pk, 7",. • • 1 7 m forms a 2?-basis of V. Furthermore, let P'l = Pi-(c*k,Pi)Pk, 48 P'k-i = 0k-i-(ak,Pk-i)Pk, 0k = Pk- Then a\,...,ai, (3[,..., B'k, 7 " , . . . , 7 M is also a £>-basis of V. We shall verify that a\,..., ak, 8[,...,0'k form a (A;,A;)-normal set by using (4.8) and (4.9) Case 1. For i,j = l,...,k — l, (auftj) = (ai,8j - (ak,8j)0k) = {ciuPj) - {ak,8j)(ai,8k) m+l = (au0j) ~ (ak,8j) ^2 c s (a , , 7^) = (ai,8j) = 6ij. s=l Case 2. For i = 1,. . . , k, j = k, m+l ik- s=l Case 3. For i — k, j = 1,..., k — 1, = <afc,/3,-> - (ak,83)(ak,8k) = 0. Case 4. For jf = 1,.. . , k — 1, m+l = (8j ~ (ak,8j)8k,8k) = (faPk) = E cs(M) = °- s=l This completes the proof. • Proof of Theorem 5. Without loss of generality we can assume that k < I. Let V\ be the symplectic subspace generated by ai,..., ak, 01,...,8k, and V2 = Vr. IfveV, let k k = v- ^2(v,0i)ai + Y2(viai)®i-w i=l i-1 It is easy to see that w € V2. Hence V = V\ + V2. By Proposition 4.1, we see that V2 is a symplectic subspace and V = V~i * V2. 49 If k < I, then a i f c + i , . . . , « / form a (I — k, 0)-normal set of V2. By Lemma 4.10 we can find 0k+i, • • •, 0 i in V2 such that a^+i, • • • , aj, 0 k + \ , • • •, 0 i form a (I — k, I — fc)-normal set. Then a i , . . . , on, 0i,...,3i form an (/, /)-normal set. So we can suppose k — I. If k = I then a combination of a>i,..., ak, 0 i , . •. , 0 k and a symplectic basis of V2 is a symplectic basis of V. • Remark. This theorem gives another way to prove that every normal array can be completed to a matrix in SP2n(V), see [29]. 4.2 Symplectic Transformations Definition 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\. By Theorem 5, there is a non-trivial subspace V2, such that V = Vi * V2. V2 is cr-invariant since (cr(Vi) ,Cr(V 2 )) = ( V i , V 2 ) = 0 , • 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,...,V2n) = (cr(vi),...,CT(v2n)) = ( v i , . . . , U 2 n ) ^ - (4.10) We know that the inner product matrix of o(v\),..., a{v2n) is M{o(v\),..., a(v 2 n )) = X'JX. Hence a is a symplectic transformation if and only if X G SP2n (V). Suppose a is a symplectic transformation. Let v\,... ,V2n and w\,... ,w2n be two symplectic bases of V. Then there is a symplectic matrix Q G SP2n(T>) such that (w\,..., u;2n) = (v%,..., v2n)Q. Let X and Y be the symplectic matrices of a with respect to the bases v\,... ,V2n and w\,... ,W2n- A simple calculation tells us Y = Q~lXQ, that is X ~ Y. Proposi t ion 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,...,Vm are a-invariant symplectic subspaces of V, and V = V\ * • • • * Vm. Then X ~ X\ * • • • * Xm where X\,...,Xm are the matrices of a\V\,..., a\Vm respectively. Proof. Let rank (Vj) = 2nj, and an,..., a.ini, Sn,..., Qini be a symplectic basis of V}. Let Xi be the matrix of a\Vi with respect to the basis an,..., aini, 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\ * • • • * Xm. For the converse, we assume that X — X\ * • • • * Xm. Let V] be the subspace generated by (i>i,... ,f2n)[0 * • • • * Xi * • • • * 0]. It is easy to see that V* is a a-invariant symplectic subspace of V and Vy + • • • + Vm = V. Thus V = Vx * • • • * Vm. • 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, and let one of them be an S-polynomial. If a, 0 G V are such that p(a)(a) = 0 and q(a){0) = 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 (a,0) = (v(o)q(a)(a),0) = (v(a)(a),q{o~l){0)) = (v(a)(a),0) = 0 since q(a)(0) = 0, and q{o~l) = a~2mq(a), where m is the degree of q(x). Here we use the fact (a(a),0) = {a,a-H0)). • Let V be the canonical symplectic space V2n. Given any X £ SP2n{V), we can define a symplectic transformation a as follows, a{a) = Xa (for a € V2n) . It is well known that the matrix of a with respect to the standard basis e\,..., e-m is X. Corol lary 4.2. Lei K, be an extension field of J7 and X, u € JC with A ̂ u and A/j ^ 1. 7/ X € SP2n(K) and a, 0 G / C 2 n are suc/i that (X - \I)ra = 0 and {X - til)s0 = 0, for some integers r, s, then a'J0 = 0. Proof. We apply Lemma 4.12 to X. Note that (x — A) r and (x - n)s(x - j^)s are mutually 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[Pi(x) 1=1 where pi (x) , . . . ,pm(x) are mutually coprime S-polynomials. Let qi{x) = f(x)/pt(x). There are m polynomials, Ui (x ) , . . . , um(x) € T[x], such that ui(x)qi(x) + • • • + um(x)qm(x) = 1. (4.12) 52 Suppose X ~ Xi*- • - * X m , where X , G M P i (for % = 1, . . . ,m). There is Q G SP2n(V) such that X = g - ^ X i * • • -*Xm)Q. Then c/(X) = Q~l\g(Xi) * • • • * g(Xm)]Q, for any polynomial g(x). By (4.12) and the fact that Pi(Xi) = 0 (for i = 1,... ,m), we obtain 7, i = j, 0, t ^ j . Hence Ui{X)qi{X) = Q - 1 [ 0 * • • • * J * • • • * 0]Q G M2n{V). For the converse, we regard X as the symplectic transformation a ->• of the canonical symplectic space V2n. Let Vi = ui(X)qi(X)(V2n) for i = l , . . . , m. (4.13) Then for each 1 < i < m, we have 1. Vi is submodule ofV2n, because Ui(X)qi(X) € M2„(-D); 2. Vi is X-invariant, for X(V*) = X (ui(X)qi(X) [V2n)) = Ui(X)qi(X) (X ( P 2 n ) ) = 3. V2n = Vi + • • • + V m , for •£Ui(X)qi(X) = I; 4. Vi 1 (i / j), by Lemma 4.12 and Pi(X)Vi = {0}. Applying Proposition 4.2, we can complete the proof. • Corol lary 4.3. Suppose f(x) and g(x) are strictly coprime S-polynomials, and X G Mfg. Then X is decomposable. Example . Consider the case D = Z. Let A i 0 0 - 1 0 0 0 0 1 1 0 - 1 0 \0 - 1 0 I) and X2 = 0 1 0 0 0 0 1 0 0 0 0 1 - 1 0 - 1 0 53 XuX2e SP 4 (Z), and fXl(x) = fX2{x) = (x2 + x + l)(x2 - x + 1). We know that %(x + l)(x2 - x + l)-±(x- l)(x2 + X + 1) = 1. Clearly, Xx is decomposable and \ (Xx +1) (X2 - Xx +I) £ M 4 ( Z ) . But X2 is indecomposable, since \{X2 + I){Xl -X2 + I)£ M 4 ( Z ) . Example . Let f(x) = (x2 + l)(x2 ± x + 1). Any X € Mf is decomposable, since (x ± l)(a;2 + 1) - x{x2 ± x + 1) = ± 1 . 4.3 Symplectic Group Spaces Definit ion 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 Gm, generated by a fixed element g of order m, where m is a finite integer or infinity. To specify a Gm-space V , it suffices to give a symplectic matrix X. The characteristic polynomial of X is independent of the representation, we call it the characteristic polynomial of the C7m-space. The set of all symplectic (7m-spaces with characteristic polynomial f(x) is denoted by Vf. Definition 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 c GxW > 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 Proposi t ion 4.3. V is decomposable if and only if it is reducible. Example . 