UNITS IN I N T E G R A L CYCLIC GROUP RINGS FOR ORDER by RONALD AUBREY FERGUSON B . S c . (Mathematics) Dalhousie University, 1967 A THESIS S U B M I T T E D IN PARTIAL F U L F I L L M E N T O F THE REQUIREMENTS FOR T H E DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of M a t h e m a t i c s We accept this thesis as conforming to the required standard T H E UNIVERSITY OF BRITISH C O L U M B I A June 1997 © R o n a l d A u b r e y Ferguson, 1997 LP R S In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Abstract For a finite abelian group A, the group of units in the integral group ring 2ZA may be written as the direct product of its torsion units ±A with a free group U2A . Of finite index in U2A is the group £IA, the elements of U2 which are mapped to cyclotomic units by each character of A. The order of U2A/SIA depends on class numbers in real cyclotomic rings ~%.[C,d + C^ ]1 Of finite index in QA is the group of constructible units YA, for which a multiplicative basis may be explicitly written. The order of QA/YA is the circular index c(A). In many cases, for example where A is a p-group with p a regular prime, this index is trivial. This thesis develops an inductive theory for determining c(C ) where C is a cyclic group of order n = l p , with I and p distinct primes, and also for giving some description of the group Q,C jYC . This is a continuation of the work of Hoechsmann for the case n = Ip. It turns out that the methods required for n = l p are, in general, very different from the ones used for r = s = 1. n r n s n r s ii n Table of Contents Abstract ii Table of Contents iii Acknowledgement v Chapter 1. Introduction 1 Chapter 2. Circular Units 2.1 2.2 11 Constructive Units Cyclotomic Units 2.2.1 A r i t h m e t i c 2.2.2 N o r m Relations 2.2.3 R e a l C y c l o t o m i c U n i t s 11 14 14 18 23 Chapter 3. Setup 3.1 3.2 26 D e r i v a t i o n of P u l l - b a c k s D e s c r i p t i o n of the F i n i t e R i n g s 26 32 Chapter 4. The Case n = 4p 4.1 4.2 4.3 39 L i f t a b i l i t y .'. The Kernel C o m p a r i s o n of Images 40 43 46 Chapter 5. Liftability 5.1 52 C o n s t r u c t i b l e units i n f i ( ( ) 5.1.1 n o d d 5.1.2 n = 2 p" 5.1.3 T h e index for constructible units 52 56 60 64 L i f t a b i l i t y a n d non-constructible units 5.2.1 n = lp 5.2.2 n = l p , 64 64 68 n r 5.2 r s Chapter 6. Setup (2) 73 Chapter 7. Calculation of Group Orders 80 7.1 7.2 Images of constructible kernels i n QC /i Images of constructible kernels i n £lC /i 7.3 R e s t r i c t e d circular indices and O C / n n n p 80 92 99 Chapter 8. Comparison of Group Orders 8.1 p Defects 102 104 iii Table of Contents Bibliography Acknowledgement I would like to thank Klaus Hoechsmann for suggesting this problem to me and as well for his support through out my program. I would as well like to acknowledge many colleagues for their continued encouragement. I include, in particular, Larry Roberts. The pull-back and other commutative diagrams in this thesis were typeset using the kuvio program developed by Anders Svensson. v Chapter 1 Introduction Consider the operation of multiplication in the whole numbers, 7L. There is a single element, the identity 1, which is neutral — multiplication of any number by 1 produces no change. A unit is any element whose multiplicative effect may be canceled, that is, it is a factor in some product giving the value 1. In Z the only units are 1 and —1. The polynomial ring, Z[x], consists of polynomials in the variable x with coefficients in 2. Each polynomial has a degree which gives the highest power of x occurring in its expansion. The degree of a product is equal to the sum of the degrees of its components. Since 1 is a degree 0 polynomial, it can only be the product of other degree 0 polynomials. Again, the only units are 1 and —1. The situation changes when, for example, we impose the condition that x = 1. All elements 5 now become expansions in terms of the basis elements 1, 3 , and x . In fact, each of 4 these elements is now a unit, along with its product with —1. These elements have finite order and are called torsion units: some power of each is the identity element. Notice as well that (-l + x + x ) ( - l + x + x ) = 1 - x - x - x + x + x - x + x + x 4 2 3 A 2 3 6 3 4 7 = 1- x - x - x +x +x - x + x + x A 2 3 3 4 2 = 1, so that — 1 + x + x and — 1 + x + x are units. 