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 Gm on a by g o x = x/(, for all x G o. Note that a = aAa'. Let a = ( a i , . . . , a.2n)', where ai,...,a2n is a J-orthogonal basis of a with 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 ' , 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. Tr is additive, so we only prove the special case where 6 = 0. Let xi,...,X2n be a P-basis of a. We obtain a system of 2n equations in the aW's, Therefore we obtain a symplectic space, denoted by [a, a], and a ( D x W + . . . + a(2n)x(2n) = ^ 55 « ( 1 )4 ) + - + a ( t e ) 4 a , ) = 0 , which only has the 0 solution. Hence = • • • = o^2™) = 0, so a = 0. • 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 / ( 2 , . . . , l / ( 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 Gm-space [ai, ai] * • • • * [ar, ar] to [bi, &i] * • • • * [bs, bs]. Thus there is an r x s matrix Q = (qij) with entries in 5 so that yi Q \xr J for all ( x i , x r ) ' G ai © • • • © o r, and (y\,..., ys)' G bi © • • • © b s . Since a is an isomorphism, Q has an inverse, and hence r = s. If we choose all x\,... ,xr to be zero except Xj, we obtain qijXj G bi. Thus q^aj C bj for i, j = 1,... ,r . If a = ( a i , . . . ,ar)' and /? = (/?i,... ,/3r)' are in [oi,ai] * • • • * [a r,a r], then t fx \ \ i=l V ai 0 (4.15) 56 and similarly (a(a),a(0)) = Tr V 61 V 1_ (4.16) 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 \Xr J a is a P-linear mapping from a\ © • • • © ar to bi © • • • © b r and preserves the inner product, hence a is isomorphism by Lemma 4.3. • Coro l lary 4.4. If [01,01] * • • • * [a r,a r] = [bi, 61] * • • • * [b r ,6 r ], then (01 • • • a r , a i • • • a r) = (bi • • • b r , 61 • • • br). Proof. For each generator a\ • • • ar of 01 • • • a r , the product (det Q)a\ • • • ar can be expressed as the determinant of the product matrix Q ai 0 0 a2 \ 0 0 ••• 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 l)bi • • • b r C ai • • • a,. 57 Multiplying this last inclusion by detQ and comparing, it follows that bi • • • b r is equal to (detQ)ai • • • ar; and it is easy to verify that 61 • • • br = (detQ)(detQ)ax •••ar. This completes 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. By 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\,..., V2n be a symplectic basis of V. The action of g on V has a representative X e SP2n{V). We choose (a, a) € Vf such that * ( X ' _ 1 ) = (a, a), suppose X 7-1 \X2n J I- \ Xi \X2n J where x \ , . . . , X 2 n is a J-orthogonal basis with respect to (a,a). We define the isomorphism (f> : V —> a by 4>(VJ) = Xj. It follows that (x{,Xj) = Tr (-^XiXj) = 5ij = (vi,Vj). That is, 4> preserves the inner product. • Furthermore, we have one-to-one correspondences ^ between M/ and Vf and E between Vf and Mf. More precisely, we have Proposition 4.4. The correspondence * o S o $ is the identity ofVf. 58 Proof. Let (a,a) G Pf, and a = (ai,..., a2n)' be a J-vector with respect to (o, a). Then a\,..., ai2n is a symplectic basis of [a, a]. Let X be the matrix of g with respect to a\,..., a 2 n . We need to prove that ${X) = (a,a). S inceg°{o t \ , . . . ,a2n) = \(aii • • • ,a2n) = (ai, • • • , Q ; 2 n ) X , and X / _ 1 a = {a, we get = (a, a). Hence = = (a, a). • The following proposition gives a method to compute n o S(V), for a symplectic Gm-space V € Vf without needing to know a symplectic basis of V. Proposition 4.5. Suppose V £ Vf. Let a-i,... ,a2n be a V-basis ofV, not necessarily symplec- tic. Let M be the inner product matrix of a\,..., a2n, and X be the matrix of g with respect to a\,..., a 2 n . Let a — {a\,..., a2n)' € V2n be an eigenvector of X with respect to Then ^ o S(V) = (a,a), where a is the ideal generated by ct\,... , a 2 n and a = A~la'Ma. Proof. We choose a symplectic basis v\,..., v2n of V and let Y be the matrix of g with respect to vi,... ,v2n- There is Q G GL2n(V) such that ( 0 1 , . . . ,a2n) = (v\,... ,v2n)Q. It follows that Y = QXQ-1 and M = Q'JQ. If 0 = Qct, then Y0 = QXQ~l{Qa) = 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~la'Ma = A~la'Q'JQa = A~l0'J0. So * o E ( y ) = #(Y) = (a,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 ™ + Vl = l. (4.17) a b 59 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 qn7l C a, q2i7Z C b and (4.18) Set u — qn, v = q2\ and then compare the top left entries of both sides of Equation (4.18). 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 |ab 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 7lm be as in Section 3.5. Then [7lm, -1] * [7lm, -1] ^ [7lm, 1] * [7lm, 1]. 60 Chapter 5 Order p elements in S P p _ i ( Z ) First, in Section 5.1 we will give examples of elements of order p in SPp-i (Z). Then in Section 5.2 27T1 we will discuss the cyclotomic units of the cyclotomic field Q[C] , where ( = e p . And 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 SP2n(V) with characteristic polynomial f(x) irreducible and separable in T>. Suppose we have an S-pair (a, a) and a basis (3\,..., (32n of a, which is not necessarily J-orthogonal. The following steps will find a symplectic matrix X G SP2n(V) such that ^(X) = (a, a). 1. Find the dual basis 71 , . . . , j2n of 0i,..., (32n, that is solve the linear system l'0ii)=6li (5.1) where (3 = (f3u . . . , fan)' and 7 = (71,... ,7 2 n ) ' ; 2. Find the integral matrix Y G GL2n{T>) such that Yd = C # 3. Find the skew symmetric matrix M G GL2n(V) such that M/3 = (2A7; 4. Find a matrix Q G GL2n<(D) such that M = Q'JQ; 5. Let X = QYQ-1. Then X G SP2n{V) and * ( X ) = (a,a). Let 71 = Z[(]. We shall apply this method to find X in fiPp_i(Z) of order p and such that 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 71, . . . , 7 P _ i , where Jr (5.2) Proo/. By Lemma 3.2, we need to verify where /(x) = xP'1 + • • • + x + 1, and /'(() = Let 70 = 7 P = 0. P - 2 p - 2 p - 2 p - 1 p - 2 (x - 0 £ 7.-+1** = E 7*+i* ' + 1 - E 7.-+iC^ = E - E 7<+iC^ i=0 1=0 i=0 1=1 t=0 i=0 i=0 p Thereby proving our assertion. • Let 0 — ( l , C , . . . , (P 2 ) ' and 7 = (71,.. . , 7 p - i) ' . Then Y is the companion matrix Cp_i 0 1 -1 -1 . .. -1 and 0 = ( 1 \ and 7 = C - i p -1 -1 -1 -1 -1 . .. - P=^Lp^p (5.3) where L„ is the n x n matrix whose entries above the diagonal are 0 and the others are — 1. Since CP = Cp-iP we have ( 0 = C'^0. Note that A = K ( ^ ) / 2 w e see that 62 .E+l Let M = Lp-iC f . By a long but routine computation, we see that Af = V 2 'L'p-i 2 is a skew symmetric matrix, and M = Q'p_lJp-iQp-i, where Qp-\ = I + L^-x G G L p _ i ( Z ) . Therefore we have shown Proposi t ion 5.1. Let ' 0 1 Xp — Qp-iGp-xQp^i 1 1 -1 1 -1 -1 1 (5.4) where each block is a ^ x ^ m a i r * a ; - Then Xp G S P p _ i ( Z ) with order p and $(Xp) = (ft, 1). (0 - l \ Example . When p = 3, we see that X = is an element of order 3 in SP2(Z). \1 - U In Section 5.3 we shall see that all Xp are realizable if p > 5, that is X p is the matrix of T* with respect to some canonical basis of Hi(S), for some analytic automorphism T of some compact connected Riemann surface S. 5.2 Cyclotomic Units The cyclotomic units in ft are sin^r Uk sm f , for(fc,p) = l - (5.5) 63 Since 1 - CK = A f t luk, where A = -(^ (5.6) i - C 1 c*̂ and is a unit, we conclude that uk E U+. The following properties of the cyclotomic units are easy to verify: ui — 1 and ump+k = — u m p _ f c = ( - l ) m u f c (5.7) uk>0, l < k < p - l , uk < 0, p + l < f c < 2 p - l . L e m m a 5.2. Y!j=\ u2j+l = u k u k + i + l . Proof. We use the trigonometric formulas, * * s i n i 2 i ± ^ r s i n E E ^ = E s i n 2 f 0=1 J = l P _ J _ V -* 1 ' (2j + / - l ) 7 T (2j + Z + l ) v T cos cos cos - cos ̂ ±m)E 2 s i n 2 £ s i n M s i n i ^ ± l l l V P r-2iF = ukuk+i+i From now on we let the i-th conjugate of ( be ( \ We have L e m m a 5.3. uf = {-lfk-l^i+l^ uikujl. (5.8) • Proof. Using (5.6), we see that uf = ( - C ^ ) ^ - 1 ^ 64 l - C i - C * = ( _ i ) ( ^ i ) ( H i ) ^ u r i • Lemma 5.4. A ^ = ( - l ^ u ^ A . Proof. Since A = A « = We obtain ^ = ^ ( L ^ ¥ ± n = {-iy^ur\ • Lemma 5.5. Suppose X € SP p _i(Z) /ias order p, and * ( A ) = (o, a). T/ien f ( I k ) = ( / > , ( - l f ' \ / ) ) , where \ < k <p — \, k' is the inverse of k modulo p, and ô fc'̂ = a € a | . Proof. Suppose a is a J-vector with respect to (a, a) and Xa — (a. Then a = A _ 1 o ; ' Ja and X k a ( k ' ) = Qkk'a{k') = h e n c e = ^a(fc'>,a*), where a f c = A - V ( f c , ) J a ^ = ^ ( A - V J o ) ^ ) = (-1)*'" V ^ * 0 (By Lemma 5.4). This completes the proof. • Lemma 5.6. uk C, for 2 < k < p — 2. Proof. We only consider 2 < A; < p - ^ . Case I: k is even. For 4 < 2A; < p — 1, we get ujj2^ = —U2kU21 < 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 k i u 7 l < 0) hence Uk^C. • Lemma 5.7. Ukuf1, ukui ^ C , /or 1 < A;, / < and k ^ I. 65 Proof. There is 2 < i < p - 2, such that il = k (mod p). Then Ukuf1 = ±uuu~1 = ±uf\ But ±uf^ 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 ukut = (ukull)uf £ C (since uf G C). • By Lemma 5.5, Lemma 5.6 and Lemma 5.7, the following corollary and Proposition 5.2 are easy to prove. Corol lary 5.1. The p—l elements [ ± l ] , [ ± i t 2 ] , . . . , [ iu^- i l are distinct in U+/C. 2 Proposi t ion 5.2. Let Xp be the matrix given by Equation (5.4). Then Xp,Xp,..., X p _ 1 are not similar to each other. Proposi t ion 5.3. If Pj^- is odd, then there is an X G SP p _i(Z) of order p, such that there are just two different classes amongst X,... ,Xp~l. Proof. Let a = u2 • • • u^i. There is X G SP p _i(Z) of order p such that $ (X) = {71, a). Suppose a G TV'1 (a ^ 0), Xa = C,a and a = A'la'Ja. From Lemma 5.5 and the fact that 71^ = 71 we get * ( X f c ) = (K,ak), where ak = (-l)*' - 1 ^, 1 ^*') and k' is the inverse of k. Note that ffc'1 —l —l — 2 a v > = ±u2k'Uk, •••uE=ik,uk, = ±m • • -Up-iuk, E+l hence a/ak = ±uk? G C U ( - C ) . Therefore * P 0 , if a/akeC, ±auk, 2 <b(xK) = { ^ ( X - 1 ) , ]fa/ak€-C. i.e. X,..., X p are in two different classes. Example . Let p — l. Let X = 0 0 0 -1 1 - 1 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 0 0 0 °1 -1 c3+c-i -1 -c6 a = 1 C 2 + i 0 c6+c °J I 1 J 66 • Then one can easily check that Xa = (,a, X G SPe(Z) and X7 = I. One can also check that a = A _ 1 o ; ' Ja = C 6 + C = u^u^. By computing, we get X ~ X2 ~ X4 and X 3 ~ X5 ~ X 6 . Proposi t ion 5.4. Suppose p = 1 (mod 3). T/iere is X G SP p _i(Z) o/ order p sitc/i £/ia£ X ~ Xk, where k is the least positive solution of k2 + k + 1 = 0 (mod p). Proof. Since p = 1 (mod 3), then x 2 + x + 1 = 0 (mod p) has a solution. Let k be the minimal positive solution. There is an X G £ P p _ i ( Z ) , of order p, with \I/(X) = (TZ,UkUk+i). Then by applying Lemma 5.5 we get ty(Xk) = (lZ,u), where = ( - ^ - ^ ( - 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: Proposi t ion 5.5. There are integers ki,...,kn, such that 2 < ki < • • • < kn < PJY~, and Ukx • • • Ukn G C if and only if hi, the second factor of the class number of TZ, is even. Proof. Let C\ be the group generated by ± 1 , U 2 , • • • ,uzz±• Then [U+ : C\] = /12, see [20]. Suppose ukl • • -Ukn = u2 G C and u G U+. We see that u £ C\ since u2, • • • ) n £ ^ i a r e f r e e generators. Let C2 be the group generated by ± 1 , u,U2,. • • , • Clearly, C\ C C 2 C U+ and 2 [C2 : Ci] = 2, so 2\h2. rP-i If / i 2 is even, there is u G U+, u C\, but u2 G C i . Then u 2 = u^1 • • - u ^ where not 2 all of rj are even. Thus u2 = ukl • • • ukn v2 for some distinct integers 2 < kj < Pj^- and some v G Ci. It follows that • • • Ukn G C. • 67 Remark. In case that h2 is odd, the 2 ^ elements [ i u ^ u ^ • • • Ukn], where 2 < ki < • • • < kn < P j ^ , are all distinct. They are in fact the elements of U+ /C. 5.3 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 p ->• 1 where T(0;p,p,p) — (Ai , A2, A 3 | A 1 A 2 A 3 = A\ = A\ = A\ = 1). If II is torsion free, then we get an action of Z p on S = U/II, with genus Now we indicate how to find all epimorphisms 9 with torsion free kernel. The epimorphism 9 : Y —>• Z p is determined by the images of the generators. The relations in T must be preserved and the kernel of 9 must be torsion free, therefore 9 is determined by the equations 9:1 A i -»• T ° , A 2 -> T 6 , A 3 ^ T c , where T is a fixed generator of Z p , l < a , 6, c < p — 1 and o + 6 + c = 0 (mod 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 Zp-space H\(S) where the action of T on H\(S) is given byT*. ThenS(V(a,6,c)) = M(a,b,c). The proof of Theorem 8 is based on Proposition 4.5. Suppose a i , . . . , a p _ i is a basis of Hi(S), and M is the intersection matrix of a i , . . . , a p _ i . Let X be the matrix of T* with respect to a i , . . . , a p _ i . Let a = ( a i , . . . , a p _ i ) ' € ftp_1 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~la'Ma), where a is the ideal generated by a 1,..., cc p_i. Remark. If we prove the special case where a = 1 and 1 < b < , that is if we show that tt(M(l,M) = (n,ubub+i), 68 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)a-lua{ublubl+l)^) and by Lemma 5.3, we could then have u = {-l)a-lua{ublubl+l)^ = ( - i ) f l - 1 « a ( - i ) ( 6 i - 1 ) ( * + 1 ) « 6 1 a u r 1 ( - i ) 6 i ( a + 1 ) t * ( 6 l + 1 ) 0 « r 1 = ua ump+bump+a+b = ^ o 1 ( - 1 ) m u h ( - 1 ) m ' u a + 6 = U~lUbUa+b where m satisfies b\a = mp + b. We see that u/uaubua+b = u~2 G C. Thus we assume a = 1 and 1 < 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 2 , A 3 , where A i , A 2 , A 3 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 2j_i + e2j (for i = 1 ,p). Then r / i , . . . , r)p are closed paths on S and [r/i],..., [rjp-i] forms a basis of H\(S), see [24]. The intersection matrix of [771], . . . , [r?p_i] is somewhat complex, so we need to find another basis. Since fl^+^A^A*-') = 1, then 7 = A[+i~lA^1 A\~{ G II is a boundary substitution of D and so [e2j_i]n = [—e2C+2i]n- In the interior of each side ej, we choose a point Ei such 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- [rjc+i] and [ni] H \-[r)p] = 0 in the homology group Hi(S). 69 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 -1 0 } p - c - l 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 — 1, 0, otherwise. 70 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 Lc+i,p-c-i, 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 _ c _ i | = 1 (See Section 2.3). Hence [ft],..., [ft-i] is a basis of Hi(S). L e m m a 5.8. The matrix o/T* with respect to [ft],.. . , [ft>-i] is /o - l \ V • l - l Proof. Let f2p+i = fi and ft+fc = ft. Since 0(Ai) = T, we get T([/;] n) = [Ai(fi)]n = [/i+2]n, for i = l , . . . , 2 p . Then T([€k]n) = T([f2k-i]n - [/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 This proves the lemma. = [6], r.fl&D = [6], r*(&-2]) = KP-i], r.([eP-i]) = -[ei]-K2] KP-i]- • Now we compute the intersection matrix M of [ft],..., [ft-i]. Let mj 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, mi,j = & • ij = p*di) • T*Hj) = ii+i • ij+i = mi+ij+i. Iterating this formula we see that mi i P _j+i = mj+i ) P+i = rrij+i^ = —m\j+\. Let kj — mij+i . Then m-ij+j = kj. Hence the intersection matrix is of the form M = fciMi + • • • + A ; P _ 2 M P _ 2 , where the Mj is the (p - 1) x {p - 1) matrix / o ... 0 1 0 A 0 -1 0 V 0 - 1 0 The entries x$ of Mj are given by x kl 1, l - k = j, - 1 , k - l = j , 0, otherwise. By Lemma 5.9, we see that kj •- mij+i = — m i i P + i _ j = —kp-j, and therefore M = feiMi + k2{M2 - M P _ 2 ) + • • • + kj^i ^ M H - I - j . L e m m a 5.10. kj — 1, l<j<p-c-l, 0, p - c < j < ^ i . 72 Proof. It is clear that the intersection of ft and ft+1 (j = 1,..., ^) is only one point, namely the vertex vx. The verification of (5.9) is straightforward by referring to Figure 5.2 and 5.3. • r2j+l Figure 5.2: p - c < j < (p - l)/2 Let a = ' l + < + ••• + cp~2^ l + c + • • • + C p~ 3 i + C V i ) a is an eigenvector of C'p_x with respect to the eigenvalue ft that is C p _ 1 a = (a. L e m m a 5.11. Let A~La'Mia, Vj A-'a'iMj-M^a, j = 2 , . . . , ^ i . Then yj = u2j. Proof. Let 0 = (1 - ()a. We see that 0k = l- Cp~k- p—ip—i 0'MJ0 = E E = E M - E A ft A;=l ;=1 i-fe=j p-i-j p-i p-i-i p-i-j = E - E = E fofo+j - E fo+jPk k=l k=j+l k=l k=l 73 p-l-3 , v , P-1-3 = E (i-c*-fc)(i-r*-'")- E (i - c- f c^)(i-r f c) *=i A;=i ="E (I - cp-fc - rk'3+cj) - P~E (i - c^-k - rk+v) k=l k=l = " i f (r f c - cp- fc+c^- fe - r''-*)+(? -1 - i) (V - ?) = D(C*-?) + ( P - I - J ) (c-?) = E 2(c f e-c f c) + ( P + i - i ) (c-?). fe=i Hence for j = 1, /3'M{p = p ((-(). For j = 2,..., we have pMjfl - V'M^P = J2 2 (Cfc - C") + (p ~ 1 - j) (V - ?) fc=i - ' £ 1 2 (cfc - ?) - o>+1 - P+i) (<r' - =p{tj-?)-P~iz ck) =P (cj - ?) k=j+l So we get , ^ (ri 7*\ C ^ C - J (C2 j ~ 1) ,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~la'Ma = u^u^i. This completes the proof of Theorem 8. • 74 Chapter 6 Tors ion i n SP4(Z) 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 SP2n{Z) having order d? If X has order d, then its minimal polynomial mx(x) is a factor of xd — 1, i.e. mx(x) is a product of some different cyclotomic polynomials, and its characteristic polynomial fx (x) is a product of some cyclotomic polynomials. Suppose d = pf1 • • -pp where pi,... ,pt are different primes. According to a result of D. Sjerve [34], the degree of fx(x) is not less then 4>{pSi) H h 0(pj() — 1, so 0(Pi1) + --- + 0 (p f )<2n + l . 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 - l \ Let Wx = and W = W i . Clearly, W _ A = -W'x and W 0 = - J 2 . V -v Proposi t ion 6.1. Suppose X € SP^CL) has order 3, 4, or 6. T/ien /x(^) = r n x ( x ) = + Ax + 1, and A ~ or 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.n{Z) [30]. We state the special case for T2 here without proof. 75 Proposi t ion 6.2. Any X € T 2 is conjugate to one of the three following matrices -h, h*(-h) or U + U' (6.1) where U 1 0 1 - 1 , Now we suppose that d > 3. Let X € Tj. The possible minimal polynomials mx (x) and characteristic polynomials fx(x) are as follows: When d = 3, m(x) = (x2 + x + 1), m(x) = (x - l)(x2 + x + 1), /(x) = (x 2 + x + l ) 2 , /(x) = ( x - l ) 2 ( x 2 + x + l). (6.2) (6.3) When d = 4, m(x) = (x 2 + 1), m(x) = (x- l)(x2 + 1), m(x) = (x + l)(x2 + 1), /(x) = (x 2 + l ) 2 , /(x) = ( x - l ) 2 ( x 2 + l) , /(x) = (x + l ) 2 (x 2 + l). (6.4) (6.5) (6.6) When d = 5, m(x) = f(x) = x4 + x 3 + x 2 + x 1 + 1. (6.7) When d = 6, m(x) = (x 2 - x + 1), = ( x 2 - x + l ) 2 , (6.8) m(x) = ( x - l ) ( x 2 - x + l), fix) = ( x - l ) 2 (x 2 - x + 1), (6.9) m(x) = (x + l)(x 2 - x + 1), fix) = (x + l ) 2 ( x 2 - x + l), (6.10) m(x) = (x + l)(x 2 + X + 1), fix) = (x + l ) 2 (x 2 + x + l), (6.11) m(x) = ( x 2 - x + l ) ( x 2 + x + l), fix) = ( x 2 - x + l)(x 2 + x + l). (6.12) 76 When d = 8, m(x) = f(x) = x 4 + 1. (6.13) When d = 10, m(x) = /(x) = x 4 - x 3 + x 2 - x + 1, (6.14) When d = 12, m(x) = /(x) = (x 4 - x 2 + 1), (6.15) m(x) = /(x) = (x 2 + l)(x 2 + x + l), (6.16) m(x) = /(x) = (x 2 + l ) ( x 2 - x + l). (6.17) 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 Proposi t ion 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); (6.18) J2*W, J2*W, J2*W, J2*W; (6.19) J2*(-W), J2*(-W), J2*(-W), J'2*{-W')\ (6.20) 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. In Section 6.2 we discuss the 77 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 A'JB h 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)-array can 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 X h i a A 0 A & B 0 0 1 0 1° C 0 where Y = with A B , C D, G SP 2(n-i)(Z), fx(x) = (x- l)2fY{x), a G Z , and a, f3, 7, 8 G Zn~l a P 7 8 AS - £ 7 , C6 - D7, Ca - A'0, D'a - B'0. (6.21) 78 Furthermore, ifY~Yy = Ai 5 i X then ( \ 7i ai 5[\ 0 A i a i Bi 0 0 1 0 \0 C i 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 i a S'\ 0 A a B 0 * b * \0 C p DJ P~lXP = Xi = € S P 2 n ( Z ) . (A B\ By computing we can see that the *'s are 0, b — 1, Y = I G 5P2( n_i)(Z), and a, P, 7, \C D) 6 satisfy (6.21). Thus fx(x) = {x - l)2gY(x). The second part is easy, merely conjugate by I*Q, where Q € 5P2(P) and Q~1YQ = Y\. • L e m m a 6.2. Suppose X € SP4(Z), mx(x) = (x — l)(x2 + Xx + 1) where A = 0, ± 1 . Then X is conjugate to one of I*WX and I*W'A, Moreover, these matrices are not conjugate. Proof. It is clear that I *W\ >* I *W'X (cf. Lemma 2.2). 79 X ~ X\ By Lemma 6.1, we get 0 A di B 0 0 1 0 \0 C ei D) (A B\ where Y = € SP2(Z) with / y (x ) = x2 + Xx + l. Then, from Proposition 6.1, Y ~ W A or Wj .̂ Without loss the generality we assume V ~ W\. Then X ~ x2 = 1 a2 62 C2 0 0 a2 -1 0 0 1 0 X , = ^0 1 Aa 2 + c 2 -X) 0 6 c 0 0 0 -1 0 0 1 0 \0 1 c - A y /1 - a 2 \ . / 1 0 \ where the last conjugacy is achieved by Q = I +1 ^ SRi(Z). We obtain v° 1 / w 7 (A + 2)6 + c 2 = 0 since mx(x) = (x — l){x2 + 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 k ^ - 1 —k 1 k - 1 where k — e Z . It is easy to check that P - 1 X 3 P = I * W\. • P = V € SP 4(Z) 7 Similarly, we have L e m m a 6.3. Suppose X € SP 4 (Z), m x (x) = (x + l)(x2 + Xx + 1) where X = 0, ± 1 . T/ien X is conjugate to one of (-1) * Wx and (-1) * W'x, and these matrices are not conjugate. 80 Proof. Since m_x{x) = (x - l)(x 2 - Xx + 1), we have -X ~ I * or 7 * W i A . Note that -W\ = W'X. This complete the proof. • 6.2 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). It is clear that q(X) is a conjugacy invariant, for if Y = Q~XXQ, where Q G SP2n{Z), then o(Y) = q{Q~lXQ) = {n'JQ-lXQn | 77 G Z 2"} , and so putting £ = Qrj gives (,'JXi = n'Q'JXQn = n'JQ^XQn = n'JYr). Thus q(Y) = q(X). 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 0 JXT7O | is least. If both n'0JXr]o and —r]0JXr)o = if^JXni occur, we resolve the ambiguity by choosing the non-negative value. We write u(X) = r)'0JXr]o- Clearly, if u(X) 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'0JXr)o = 0. Example . 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) = fx(x) be the characteristic polynomial of X. Then ,„m l < (I)""* ! /Wfc !> !± . „ 2 2 , Proof. Note that rj'JXrj is a quadratic form over Z . If M is a symmetric matrix belonging to M n ( Z ) , and a — min {JT/MTJI 177 G zW,i) ^ 0}, then -J 2 | d e t M | » . 81 See [26]. Clearly, it is also true if M is a rational symmetric matrix. We know that rj'JXr) = \r,'{JX + (JX)')r], where \(JX + (JX)') is a rational symmet- ric matrix. Because [JX)' - X'J' = -X'J = -JX'1, and \J\ = \X\ = 1, we see that \JX + (JX)'\ = \JX - JX'1] = IJHX- 1!^ 2 - 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 SP 2 n (Z) , and 1 G q(X). Then / n n _1 n \ X 0 0 - 1 0 0 A a B 1 V a 6' \0 C 0 DJ , A B where \ I G S P 2 ( n _ x ) ( Z ) , a G Z , and a, 0,J,8 G Z " " 1 aatfa/y (6.21). Proof. Since there is a primitive vector rj G Z 2 n 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 P-XXP = Xi = rj * Xrj * A) * b *^ 0 A a B 1 i a 6' \0 C 0 DJ e 5P 2 n (Z) . 