4 2 3 No positive power of — 1 + x + x gives the identity. It is torsion free. The element — 1 + x + x 4 2 we can write as (—l + x + x ) . With the positive and negative powers of — 1 + x + x we have 4 - 1 4 1 3 Chapter 1. Introduction an infinite number of units. Each element of the above basis is a power of the single generator x, making this set a cyclic group of order 5. What we have formed is the integral group ring ILC5, where C5 designates this cyclic group and coefficients for expansions in term of elements of C5 are taken from 7L. In the same manner, we form the group ring Z C , where C designates a cyclic group of order n. n n We now take a more careful look at the condition x 5 - 1 = (x - l)(x 4 +x 3 + x + x + 1) = 0 2 used to create the ring ZC ~ Z [ x ] / ( x - 1). 5 5 For an element to be a unit in ZC5, it must as well be a unit when we apply the further condition x — 1 = 0, which brings us back into the ring of integers Z . The substitution x = 1, the augmentation map A : x —» 1, must therefore result in either the value 1 or the value —1. The other restriction $ ( z ) = x + a; + x + :E + l = 0 4 3 2 5 brings us, in this case, into the ring which has as a basis the four roots of this equation, the so-called fifth roots of unity, (5,(5,(5 an( ^ CsS where we designate one of these roots as (5. Algebraically, these four roots are indistinguishable. There are four operations which interchange these roots, namely ri(= 1),T2,T3 and T4, where for example T2 : (5 -> C5 • $5(2;) is the fifth cyclotomic polynomial, giving rise to the ring Z[£s] of cyclotomic These four operations comprise the galois group of integers. designated G5. This group also has a similar action on C5, interchanging its four generators, and is a very powerful tool in gaining an understanding of arithmetic operations. The character map 5 : Z C -> Z[Cs], X 5 2 Chapter 1. Introduction w h i c h substitutes £5 for x, must send a unit of Z C 5 to a unit of Z [ ( s ] . K n o w l e d g e of c y c l o t o m i c units, i n this case the units of Z [ £ s ] , thus gives us a starting point for d e t e r m i n i n g units of a group r i n g Z C . T h i s is a n o l d a n d well studied field, g i v i n g . u s , i n m a n y cases, enough n i n f o r m a t i o n to give a complete description of units i n cyclic group rings. If a u n i t i n Z a n d a unit i n Z f i ^ ] gave rise to a unit i n ZC5, our task w o u l d then be easy. T h i s is i n fact the case w h e n we use fractions, the set Q of r a t i o n a l numbers, for coefficients. A l l non-zero numbers i n Q are units, m a k i n g this case quite different. W i t h Z for the coefficients, we adopt a procedure s u m m a r i z e d i n the pull-back ZC diagram Z[Cs] 5 A Z HF . 5 W e o b t a i n IF5 by reducing Z m o d u l o 5. T h e m a p Z [ ^ ] —>• IF5 first sends £5 to 1 a n d hence 5 to 0 since C + C + C + C5 + 1 = 0 . 4 5 3 5 2 5 U n i t s i n Z m a p to 1 or 4 i n IF5. Hence, a unit i n Z [ ^ ] gives rise to a unit i n ZC5 o n l y i f its coefficient s u m is 1 or 4 ( m o d 5). T h e torsion units of Z [ ^ ] form a cyclic group of order 10 generated by —£5 = X s ( — x )- N o t i c e that 1 1 /• 1 /-2 _ 1 ~ Cs _ (1 ~ C5) 1 + 0> + C5 - --, T- = —j 7 3 T3 _ , x _ l - (1 ~ C5j , FA T 3 where the exponential notation is used to denote the action of G5, a n d that are u n i t s , since their product is 1. T h e torsion unit —£5 a n d positive a n d negative powers of 1 + C5 do generate a l l units i n Z f ^ ] . 3 However, the coefficient s u m of 1 + C5 is 2 (mod 5) , so it does not give rise to a u n i t i n Z C . 3 5 Its square, however, has coefficient s u m 4 ( m o d 5), so it does give rise to a u n i t . N o t a l l units i n Z[Cs] are liftable to units i n the group ring, though some power of each is liftable. 3 Chapter 1. Introduction For this unit (1 + C s ) , we find 3 2 ( i + c ) = i + 2c 3 2 5 + c 3 5 6 5 = -c 3 5 ( - 2 - ci - c ) 3 5 = -C 5 = -C 5 3 ( - 1 + C + C - ( i + Cs + c 4 5 3 5 2 5 + C + C )) 3 4 5 5 (-i + C + C ) 4 5 = - i + Cs + C 5 4 5 m o d u l o torsion i n Z ^ ] . T h i s is the image of the unit —1 + x + x 4 i n Z C 5 , so w i t h — 1 + x + x a n d —x we c a n generate a l l units of Z C 5 . T h i s m e t h o d extends to the group rings 7LC for p r i m e p. V — 1 = (x — l)<J> (x), a n d we x p p are again searching i n a cyclotomic r i n g , Z [ £ ] , for units liftable to the group r i n g . T h e element p (l-a,) » T where r a = l + x + x + ...x _ 1 2 a _ 1 , is i n G , maps to a unit i n Z [ £ ] but fails to have coefficient s u m ± 1 unless a — 1. W e p p c a n rectify this i f we are able, i n the group ring, to d i v i d e by another element h a v i n g the same coefficient s u m . T h i s leads us to consideration of quotients of the type ( 1 k for r a n d a - X)(r.