82 The remainder of the proof is similar to that of Lemma 6.1. • Corol lary 6.1. Suppose X G SP 2 n (Z) , mx(x) = x2+Xx+l, with 1 G q{X). Then X ~ WX*Y, where Y G SP 2( n-i)(^) with my (a;) = mx(x). Proof. Since X2rj — -XXrj-rj, we see that the entries of the matrix in Lemma 6.5 are: a = — A , a = 0, 0 = 0, and so 7 = 0, S = 0. • L e m m a 6.6. Suppose X G SP 2n(Z), and p(X) = 0. Th en X ^0 A a 1 7' a 6 0 C 0 D ^0 0 1 0 J A B where I ] G 5 P 2 ( n _ 1 ) ( Z ) , a G Z , and a, 0,7,8 G Z n _ 1 sate/?/ (6.21). . C -D < Proof. Note that we have a normal (2,0)-array (n,Xri), where 77 G Z 2 n is primitive. L e m m a 6.7. Suppose X G SP4(Z), with mx{x) — x2 + Xx + 1, w/iere A = 0 , ± 1 . T/ien L If u{X) = 1, Men X ~ W A * W A . 2. If p{X) = - 1 , Men X ~ W A * W j . 5. 7 / / x ( X ) = 0 and X = ±1, then X ~ W A * Wj . 4. 7 / M ( X ) = 0, A = 0, and 1 G q(X), then X~W0*W{> = ( - J 2 ) * J 2 . 5. 7 / / x ( X ) = 0, A = 0, and 1 £ g ( X ) , Men X ~ W 0 + W 0 = (-^2) 0 2̂, • Proof. (1) If / i(X) = 1, then by Corollary 6.1, X ~ Wx * Y , for some Y G 5P 2 (Z), with mY (x) = x2 + Xx + 1. From Proposition 6.1, Y ~ W A or Wj[. Then X ~ Wx * Wx or W A * W'x. But ji(W A * W'x) = 0, hence X ~ Wx * Wx. 83 (2) If n(X) = - 1 , then (i{-X) = 1. It is clear that m_x(x) = x2 - Xx + 1, hence -X ~ VF_ A * V7_ A , and thus X (W"_A * W_ A ) = Wj[ * W'x. (3) -(5) In the following we assume that p,(X) = 0. By Lemma 6.6 we get X ~ X\ - a b where Y = | ), a, b € Z . Let P 6 Xb — a, X(a) = 0 -1 a 0 1 - A 0 - a 0 0 - A -1 y 0 W'~x A 0 0 0 1 0 6 0 0 1 0 0 0 V Then P^XiP = X{a), where yo 0 1 0 obtain X ~ X(0) or X ( l ) . Let Q - / l 0 0 1 0 1 1 0 0 0 1 0 \0 0 0 1 . Then Q~lX{a)Q = X(a - 2). So we 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. Proposi t ion 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\ W* W'; h *W, h* W. with respect to characteristic polynomials (6.2), (6.3). (6.23) (6.24) Proposi t ion 6.5. The number of conjugacy classes in T4 is 8. A complete set of non-conjugate classes is given by J2 * J2, J2 * J2, J2 * J2, (-h) 0 h\ 84 (6.25) h * J2, h * J2\ (6-26) ( - / 2 W 2 , (-h)*J2- (6.27) with respect to characteristic polynomials (6.4), (6.5), (6.6). 6.3 The Case of f(x) = xA + x2 + 1 In this section we discuss the case that X £ SP 4(Z) has fx{x) = xA + x2 + 1. From Theorem 6 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), W*{-W'), W'*(-W), W'*{-W). (6.28) Note that mx2(x) — x2 + x + 1, hence X2 is conjugate to one of three non-conjugate matrices W * W, W2*W2, W* W2. Without loss of generality we assume that X2 = X\ * X2, where X\ and X2 are either W or W2. We can express X as X = Pi * P2 + P 3 o P 4 (6.29) where the Pj's are 2x2 matrices. Then X3 = X(Xi * X2) = P1X1 * P2X2 + P3X2 o P4XU X3 = {Xi * X2)X = X1P1 * X2P2 + X1P3 o X 2 P 4 . Note that X has order 6. Then (JX3)' = X'3J' = -JX'3 = —JX3. Therefore we have P i = a X 2 , P2 = -aXl P 3 P 4 = ( l - a 2 ) X i , P4P3 = (1 - a2)X2, (6.30) and det P3 = det P 4 = 1 — a2 for some a € Z. Also, since X £ SP^Z), we have P[JPX + P'AJPA = J, P'2JP2 + P'z J P 3 = J , and ^ Pi 'JP 3 + P^JP 2 = 0, PXJP[+P3JP^ = J, P2JP'2 + PiJP't = J, P\JP[ + P3JP2 = 0. (6.31) 85 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 = W2P, then P = ( ° 6 V Clearly, if P = al + bW, then det(P) = a2 - ab + b2. Now suppose that X2 = Wl * Wl. From Equation (6.30), we see that P3 = bi + cW, where b2 - be + c2 = 1 - a2. Hence a = -1,0,1. If a = ± 1 , then b = c = 0, thus X is decomposable. If a = 0, then Px = P2 — 0, hence X = P3 o P 4 is quasi-decomposable. We know that the Diophantine equation b2 — be + c 2 = 1 has six integral solutions. 1. b = 1, c = 0, then P 3 = 7, P 4 = Wl; 2. 6 = 1, c = 1, then P 3 = -W2, P 4 = - I v ^ 1 ; 3. b = 0, c = 1, then P 3 = W, P 4 = W ' - 1 ; 4. b = 0, c = - 1 , then P 3 = - W , P 4 = - W ' - 1 ; 5. 6 = - 1 , c = 0, then P 3 = - 7 , P 4 = -Wl\ 6. 6 = - 1 , c = - 1 , then P 3 = W 2 , P 4 = V F m . By Lemma 2.1 and I o W' ~ W 7 2 ' o IF 2 ' (use J * W' as the conjugating matrix) we see that the matrices P3 o P 4 , in all 6 cases, are conjugate. So we obtain Lemma 6.12. Suppose X2 ~ Wl * Wl, I = 1,2. If X is indecomposable, then it is quasi- decomposable and conjugate to I o Wl. 86 Now we consider the case that X2 = W * W2. Lemma 6.13. Suppose X2 ~ W * W2. Then X ~ X(a,b,c), where (a b —a c\ —c 0 b + c —a a b + c 0 -b a c -a) X(a,b,c)= (6.32) for integers a, b, c satisfying a2 — 1 = b2 + be + c2. Proof. From (6.30), we see that X = {-aW2) * {aW) + P 3 P 4 , where P 3 P 4 = (1 - a2)W and P3W = P3W2. Applying Lemma 6.11, we get 6 c \ I—c b + c' P 3 = ( and P 4 = Kb + c -b) \b c It is clear that det P 3 = -(b2 + be + c2) = 1 - a2. • Remark. For any integral solution of a2 - 1 = b2 + be + c 2, X(a, b, c) 6 5P 4 (Z), and its charac- teristic polynomial is (6.12). Clearly, a ^ O . Remark. A n easy calculation proves that X5(a, b, c) ~ X(—a, b, c). 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 3 - I) € M 4 ( Z ) if and only if a is odd. 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 -2c -b + c —a -b -2c 2a Its principal minors are: M i = 2a, M 2 = det / M 3 = det 2a 26 + c 1 26 + c 2a j 2a 26 + c 26 + c 2a —a -b + c = 9. 4a2 - 462 - 46c - c 2 = 4 + 3c2 > 0, \ = 6(a3 - a62 - a6c - ac2) = 6a, —a -6 + c 2a Hence M is positive or negative definite dependent according as a > 0 or a < 0, • Corol lary 6.2. X(a,6 , c) is quasi-indecomposable. Corol lary 6.3. X ( a i , 6 i , c i ) X(a2,62,c2) « / a x a 2 < 0. If a is even, then X(a , 6, c) is also indecomposable. It is known that the Diophantine equation a2 — 1 = b2+bc+c2 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 — y x 0 — 2x —x —z z x —x z \-x y V x J where r x = a — b — c, y = 2a — 26 — c, ,z — 2a — b — 2c, Then V(x,y,z) = QX(a,b,c)Q~1, where or Q = a = — 3x + y + z, b - -2x + z, c - —2x + y. \ (l 1 -1 0 0 - 1 - 1 1 1 1 0 0-1 v° 0 1 - v It is easy to see that a 2 — 1 = 62 + 6c + c 2 if and only if yz = 3x 2 + 1, and a is even if and only 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 — 3x2 + 1 and x + y + z is even. Then 1. Ify> 0, then V{x, y, z) ~ V{0,1,1); 2. Ify< 0, then V(x,y,z) ~ V ( 0 , - 1 , - 1 ) . Proof. Suppose yz = 3x 2 + 1, and x + y + z is even. If y is even, then y = 4k, where k is odd. The reason for this is that x is odd, and then z is odd and 3x 2 + 1 = 4/ where / is odd. If p is an odd prime and y = 0 (mod p), then p = 1 (mod 3). This is because p ^ 3, and 3x 2 + 1 = 0 (mod p). Thus we see that y has the form y = ±4rpri1-P? 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 = 3u2 + v2 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 2 + b2 — y, which implies (a — xb)(a + xb) = a 2 — x 2 6 2 = a2(3rc2 + 1) — yx2 = 0 (mod y). Hence either a — xb = 0 (mod y) or a + = 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 = yiy2, and (UJ, v{) are solutions for yi (for i = 1, 2), that is ?