-D(r4-D ' in G . l + x" X " ...+X^-l) 1+ x + x + + 2 = + + 2 ( l - x ) ( l - ^ ) (l-x )(l-x )' a 6 K } T h e numerator a n d demominator here b o t h m a p to units i n Z [ £ ] . W h e r e a p p a n d b are relatively prime, we c a n actually perform this d i v i s i o n i n the p o l y n o m i a l r i n g Z [ x ] to produce a unit i n iY.C . B y m u l t i p l y i n g this unit by a suitable power of x we make it invariant p under the s u b s t i t u t i o n x K> X _ 1 = x p _ 1 . U n i t s arising i n this way are called constructible units, a n d f o r m the m u l t i p l i c a t i v e group WC . p elements. N o t e that since r p+a WC is a torsion-free group; it contains no t o r s i o n P — T , we can start w i t h any two elements r 0 a a n d Tfc of G p but adjust the values of a a n d b so that they are relatively prime. I n Z C , where G P p u(x), u(x) ,... Ta is cyclic, a l l units of this type are generated by (p — 3 ) / 2 ,u(x)( ) T a ( p 5 ) / 2 , of the single generator 4 conjugates, 4 Chapter 1. where r a Introduction a n d Tb each generate G . (p — 3 ) / 2 is i n fact the m i n i m u m number of generators, the p rank of WC . p I n the case p — 5 -1 + X + X 4 = X (l 4 -X + X )= 2 X {1 4 - )(T -V(T -1) X 3 2 is i n WC5, so that, for Z C 5 , WC5 is the full set of units, m o d u l o torsion. T h i s is the p a t t e r n for the other group rings Z C . p A more general setting is the consideration of units i n 7LA, where A is a n finite a b e l i a n group. It was first k n o w n that the torsion group of TLA is ±A a n d t h e n that the u n i t group, UA is the p r o d u c t of ±A w i t h a free group composed of elements u(x) w h i c h are inversion invariant, that is, u ( a ; ) = u(x). _1 A later result [2] was the presentation of a subgroup of full rank. A l t h o u g h the r e s u l t i n g group was of finite index i n UA, this index could s t i l l be quite large. I n [6], H o e c h s m a n n defines the constructible units, YA, of order d contained i n A, w i t h x as a generator. (1.1) where r Q and for ~ZLA. L e t d be a cyclic subgroup WCd is generated by elements of the form are i n Gd (a a n d b are b o t h p r i m e to d), a n d has r a n k [4>(d) — l ) / 2 . T h e n YA = JJ WC, C cyclic, CCA the direct p r o d u c t over a l l cyclic subgroups of A. T h i s has more t h a n one factor even i f A is cyclic, p r o v i d e d the order of A is not p r i m e . T h e rank of YA WCs. is the s u m of the ranks of the I f Gd is cyclic, we can find a multiplicative basis, a m i m i m a l set of generators, consisting of conjugates of a single generator [6]. W h e r e Gd is not cyclic, we can use L e m m a 2.2 below to find a basis. YA is a n a t u r a l choice for a reference set of units for 7LA, a n d has a n analog i n the set of cyclotomic units for a cyclotomic r i n g Z [ £ ] , where ( is some root of u n i t y [14]. I n m a n y cases YA together w i t h ±A gives the full set of units. A group intermediate between YA a n d UA is Q.A, the elements ofUA w h i c h m a p v i a a char- acter Xd hito the real cyclotomic units, fi(Cd), of Z[£<$]• For p r i m e p, flC p = YC . p F o r a p-group A, c(A) = | f L 4 / Y . A | , the circular index, turns out to be t r i v i a l for regular p [9]. W h e n p is irregular, VtA/YA is a p group, n o n - t r i v i a l w h e n \A\ > p. A m e t h o d for p r o d u c i n g 5 Chapter 1. Introduction generators of this factor group for \A\ = p and p irregular is o u t l i n e d i n [5]. 2 W h e n we t u r n to groups whose orders have different p r i m e factors, we enter quite different territory. T h e case where A is Ci , a cyclic group of order Ip w i t h I a n d p distinct primes, was p first given a general treatment i n [6] for even order a n d i n [7] for o d d order. T h e m e t h o d o f analysis is t h r o u g h another pull-back w h i c h splits off the c y c l o t o m i c r i n g Z [ £ / ] : p nc ^ lp >nCi ] P YC xYCi HF,[C ] x IF [C,]. p P P T h e character xip splits the discussion into two separate considerations — its image i n a n d its kernel, the subset of Q,Ci m a p p i n g to 1. T h i s gives c(Ci ) as the p r o d u c t o f factors P p arising from each. T h e image of this kernel has a certain index i n YC x YC\. T h e failure of the order of the image of P YC x YCi i n the finite r i n g \Fi[( ] x VF [Q] to m a t c h this index measures a defect i n this r i n g , a n d P p p defines the invariants di(p) a n d d (l), closely related to the di(( ) a n d d (Q) i n [7]. T w o other p invariants p p ei(( ) a n d e (( ), compare the orders of the images of YC x YC\ a n d £1{( ) x fl(Ci) p p p P P i n these finite rings. A n easier invariant w h i c h comes into prominence is ro = (I — l,p — l ) / 2 w h e n Ip is o d d a n d = 1 for Ip even. I n [7] i t is shown that c(Ci ) = di(( )d (Q) w h e n m = 1. p p p However, w h e n m / l , only a d i v i s i b i l i t y statement was obtained. I n C h a p t e r 5, we are able to get a precise c a l c u l a t i o n for c(C )=md {nll)d {n/p) lp l (^, l \Q((i )/Y(Q )\ p p ^) . I n the following table is a survey of recent results showing where n 65 74 c(C ) 2 3 2 n 209 217 221 226 19 3 4 9 n c(C ) n 85 91 3 133 p 145 4 7 235 247 253 259 265 291 301 305 3 3 11 3 2 73 24 8 3 6 183 c(Ci ) is n o n - t r i v i a l . 143 3 146 leading to 185 44 187 2 194 3 7 202 205 3 4 365 394 2 3. Chapter 1. Introduction In some of these cases, c(C ) is s i m p l y equal to m. A final d e t e r m i n a t i o n of the c i r c u l a r i n d e x n seems to depend o n a case by case evaluation of group orders i n finite rings. P a r t of the procedure used here involves factoring integers w h i c h often become quite large as n increases. F o r these calculations M a p l e V was used. T h e m a i n focus of this thesis is i n extending the theory to encompass Z C „ , where now n = lp, r s a p r o d u c t of a r b i t r a r y powers of the distinct primes I and p. A g a i n , the analysis proceeds v i a a p u l l - b a c k (PBO) s p l i t t i n g off the highest cyclotomic component Z [ £ ] : n TLC ^ n • Z[Cn] 6[ SL ^n/l P n/lp • ~ x • x x T h i s is the subject of C h a p t e r 3. T h e entry i n the lower left is as well a p u l l - b a c k ( P B 1 ) , g i v i n g a direct p r o d u c t o n l y w h e n the group Q,C /i n p is t r i v i a l . T h e finite r i n g i n the lower right is no longer semi-simple for rs > 1. T h e case n = 4p is worked out i n C h a p t e r 4. Here P B 1 gives the direct p r o d u c t SlC^p x 1, allowing a first peek at the s i t u a t i o n involving higher powers i n a somewhat s i m p l e setting. T h e m a i n result here is T h e o r e m 4.7, g i v i n g c ( C ) = d (Cip) 2 4 p 2 d (p). 4 T h e group ( I F 2 [ i ] / ( $ ( t ) ) ) is the direct product of a group of o d d order isomorphic to IF2[Cp] 2 x p a n d a n elementary 2-group. T h e new invariant di(p) is a 2-power related to the order of the intersection of the image of VLC p a n d this 2-group. A m e t h o d for its c a l c u l a t i o n is o u t l i n e d 2 a n d a table of values for the first 100 o d d primes is given. C h a p t e r 5 deals w i t h the surjection YC n ~* Y(C„)" 7 Chapter 1. Introduction i n d u c e d b y Xn- T h e first section looks at Y(( ). There is a difference i n the a r i t h m e t i c between n the even a n d o d d cases necessitating separate treatments. T h e invariant m " , i n t r o d u c e d i n [7], shows u p i n the calculations. T h e invariants r ( — 1) 1\ pm" \ 4mm" VP for n o d d a n d <t>(2n<i>(p ) f Tr ( —1) 8mm" S P for n = 2 p , r s r > 2, become i m p o r t a n t . F o r n o d d we define m! to be m " / 2 i f this invariant has the value —1 a n d m" i f i t takes the value 1. F o r n = 2 p , r m"/2 i f this invariant has the value —1 a n d p = 1 (mod 2 the number m' has t h e value s ) . P r o p o s i t i o n s 5.5 a n d 5.7 t h e n r _ 1 establish the index = Y(Cn) T h e second section looks at L(( ). n of the index \Cl(^ )/L(( )\. n m m'(f>{l )(f>{p ). 2 r s W h e r e ( m , Ip) = 1, we are able to give a n exact formulation I n T h e o r e m 5.13 we have the result n HCn) Y((n) = I( mm m m \ O n l y i n the case n = 8p are we able t o give a satisfactory extension t o a case where I \ m. s P r o p o s i t i o n 5.10 gives: L(Cn) Y((n) = l I m m \ mm \,e ,e (4p)' e (8)J (8) ' 2 2 p p / if 2 f \UC4 /YCip\. T h e methods used i n the proofs can give upper a n d lower bounds for the P required index, b u t , at this stage, we cannot give a n exact answer. H a v i n g dealt w i t h the image o f the m a p induced b y Xn o n £lC /YC , n n we must n o w look at its kernel. B y the pull-back property, the latter is isomorphic t o ker/? im[L(.]Yc where im^/.jyjc is the image of a certain group Yfz of constructible units under the left vertical arrow. Its order w i l l be computed as the quotient of ftCn/l n/lp ^ C n / p x im[;./.]