/j = 3uf + v 2 and U{ + xu; = 0 (mod y). Let U = U\V2 + ^2^1, u = V\V2 — "&U\U2. Then 3u 2 + v2 = y and (u + xu)a; = (^1^2 + u2vi)x + (v\V2 — 3uiu 2 )x 2 = xv2(u\ + xv{) + u2v\X + U\u2 (mod y) = (u\ + xv\)(u2 + XW2) = 0 (mod y) 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 = 3u 2 + v2 and u + vx = 0 (mod y). Then v - 3xu = v + 3x 2v = (3x2 + l)v = 0 (mod y). Let P = u —u \ u+xv v—Zxu V—diXU u+xv y y y y u+xv 3m—v r v 2(u+xv) y y y J — v u 2 u 0 T h e n P e S P 4 ( Z ) and PV(0 ,1 ,1JP" 1 = That is V(0,1,1) ~v(x,y,z). The second part is similar. • Remark. The u, v in the proof are coprime. We see that there is a primitive solution of the Diophantine equation 3u2 + v2 = m if m is a product of a power of 4 and odd primes of form 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 Proposi t ion 6.6. Any X € TQ, is conjugate to one of following matrices ~{W*W), -(W'*W')} -{W*W')\ h * (-W), h * (-W); - (72 * W), - (72 * w'y {-h) * W, ( -J 2 ) * W; W*(-W), W*(-W'), W'*(-W), W'*(-W); IoW, IoW', V{Otl,l), V(0,-1,-1). with respect to characteristic polynomials (6.8), (6.9), (6.10), (6.11), (6.12). 6.4 Realizable Torsion (6.33) (6.34) (6.35) (6.36) (6.37) 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. Proposi t ion 6.7. A complete list of realizable classes in SP 4(Z) is as follows Order 2, Order 3, Order 4, Order 5, Order 6, Order 8, Order 10, where U 1 0 1 - 1 . - h , U + U'; W * W'; ( - J 2 ) * J 2 ; Y, - (W*W), z, -Y, A) 1 0 0 0 0 - 1 0 0 0 - 1 1 \i i - l oy Y = Y\ no, i.i), -Y\ \ , and Z = Y\ Y\ ^(0, -1 , -1) ; -Y\ ( 0 - 1 1 0 ^ - 1 0 1 1 -1 1 0 0 o - l o oy - Y 4 (6.38) (6.39) (6.40) (6.41) (6.42) (6.43) (6.44) 91 Consider the short exact sequence of groups 1 n - > T A G - > 1 (6.45) where F = T(gQ;mi,.. . ,m t ) , G is a cyclic group and IT is torsion free. Recall the Riemann- Hurwitz formula 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, mi = 3, m 2 = m3 = 6, (ii) t = 4, mi = mi = 2, m3 = m 4 = 3, (iii) t = 4, mi = m 2 = m$ — 2, m 4 = 6. 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, mi = 3, m 2 = 7713 = 6. That is where g is the genus of U/TI. For g = 2 the Riemann-Hurwitz formula becomes (6.46) r = T(0; 3,6,6) = (A, Bu B2\A3 = B{ = B% = ABXB2 = l ) . There are two epimorphisms F —>• Z§. 01 : Bi -»> T , or 92:< Bi -»• T 5 , B2^ T, £ 2 - * T 5 , 92 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 Bk (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). 9\ induces a homomorphism T* : H\(S) ->• Hi(S) given by r)3 -> ~m + Vi, «4 ->• -V2+V3- Hence the matrix of T* with respect to [771], [772], [773], [774] is V(0,1,1), and so V(0,1,1) is real- izable. Similarly, consideration of 92 proves that V(0,-1,-1) is realizable. Case II, t = 4, mi = m 2 = 2, m 3 = m 4 = 3. That is r = r(0; 2,2,3,3) = ( A i , A2, BuB2\Al = A\ = Bf = 523 = AXA2BXB2 = 1> 93 There are two epimorphisms 6 : T ->• Z 6 0:1 Ai -»• T 3 , A 2 -»• T 3 , S i ->• T 2 (resp. T 4 ) , £ 2 -> T 4 (resp. T 2 ) . Each # induces an action, denoted by T , on some Riemann surface S. Consider that epimor- phism 6 such that 0(B\) = T2. Let X be the symplectic matrix of T* with respect to a canonical basis of Hi(S). From a result of Macbeath[21], we see that T is fixed point free, and therefore ti(X) = 2. Then X must be conjugate to one of the three matrices -(W*W), -(W'*W'), -(W*W). See Proposition 6.6. On the other hand, X2 ~ W * W. Hence X ~ —(W * W), and so — (W * W) is realizable. The other epimorphism leads to the same conjugate class. This completes the proof of the case of order 6. 94 Chapter 7 The Eichler Trace of Z p Ac t ions on R i e m a n n Surfaces 7.1 The 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 4 1 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 * 1 is the trace of T _ 1 : 5 —>• S. Therefore, if A were a subgroup we would have X + A7 = 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 . Proposi t ion 7.1. A is a subgroup ofZ[Q. Proof. Suppose x i and X2 are in A, say * 1 " 1 X 1 = 1 + ET*73I' X 2 = 1 + E77—T- j=i 13 j=i s 95 Therefore x i + X2 = X, where X = 1 + E$=i + E jU ^ijzp I f Xi and X2 are repre- sented by T i : Si —> S i and T2: S 2 —> S 2 respectively, then x c a n be represented by the equivariant connected sum of 7\ and T 2 . Namely, for j = 1,2 find discs Dj in Sj such that Dj,Tj(Dj),... , T j _ 1 ( D j ) are mutually disjoint. Excise all discs Tk(Dj), k = 0,1, . . . ,p - 1, from Sj , j = 1,2, and then take the connected sum by matching d(Tk{Di)) to d(Tk{D2)) for fc = 0,1, . . . ,p — 1. The resulting surface S has p tubes joining Si and S 2 . The automorphisms T i , T 2 can be extended to an automorphism T: S —»• S by permuting the tubes. The Eichler 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. Definit ion 7.1. Define elements 9\, 92, ... ,9m in Z[£] by p-2 9i = C+ Yl & a n d °k = Ck - C k , 2 < k < m. j=m+l Proposi t ion 7.2. A basis of B is given by the m + 1 elements 1, 9\, 92,... , 9m. Proof. Suppose x = Ej=o aj& £ ̂ [C]- Then a short calculation gives p-2 X + X = 2ao - «i + Yl (aJ + av-i ~ a i ) C j ' and therefore x £ -B if, and only if, a, + a p_j = ai , 2 < j < p — 2. Solving for a m + i , . . . , a p _ 2 in terms of a i , . . . , a m and substituting into x gives X = ao + ai#i + 02̂ 2 -I r- a m 0 m . Thus the elements 1, 81, 92,... , #m form a spanning set for 73. 96 Now suppose some linear combination is zero, say ao + a\9i + a 2 0 2 H + a m 0 m = 0. It is easy to see that this is equivalent to a 0 + a i C + • • • + a m C m + (a x - a m ) C m + 1 + • • • + (ai - a 2)C P" 2 = 0. Thus we get ao = a\ = a 2 = • • • = a m = 0, that is the elements are linearly independent. • Remark. Every integer n G B since 6\+ 0\ = —1. We also have £ — £ _ 1 G B; in fact C - C" 1 = i + 20i + 0 2 + • • • + em. It follows that the elements 1, £ — £ 2 — £ ~ 2 , . . . , £ m — Q~m form a basis for an index 2 subgroup of B. A n immediate corollary of Proposition 7.2 is Corol lary 7.1. is a free abelian group of rank (p — l)/2. .4 basis is aiuen bjy Me elements 01, 0 2, • • • , 0m- 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 u 1 1 + E _ x = 1 + E £J,- _ ! • j=i s j=i s This leads us into consideration of when certain linear combinations of the elements TJEZI A R E zero, that is we want to solve the equation 23̂=1 jd̂ x = 0 f ° r * n 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 \jk=n jk=-l J 97 Proof. We use the identity ^ = I E?=i JC* W _ 1 ) and get p- i , P p - i fc=i ^ y j=i fc=i 3=2 k=l ^ ^ n = l \jA:=n / = ^ 1 + - - - + X ^ ) + J E [ E « + 1 H C B + M E + I (T 1 - ^ ^ n = l \jfc=n / V \jk=-l J Now substitute (J?*1 = —1 — £ — — £ P - 2 into the last term to see that ETFTT = ^ i + - + v 1 ) + f E E ^ 1 ) 1 * - E O' + D ^ K k=l S ^ ^ n = l \jfc=n j'fc=-l / - ; E u + v*k F jk=-l 1 1 P~2 I = - - E 3xk + - E E ̂ * - E i ? jk=-l n= l Yj'fc=n jk=—l • As a corollary we get Corol lary 7.2. ^ . = 0 i / , and on/y t/, Zjk=njxk = 0, / o r 1 < n < p - 1. Now it is convenient to change the variables x\,... ,xv-\ to new variables y i , . . . , y p - i according to the equation yi = xk, where kl = 1 (modp). (7.1) Then Corollary 7.2 becomes Corol lary 7.3. Y%Z\ = 0 if, and only if, £fc=l i?(nA;)yfc = 0, /or 1 < n < p - 1. 98 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 yk we apply a sequence 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 ith row to the (p - i)th row, 1 < i < m, yields the matrix Mi 2 4 * l?(2t) . m i?(2m) . p p ^ p p m 2m m + l 1 m + 2 3 R(mi) R{(m + l)i) R((m + 2)i) . R(m2) R({m + l)m) R((m + 2)m) . V P P P P P p - 1 p - 2 R((p-l)i) R((p-l)m) P P 2. Adding the jth column to the (p — j ( , M 2 = 2 4 i R(2i) m R(2m) P P P P ) t h column, 1 < j < m, yields the matrix \ m p p 2m p p R(mi) p p R{m2) p p p 2p 2p p 2p 2p . . . 2p P P P P 2p J 3. Subtracting the (m + l)st row from rows m + 2,.. . , p - 1 , and then subtracting the (m + l)st column from columns m + 2,.. . , p - 1 gives the new coefficient matrix 99 / 1 M 3 = 2 4 R(2i) m R{2m) P P 0 0 0 0 m p 0 2m p 0 R(mi) p 0 i?(m2) p 0 p 2p 0 0 0 0 0 0 0 0 0 0 0 0 The variables z\. for this coefficient matrix are related to the yk by the equations Zk = Vk- Vp-k, 1 < k < m, z m + i = y m + i H h yp_i, zm+j = ym+j, 2 < j < p - 1. Examination of the last m — 1 columns of M 3 reveals that 2 M + 2 , • . . , 2 p - i are completely independent; whereas, z\,... , zm+i must satisfy the matrix equation I 1 2 . 2 4 i ii(2i) . m il(2m) . \ P P m p 2m p R(mi) p Z2 Zi . R(m2) p p 2p J / o \ 0 \ Zm+l J 0 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 ith row, 2 < i < m, yields the coefficient 100 matrix fl 2 . 0 0 . 0 0 . 0 0 . 0 0 . V p v • J 0 • ~[3j/p]p • • -[ij/p]p • • -[™j/p]p . P m 0 -[3m/p]p P ~P -2p —[im/p]p —(i — l)p — [m2/p]p — (m — l)p p 2p J 2. Subtracting j times the first column from the jth column, 2 < j < m, yields the matrix \ / 1 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 •.. • -[m2/P]p — (m — l)p \P -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% + • • • + mzm and Wj = Zj, 2 < j < m + 1. It follows that w\ — wm+\ = 0 and w2, • • • , wm are related by the equations W2 + 2wz -\ h (m - l)wm - 0, ~[3;/p]p ••• -[3m/p]p \ ( w3 \ ( 0 ^ -[9/p]p . -[Si/p]p . \ -[3m/p]p . -[ij/P]p .. [rnj/p}p .. 0 — [im/p]p -[m2/p]p J \wm J \ 0 J 101 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 ±pm~2hi, where hi is the first factor of the class number [28]. Thus Wj = 0, 1 < j < m + 1. This proves that Ylk=\ = 0 >̂ a n a - o l u y ^ Vk ~ Vp-k for 1 < A; < p — 1, and ym = -ym+2 yP-u where ym+2, • • • , 2/>-i are completely arbitrary. Translating back to the xk variables we have: Coro l lary 7.4. 2̂A-=1 = 0 */> and tf> xk = xp-k for 1 < k < p — 1, and Xm — Xm+2 ' ' ' Xp— l , where x M + 2 , - - - , 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 * 1 P _ 1 u Xi = 1 + £7fe—T = 1 + E7FTT' 3=1 S fc=lS u ^ p-1 * 2 = 1 + E 77—T = 1 + E 7FT7' j=l S fc=l S where uk is the number of times k appears as a rotation number in x i , a n d vk is defined similarly. We immediately get t = u since x i + Xi = 2 — t and X2 + X2 = 2 — u. The equation Xi — X2 = 0 gives the linear relation YX=i = 0' where xk = uk — vk. It follows from Corollary 7.4 that the vector x = (xi, • • • , xp-i) is an integral linear combination of the vectors ij = (• • • , 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 p _ i ] by replacing a canceling pair {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 * 1 .7 = 1 A=<xe z[C] where the only restriction on the rotation numbers kj is that Yl)=\ R{kjl) = 0 (mod p). If we define xk to be the number of j , 1 <j <t, such that kj = k, then we can characterize A by A = <x € Z[C] P - I p - i 1 X = 1 + E T * ^ ' X * ^ 0 A N D E ^ _ 1 ) X * = ° (modp)L (7.2) fc=i^ fc=i J In the next lemma we show that by passing to A we can remove the restriction that the xk be non-negative integers. L e m m a 7.2. The set of Eichler traces modulo Z is given by A={x<ZZ[C] p-1 p-1 \ ^ = E7>rrT' J2R^Xk = ° (modP)[. fe=l ^ k=l J Proof. First note that by choosing all xk = 1 in (7.2) we get an element x € A. In fact a short 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 xk become positive, and this does not change its value in A. • This description of A contains a lot of redundancy. In fact we have the following charac- terization of A. 103 L e m m a 7.3. The set of Eichler traces modulo Z is given by m m U — 1 a 1 i X k=l " k=l Proof. According to Lemma 7.2 a typical element X £ A can be represented by p—1 m m E Xk _ v Xk Xp-k rk _ i — Z ^ / ^ f c _ i + Z ^ /--k _ i ' ft=i s fc=i s fc=i s where the xk are integers satisfying J2k=\ R{k~l)xk = 0 (mod p). Now we use the fact that 1 1 + — r = - l to see that \ — where m E h Jfe=l ' The restriction on the integers Zfc becomes 2~Jfe=i -^(^ 1)2;fc = 0 (mod p), since p—1 m m ^Rik-^Xk = ^Rik-^Xk + ^Rdp-ky^Xp-k k=l k=l k=l m m = ^Rik-^Xk + ^ip-Rik-^Xp-k k=i k=i m = Y,R{k-l)zk (modp) k=l and YX=\ R(k~l)xk = 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, ... ,9m, 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. Definition 7.2. For integers k, n define C(k,n) = R(k~1n) + R(k~1) — p. 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 1 ) - p. (ii) C(l,n) =n + l-p, C{k,l) = 2 J R(*T 1 )-p, C(p - 1, n) = p - n - 1, and C(k,p-1) = 0. L e m m a 7.4. The elements of A are those elements £ E Z[(] of the form ^ m m x = - E(E c^' nK)^> P n = l fc=l where the only restriction on the integers zk is E f c L i R{k~l)zk = 0 (mod p). Proof. By Lemma 7.3 a typical Eichler trace modulo Z is given by x, where x = E / c L i a n d E f c L i R(k-l)zk = 0 (mod p). Using Lemma 7.1 we have p-2 * = - \ E ^* + ^ E ( E ^ * - E cn- y jk=-l y n=l \jfc=n jk=-\ / The condition Xl fcLi R{k~l)zk = 0 (mod p) can be written as E j fc=i J** — 0 ( m o d p), and so J2jk=-iJzk — Y,jk=i (P ~ J)zk = 0 (mod p). Therefore, modulo Z we have « - '*) < " - ; E f £ - E « ) < • • n=l yfc=n jfc=-l / n=l yfc=n j'fc=-l / Note that the term corresponding to n = p — 1 contributes 0 to the sum. Also note that Zjk=n 3Zk ~ E i f c = - i )*k = £ £ = i and therefore x = J (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 n, and then use properties of the coefficients C(k,n). X 1 m / m \ ^ m / m \ P n=l \k=l ) F «=1 \*=1 ' 1 m / m \ 1 m / m \ P » = 1 \jfc=l / P n=l \*=1 / . . m / m \ m / m \ - 1 E ( E c ^ « n - + E E (2i^_1) - c _ n P n=l \fc=l / ^ « = 1 \*=1 7 m / m \ ( \ m \ P n=2 \k=l J \ P k=l J 105 + ^ E ^ i ) ^ ) ( c m + 1 + ---+cp-1) 1 m / m \ ^ n = l \fe=l / The last equation follows from 9i = ( + £ m + 1 -1 (- ( P - 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,...,at] + x[bi,...,bu] = x[ai, • • • ,at,h,... ,b u]. (ii) £ [ . . . , a , . . . , p - a , . . . ] = x[. . . , a , . . . , p ^ a , . . . ] . If we define yk to be the number of j, 1 < j < t, such that aj = k, then we obtain 1 m / m \ • • •, at] = - E E C ^ _ 1 ' ^ (7-4) P n = l \A;=1 / where zk = yk — Vp-k- This is because that yk — X f t ( f c - i ) and kyk = 0 (mod p). Definit ion 7.3. Let K be the collection of m-tuples v — [z\,... , zm] satisfying the condition 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], vk = [1,... , 1 , - 1 , . . . ] , 2 < k < m - l , vm = [1,0,... ,0,2], 106 where for 2 < k < m — 1, the 1 is in the first and the kth entries, the —1 is in the (k + 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 1 m m P 71=1 fc=l Lemma 7.4 implies that L is an epimorphism. Proposi t ion 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 c ( f c' n )* f er ; P 71=1 U = l / pt{ m = E ~°n' 71=1 - m / m \ P 71=1 Vfc = l / 1 m = -Y{C(lM + C{k-\n)-C{{k + l)-\n))dn P 71=1 . m - V ((n + 1 - p) + R{kn) + R(k) -p- R({k + l)n) - + 1) + p) #n P 71 = 1 m £ 71 = 1 '(k + l)n . P . P . - l f t * m / 771 \ P 7 l = l \fc=l / ^ m = - £ ( C ( i , n ) + 2 C ( m - \ n ) ) f t ; n = l ^ 771 = - E ( C U , ") + CCrn" 1 , n) - C((m + l )" 1 , n)) ft^i 71=1 = E 71=1 (m + l )n P mn P l) 0n, 107 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 M(k,n) = (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 kth row. The resulting entries of the new kth row are '{k + l)n kn . P . 