>Ac by ker^ Chapter 1. Introduction T h e second of these indices, w h i c h is just |im/?|, is dealt w i t h i n C h a p t e r 8. T h e m a i n result of C h a p t e r 6, P r o p o s i t i o n 6.2, gives the first index as nc -, n/p YC j n/lp n/lp YC,n/lp lirii n/lp] K Y ,C YpYc 1 Y(C \C) Y(C \V) n/p n/l ™[n/l]Yv i m Y n p [n/p]*£ Y V C w h i c h shifts our attention from P B 1 t o more familiar groups. C h a p t e r 7 is devoted t o c a l c u l a t i n g orders of groups i n the above formula. T h e first two sections are e x p l i c i t , e x p l o i t i n g i n d u c t i v e methods. T h e second section introduces new invariants gi(n) a n d g (n). p gi(n) is a n /-power a n d is t r i v i a l w h e n t h e order o f I i n F is even; g (n) is a n p p p-power a n d is t r i v i a l w h e n the order of p i n IF* is even. T h e final section concerns a different question of liftability — w h i c h units of Z C ^ , where d = Z'p ', are liftable t o Z C ^ . A new index, 7 Cn{C /kY, w h i c h divides c(C p) is defined for A; = n/l,n/p a n d n/lp, t a k i n g into account only n n those u n i t s w h i c h lift t o SlC /i x /ip $lC / . n n d = Ip, namely, that the m a p O C p L e m m a 7.11 gives the answer t o liftability for n p - » 0,Ci is surjective. p p I n C h a p t e r 8 we s t u d y the image of /?. T h e defects of the order of the images of di(n/l) a n d d (n/p) are defined i n terms p (YC /i\V) a n d (YC / \C), n those units i n n p become t r i v i a l w h e n m a p p e d t o YC p . n T h e indices iy/Y-p p a n ( c YC p a n d YC j ^ *n/y n a r e which n p defined t h r o u g h the filtration P {(YC \V) n/l x (YC \C)) n/p C (3 (YC x n/l n/lp YC ) n/p C (3 (QC n/l x n/lp nC ). n/p T h e o r e m 8.1 then gives c(C ) n _ di(n/l)d (n/p) p in/Y • iy/Ypx c (C»/;) -c (C / ) L n n n n{C /i Y c n p p L gi(n)g (n) p HCn) Y(Cn) T h e first three factors o n the right m i r r o r the ones displayed at the top of this page, the first one h a v i n g been d i v i d e d by |im/?|. T h e t h i r d i t e m o n the right as well as the last (when ( m , Ip) = 1) have a n explicit formulation. T h e second concerns groups w h i c h arise earlier i n the i n d u c t i v e process. T h e four invariants i n the first i t e m are peculiar to this stage. T h e y refer t o the orders Chapter 1. Introduction of groups in the finite rings of PBO, but, at the moment, there is no good indicator available for predicting their values. 10 Chapter 2 Circular Units 2.1 Constructible Units The first part of this section summarizes results stated in [6], and picks out details which will be relevant in later considerations. Let A be a finite abelian group and A (A) be the kernel of the obvious ring homomorphism TLA-^TL ("augmentation"). An interesting subset of U(A) is U (A) = U{A) n ((1 + A A)). It 2 2 turns out that U(A) is the direct product of ±A with this free group U2(A). Consider the exponential type homomorphism from the additive group of TLA to the multiplicative group A. This map gives valuable information concerning the ideals A A and its square A A, and the unit group U2(A): 2 (i) e induces an isomorphism A A / A A ^ A. 2 (ii) U (A) - » 1. 2 (iii) (72(A) C U (A), the group of symmetric units of TLA, i.e., those invariant under the ring + involution * : TLA -» TLA which is induced by the map z~ defined on A. l At least in the case where A is cyclic, these (ii) and (iii) imply a further restriction. Lemma 2.1 If A is cyclic, then u(x) € U {A) implies that u(x) — u'(x + x~ ) for some l 2 polynomial u'. 