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. J The first column of this matrix is 0, . . . , 0,1, hence ( ^ ^ det(M) = ± d e t if] ~ [ ^ 1 V where 3 < k, n < m. 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(vr), for 1 < r < m and 1 + r + s = p. This complete the proof of Theorem 11. 108 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 p , sig: W -¥ R(ZP). Let e: i l (Z p ) —> Z[£] 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 (,k — C~k, k = 1,... , m. From the Remark earlier in this section 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 Cobordism In this section we prove Theorem 12. To begin with suppose T i : S i —>• Si and T2: S2 —> S2 are automorphisms of order p on compact connected Riemann surfaces. We do not assume that the orbit genus of either Si or S2 is 0. We start with a standard definition. 109 Definition 7.4. We say that T\ is equivariantly cobordant to T 2 , written T\ ~ T 2 , if there exists a smooth, compact, connected 3-manifold W and a smooth Z p action T : W —> W such that (i) The boundary of W is the disjoint union of S\ and 5 2 , d(W) = Si U 5 2 . (ii) T restricted to agrees with Tx U T 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\,... , at >, and if the orbit genus g = 0, we denote it by < a\,... , at >. The set of all cobordism classes of Zp actions on compact connected Riemann surfaces is denoted by Q. Addition of the cobordism classes of the automorphisms T\: S\ —> S i , T 2 : S 2 —> S 2 is defined by equivariant connected sum as follows. Find discs Dj in Sj such that Dj, Tj{Dj),... ,T?~l(Dj) are mutually disjoint for j = 1,2. Then excise all discs Tk(Dj), j = 1,2, k = 0,1,. . . ,p — 1 from S\, 5 2 and take a connected sum by matching d(Tk{D\)) to 9(T f e(/J 2)) for k = 0,1, . . . ,p — 1. The resulting surface S has p tubes joining Si and 5 2 . The automorphisms T\, T 2 can be extended to an automorphism T: i f ? —> S by permuting the 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 , . . . , at > + < h I 61,. . . , bu >=< g + h | a u . . . , at, h,... , bu > . (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 ~ aii • • • iP ~ at >• The proofs are not original, but are presented here to emphasize the relationship with A. L e m m a 7.5. < g | 01 , . . . , 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 Tk(D) in S x {1}, k = 0,1,. . . ,p — 1, attach a copy of a handlebody H of genus o by identifying the disc Tk(D) with some disc D' C d(H). Let W2 denote the resulting 3-manifold. See Figure 7.1. The action of Z p can be extended to W2 by permuting the handlebodies. The manifold W2 provides the cobordism showing that < g I a-i,... ,at >=< a-i,... ,at > . • Figure 7.1: Cobordism of g = 0 L e m m a 7.6. < a,p — a, 0 3 , . . . , at >=< 1 | 0 3 , . . . , at > = < 0 3 , . . . , at > . 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,a3,... ,at >=< 1 | a 3 , . . . ,a t > . I l l 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,... , at]) =< « i , . . . , at > . 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 homomor- phism. 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[ai > • • • j at]• Suppose T : S —>• S represents [ai , . . . ,at]. 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 [ai , . . . , at] consists entirely of canceling pairs, and hence x[ai, • • • > at] = 0 in A. • 112 7.3 Dihedral Groups of Automorphisms of Riemann 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 D2p be the dihedral group of 2p elements and Tp,T2 € D2p be two fixed generators of order p, 2 with the relations Tp = T | = (TpT2)2 = 1. Suppose there is an embedding of D2p in Aut (S). We have a faithful representation R : D2p —• GLg(C), by passing to the space of holomorphic differentials on S, assuming g > 1. We want to characterize such groups R(D2p). We denote by D2p(A, B) any subgroup of GLg(C) generated by A, B with the relations Ap = B2 = ( A B ) 2 = I. Let G{ = D2p{AuBi) (i = 1,2). Gi and G2 are said to be conjugate, denoted by G i ~ G2, if there is Q € GLg(C) such that Q~xGiQ = G2, and strongly conjugate, denoted by G\ « G2, if Q'^AiQ = A2 and Q"lB\Q = B2. A subgroup D2p(A, B) is said to be realizable if it is conjugate to some R(D2p). It is well known that the trace of an element of order 2 in GLg(C) is an integer, and the trace of an element of order p in GLg(C) is an algebraic integer in the cyclotomic field Q(C)- A subgroup G in GLg(C) is called an I-group if all elements of G have integer traces. Let X E D2p(A, B) be of order p. Then X ~ X~l, and hence tr (X) = tr (X'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 € D2p(A, B) of order p has rational trace, then D2p(A,B) is an I-group and all elements of order p in D2p(A,B) are conjugate. Proof. It is clear that tr (X) = k + ki{( + C - 1 ) + • • • + km{(m + ( _ m ) (m = ^ f 1 ) , for some non- negative integers k, k\, ..., km with k + 2(ki -\ + km) = g. But (,,... C p _ 1 are independent over the rational field Q, so we have k\ — • • • — km, say /. Therefore tr (X) = k — I is an integer. • Lemma 7.8. Suppose Gi = D2p(Ai,Bi), i = 1,2, are two I-groups. Then the following three conditions are equivalent. 113 1. G\ ~ G2; 2. C?i « G2; 3. tr (A^ = tr (A 2 ) and tr (Bi) = tr (5 2 ). Proof. For a dihedral I-group we have the following canonical form G = D2p(Ai,BXjy), where Ai = \ V (1 and Bx>y \ where x + y + (p - 1)1 = g and tr (At) = x + y - l . Since the number of blocks of Ĵ 's in Bx,y is even, tr(BX t V) =x-y. • 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 D2p(A,B) is realizable, then D2p(A,B) is an I-group with tr (A) < 1 and tx(B) < 1. Thus we complete the proof of the necessity condition of Theorem 13. To any action of D2p on S we can associate a short exact sequence of groups 1 -> II T (g 0 ;TV^Tp, 2,.. 772) 4 D2p -> 1 where V must has form T(g0;p,...,p,2,..7a) = (Xu..., Xgo, Yu ..., Ygo, Au ..., At, Bu ..., Bs) 114 with relations A\ = ••• = Apt=B2 = --- = B2 = [XuYl]---[Xgo,Ygo]A1---AtBl---Bs = l (7.6) By the Riemann-Hurwitz formula (2.16) we see that s must be even. From the results of Macbeath[21], we obtain that Fix (Tp) — 2t and Fix (T 2) = s. Hence if D2P{A, B) is realized by this action then tr(A) = 1 - t and ti(B) = 2-^. To prove the sufficiency condition of Theorem 13, we need the following lemma. Assume that D 2 p ( A , B) is an IR-group. 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;, Bx>y, as in the proof of Lemma 7.8. 5 + ( p - l ) t r ( A ) - | - p t r ( B ) = x + y + (p-l)l + (p-l)(x + y-l)+p{x-y) = p(x + y) + p{x - y) — 2px. 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) tr (A)+ptr(B)). 2p t s We define an epimorphism 0 : T(go; p,. ^p, 2,. T. ,2) —> D 2 p as follows: Case 1: If tr (A) = 1 and tr (B) = 1, then t = 0, s = 0, and o0 > 2. We set 0(X1)=0(Yi)=Tp and 0(*i) = 0(YO = T 2 (for i = 2,. . . ff0)- 115 Case 2: If tr (A) = 1, tr (73) = 0, then t = 0 and s = 2, and go > 1. We define 9(Bl) = 9{B2)=T2 and 0(X*) = 0 ^ ) = T p . Case 3: If tr (A) = 1 and tr (73) < - 1 , then t = 0 and s > 4. We define 9{Bi)=TpbiT2 and 0(X,-) = 0(Yj) = 1, where bi are integers (not all the same) with 0 < bi < p — 1 and Ei=i( — l ) l °i = 0 (mod p). 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 g0 > 1. 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