11 Chapter 2. Circular Units Proof: Let n be the order of A with x a generator of the group. Let u(x) = Yli=o °i x% ^ t e n e expansion of u(x) over the group elements. If n is odd then u(x) = u(x~ ) implies that l (n-l)/2 U{X) = CQ+ ^ Ci{x + X~ ), l L i=l which can be rewritten in the form u'(x + x~ ). However, when n is even, A has an single l element, x l , of order 2. Symmetry alone tells us that n 2 (n-2)/2 u(x) = c + c x / n 0 + 2 n/2 ^2 i=\ Ci{x +x~ ). l l Applying the map e gives e{u(x)) = (re"/ ) "/ . Since this has the value 1, it is necessary that 2 c 0 2 n/2 be even. In other words (n-2)/2 ^ n/2 -n/2 C u { x ) = C Q + {x +x )+ £ i=l + aT ) 1 and the result follows for this case as well. I The group of constructible units of A, Y(A), is the direct product of the free unit groups W(C) C U2(A), one defined for each cyclic subgroup C of A. Such a group is trivial where <j>{\C\) < 2, but otherwise has rank </>(|C|)/2 - 1. To describe the group W(C), we first fix a generator x for C. Where Gc is the automorphism group of C, and He = Gc/ < * >, there is an explicit isomorphism w :AH 4 W{C), 2 c w : a —r w (x). a We can obtain this element, w (x), as follows. Let a r : x -> x for a G Z . a a Let nc = \C\. He consists of equivalence classes of the form [r ] where (a, nc) = 1 and r ~ T\, a a if and only if o — b = ± 1 (mod nc)- For 7 = (a — 1)((3 — 1) € A H, we pick T ~ a and T& ~ (3 2 A 12 Chapter 2. Circular Units w i t h a, b € IN such that (a, b) — 1. T h e element i y ( x ) i n W ( C ) is given b y a f^-iKx-i) (z -l)(x -l)' Q b w i t h n (, = —(a — 1)(6 — l ) / 2 (mod n c ) . Since a a n d 6 are chosen relatively p r i m e , the p o l y n o a m i a l (x ab - l)(x - 1) is divisible b y (x - l)(x a - 1) i n Z [ x ] . T h e quotient is i n fact a p r o d u c t b of c y c l o t o m i c p o l y n o m i a l s . It has degree (a — 1)(b — 1) so that w^(x) is s y m m e t r i c a n d is i n fact expressible i n the form w'(x + x~ ) so that w (x) G U^iA). 1 7 A m e t h o d for finding a set of generators for W(C) where HQ is cyclic is described i n [7], w h i c h we now generalize. L e t n p H = Y[Hi, i=l c w i t h each Hi cyclic of order T?J a n d having generator OJJ. L e m m a 2.2 A basis for A HQ 2 (!) { ( " i i - l ) ( " i i - l ) ^ (2) { ( a i ! - n 1)(OJ - 2 l)a£ as a Z module is given by the union of the following : j i = 0 . . . n - 2}, n = l . . . n . 1 h p a\l : j i = 0 . . . ^ - 2, j 2 = 0... ni 2 - 2}, 1 < t i < t 2 < n . p ( p) ~ ^> i n = Jt = 1 • • • n» - 2 , i = 1... n } . (Where n a i p p > 2.) i=i Proof: O n e t h i n g to note is that ( a i - l ) « i _ so that, for example ( a ^ — l ) ( o : i — l ) ^ ™ ' " 1 1 + . . . + a i + l ) = 0, 1 1 is a s u m of terms i n (1) for each n j . (1) gives us Y^i=i( i ~ 1) elements. n (2) gives us Si<i <i <n ( ii n 1 ( n ) gives us p 2 p — l)( i2 n — nr=i( * ~~ 1) a d d i t i o n a l n !) a d d i t i o n a l elements. elements. 13 p sets: Chapter 2. Circular Units T h e t o t a l n u m b e r of elements is thus Up 17(1 + ( « i - 1)) - 1 = \H \ - 1 = # | C | ) / 2 - 1. C t=i T h e p r o o f is finished by showing that these elements do i f fact generate A n element i n A He A Hc2 can be w r i t t e n as a s u m of elements of the form (f5\ — l)(/?2 — 1)/% w i t h 2 each of the /3's i n f f c . E a c h of the /?'s is a product of powers of the oVs. R e p e a t e d a p p l i c a t i o n of the s i m p l e formulae (a/ - 1) = (a. - 1) + - l ) a i + . . . + (ai - 5 ( a i a j - 1) = ( a i - l)(a.,- - 1) + ( a j - 1) 4- ( a ^ (a, l)a/ "\ : 1), - l ) ( a j - l ) ( a j - 1) = ( a , - l ) ( a i - 1)QJ - ( a i - l ) ( a j - 1) (at - l)(aj - l)a f c = (ai - t h e n allow us to express any element i n A He 2 l)(aj - l)(a - f c 1) - (a, - l)(aj and - 1) as a s u m of elements i n the sets o u t l i n e d above. I 2.2 2.2.1 Cyclotomic Units Arithmetic W e start by i n t r o d u c i n g the following definitions: Gd = {r a : (a, d) = 1,1 < a < d}. T h i s is the galois group of 7Z.[Q]. Gdi,d = { a '• a = i-^ + l,(a,d) = 1}. T h i s is a subgroup of the galois group Gd, defined for T di | d. W h e r e d = n, this may be shortened to G' d Gpspp' — {r a : a = i2 s + 1 or even . , (a, 2p) = 1}. T h i s is the only exception to the above definition, useful i n s i m p l i f y i n g a r i t h m e t i c w h e n n is even. m = ((f>(l ),(f>(p ))/2 unless n = 2p where we take m = 1. v {(d) = Cd~(Cd r s l s a (a, d) = 1 a n d i = -(a — l) T a _ 1 i where — l ) / 2 i f a is o d d or i = —(a — l ) / 2 ( m o d d) i f a is even, w h i c h occurs o n l y i f d is o d d . V(Q) is the set of units i n Z[£n] generated by { v ( ( A } . These are "single quotients". E l e m e n t s 0 of V(Cd) are real. 14 Chapter 2. Circular Units W(Cd) - {v (Cd) ~ n • T ,n 1 a E G }. a These are "double quotients". W{Q) d Y(C ) a n d is i n fact C n the image of WCd i n Z [ £ ] . n I n the following l e m m a we describe some of the a r i t h m e t i c i n v o l v i n g these single quotients. L e m m a 2.3 (i) v {( )v (( ) a (ii) d b v -i(( ) a = v (( ) d ab = v >,{( ) (mod 6 a where Y(Cn)) d b E G. d € Y«„) 1 a (v) v (( ) » =v ((- ) (vi) v^iQ) = - 1 . T d a (mod y ( c » ) ) . d - 1) € A H then 2 a = Ylt=i{T d) ai ™a(Cd) if a is taken mod d. _1 a (iv) v {QY»- (vii) If W(Q)). = u (C<i) j where the inverse d (iii) M C n ) ) a (mod d = v (Q) Ui b = v 0 ai b = YlLi (Cd) = ± Ui=o i a = ± l ( ) mod d a ««. (Cd)- P r o o f : (i) W e have = [-«-l)+(*-l))/» Va(Cd)v (Cd) % C b = ^ ( a 6 ~ 1 _ ( a _ 1 ) ( 6 _ l ) ) / 2 ( m ° d d)] C d ) (r.-X) (7.-l) (l - + Q)(TaT -l)-(T -l)(T -l) = W 6(Cd)^[-(r -l)(r -l)](Cd)0 a 6 (ii) T h i s follows from (i), s u b s t i t u t i n g b = a - 1 . (iii) T h i s follows by i n d u c t i o n using (i). (H () v (vi) V a ( c d y ^ = (c ' { {a 1)/2 m d o d ( i - c r - r = (r — 1) + 1, so this follows from (iv). b b = & i ^ L = - l . 15 1 = « r.-i)(n-i)(c«)( b a b n d Chapter 2. Circular Units (vii) T h i s follows essentially from (i). Indeed, i n A i 7 d , 2 k k - !) = £ ( i=l r * " ) + ( < ^ - 1) - (r J ~ l)(r T ai 0 2 - 1) i=3 = < (m - ) x T E ^ n t x J'=2 =1 k = in - i) 1 «o - 1)(Ta >- 1 } +Yl v a 3=2 where ctj = — (T^- 1 a) ~ ^)( aj ~ 1) T A ! / ^ for 2 < j < e 2 T h i s means that (T;, — 1) G A i ? d , 2 so that Tft = 1 i n iT^, i.e., 6 = ± 1 m o d d, g i v i n g «6(Cd) = ± 1 U s i n g (i), we have k «d)w (Cd) Q 2 i=l = w = n?=i«* ^II ( , i , Q i^) v (Q)w (Q). b a I T h e r e are some i m p o r t a n t differences between the structures of the factor groups d e p e n d i n g o n whether d is even or o d d . Lemma 2.4 When d is odd, there is a map rG D - h y ( C d ) WIQY When d is even, there is a map G V(Q) <r _ > ~* w(( y 2d d 1 d 16 V(Q)/W(Q) Chapter 2. Circular Units Proof: F r o m (i) of the previous l e m m a we have a m a p T h e following argument uses this along w i t h the i s o m o r p h i s m G ~(Z//cZ) . x f c Now -i Vd a((d) For f d+a tf^WlZ^*- Sd = + = Q v {Q). dl2 a d o d d ( ~ ^ = (C<f) ' ~ 1» establishing the first map. d 2 1/ 2 >d For d even Q d/2 = - 1 . B u t then v d+ {Q) = -«d+a(Cd) = v (Q), establishing a m a p 2 a a ° 2 d ~* Wfcd" For d either even or o d d , Cd - - Y^ v - {Q) = d {d a l)/2{J a _ f-a-(d-a-l)/2 " Q d (1 ~ i-C- _ ja-(d—l)/ = Cd) C 2 ( 1 _ C d ) T - _i = -C " ^(C )d/2 d W h e n d is even, Vd— (Cd) = a a(Cd)- v d In particular, = vi(Q) = 1 so the second m a p of the l e m m a is established. T h i s is also sufficient to describe this c o n d i t i o n since a is o d d a n d a(d — 1) — (a - l)d + d — a = d — a W h e n d is o d d , this c o n d i t i o n gives Vd- — — a{Q)v a (mod 2d). I Remark. W h e n d is an o d d prime, the m a p described above is injective as well. However, i n some composite cases, e.g. for d = 55, the m a p may fail to be injective. 17 Chapter 2. Circular Units 2.2.2 N o r m R e l a t i o n s T h e purpose of this subsection is to translate information obtained from n o r m relations i n Z [ £ ] n to a r i t h m e t i c i n the real s u b r i n g of Z [ £ „ ] , a n d i n p a r t i c u l a r i nfi(Cn)>the real c y c l o t o m i c units. L e t kd | n. W e w i l l use n o r m relations between the units of Z [ £ „ ] i n the f o r m n (2.1) (i-d) - = ( i - o . T a=in/{kd)+l,i=Q...k-l T h i s varies from the formulation used i n [1] a n d [7], b u t is of the form used i n [4], a n d is more aligned t o the maps we are using. I n [1], Bass set out to prove a conjecture o f M i l n o r that these generate a l l relations between c y c l o t o m i c units. However, i n [4], E n n o l a shows the group generated b y these relations m a y fall short of the full group of relations b y a n i n d e x w h i c h is a power o f 2. I n our case, however, where n has only two p r i m e factors, we do not have these e x t r a relations, cf. [11] a n d [12]. Where n = l p a n d 1 < k < r we denote G' = {r : a = i(n/l ) + 1,0 < i < l }, w h i c h is a r s k lk k a subgroup o f the G a l o i s group G . U s i n g (2.1), we have n (l-Cn) E G <* =(1-Cf). T o use (2.1) w i t h the Galois subgroups G^ a n d G * of G p L e m m a 2.5 n (2.2) we need to do a l i t t l e more work. From (2.1) we get (i-c«) E G (i-ci') (2.3) = (i-Cf) (2.4) " = and (l-Cn) » E G where I 1 is taken modulo p ; p s 1 S is taken modulo V. 18 Chapter 2. Circular Units Proof. A n i n d u c t i v e proof o f the first is as follows: For n = lp we have s (1 - C ) = (1 - £ )E{T.:a=ip*+l u<,-<l} l n = n (l-C„) > E G '(l-Cn) "l T where a; = 1 ( m o d p ) b u t / | a;. T h i s means that aj = 7 • Z s ( l - C n ) » ' =(l-Cn) ' 'T T T - 1 (mod p ) a n d s =(1-C !) '- 1 T 1 7 g i v i n g the result follows for r = 1. A s s u m i n g the result for n = / r _ 1 p , we proceed t o n = l p . s T s (1 - Ci") = ( l - c )£{r«:«=ip'+i,o<.-<r} n = (1 _ ^ ^ E G j r ^ _ £ ^ \ £ { T : a = l (mod p ),l\a} 3 a = (1 _ ( ) E G r (]_ _ ^ ) / - i E t f . ^ ^ + l ^ i ^ ' - } T 1 n = (l-Cn) E G "(l-C^) '- , T 1 by the i n d u c t i o n hypothesis, finishing the proof. 9 F r o m o u r definition v (( ) = a Q - v (Q) {a d 1)/2 a A p p l y i n g the operator r — 1 to (2.2), for example, where r G G , gives a a n for some power j o f Q . Since we are now i n the real s u b r i n g of Z [ £ ] , we must have n n T h u s , it is o n l y i n the case where n is even that the p o s s i b i l i t y that = — 1 occurs, a n d the following lemmas that we calculate where this is relevant i n o u r a r i t h m e t i c . Lemma 2.6 For r G G , 1 < k < r, we have a n Va((n) '< =V (C< ), EG k k a 19 = ±1. Chapter 2. Circular Units except in the cases where I = 2, k is odd, and a = 3 (mod 4), where Proof. T h e o n l y cases we need to be concerned w i t h are where n is even, i n w h i c h case a must be o d d . P r o m we derive T h e net power of o n the left side of this equation is \ ^ 2 \ l k T h i s is congruent to 0 m o d n w h e n I is o d d or w h e n / = 2 a n d a = 1 m o d 4. However, i t is congruent to n/2 m o d n i f / = 2 a n d a = 3 m o d 4. I T h e next l e m m a works out the case for k = r using (2.3). Lemma 2.7 For I odd or for 1 = 2 and r > 1 =«a(Ci ) - '" r Va(Cn)^ ' G r 1 T 1 (2-6) Proof. F r o m a - C„) '-( 1 ) I : g " = a - cf) --" - <"'>, <T (1 T we derive cF 1 E a %,.(c„)^" = VcT)'--*-'. (2.7) N o t e that G = { r - 1 : b = i p + 1,0 < i < / , {b, n) = 1} s r r 6 = {r - 1 : = ip + 1,0 < i < Z } - {r - 1 : 6 = / • r s b r b 20 1 + ilp , 0 < i < l ~ - 1} (2.8) s r l Chapter 2. Circular Units For the second set, the inverse of I is taken m o d p , so that = 1 m o d p , but not necessarily s s m o d n. However, the 6's are = 1 m o d p . s T h i s gives for the net power of ( £(,„• + g-l on the left side of (2.7) 2 n 1 , - ' £ • n> + I ¥ -r<i ) - , - ) - 'JLLW - i=0 i=0 _ i {v -y- ) r l s P 2 = 0 since 2 | V — Z cases. r _ 1 (modn) (2.9) for any o d d p r i m e I a n d for / = 2 i f r > 1, p r o v i n g the l e m m a for a l l these I W e note that for I = 2 a n d r = 1 that MC ») = -MC„ ) 2 1-T[(pS+1)/21 2p - A n a d a p t a t i o n of (2.3) w h i c h we w i l l use, is (1 - C n ) E G " - « r ) ( l - Cn)* (r) = (1 - GO - "'" , 1 7 1 w h i c h gives II in fi(C ). n With n odd, "«0Cn) ( C n H l - Cn) )~ =(±l)v,-i 2 (Ci')" the first factor o n the right is + 1 , but for 1 n even we again have some work to determine the cases where this factor is +1 a n d those where it is —1. Lemma 2.8 We have f n «*««)] (CHI-a) )^=«,-i(cr) 2 \Ta€G,T J for (i) n odd. (ii) p = 2. (iii) Z = 2 and p = 1 (mod 4) with r > 2. s (iv) / = 2 and p = 3 (mod 4) with r = 2. s 21 _i Chapter 2. Circular Units . For the other cases with n even, we have II i.e. for (v) ««(C») (CnHl - C n ) ) ^ = - t ; , - ! ^ ) ' , 2 1 1 = 2 and p = 1 (mod 4) with r = 2. s (vi) Z = 2 and p = 3 (mod 4) with r > 2. s Proof. Since the factor ± 1 must be a power of £ „ , we have that the factor is 1 for o d d n , g i v i n g (i). W i t h p = 2, we use the galois subgroup = {r :a = i2 a + 1, (a, n) = 1}. W e then get the s + l conversion C II MCnHCn^l-Cn) ) 2 2 = (Ci')" 1 where the power k has the value ( m o d n) r _ 1 i 2 s + T—v 2 =0 Il -•l r E 1 2=0 _ r(i r + U2 1 T r _ Z (Z - Z - ) 2 r r X 1 - \)2 i {r~ l r - s+l 2 4 r 1 Z^l-Z" ) r — r-i- s+1 = 4 + z - ^-!) 1 ~2~ 7 + 1 2 z (i - z - ) r 1 2 s + 1 4 = = ^ - i)2 0(Z ) - 11+ S+1 Z ^ - ^ Z - 1)2* +1 4 = 0 T h i s is sufficient t o prove (ii) . W h e r e Z = 2, we use Gr = { 2 : a = 2ip + 1,0 < i < l ~ s Ta T x - 1}. T h e n the power k is given ( m o d n ) b y 2 r 1 — 1 or—1 i o—1 £ ^+V~ ^-i=0 22 2 r - (2 10) Chapter 2. Circular Units Since this final t e r m must be a m u l t i p l e of 2 , we must choose 2 r ( m o d p ) to be o d d for the _ 1 s simplification. For p = 1 ( m o d 4), we use 2 " = {p + l ) / 2 . T h e n (2.10) becomes s 1 2 r-2( r-l _ ^ s 2 p s + 2 r-2 _ y LZ^! = 2 r 2 ~ p 3 - 2~p s r 2 + S 2~p r 2 S = 2 ~V2r = i f r > 2, b u t 0 = n/2 i f r = 2. T h i s proves (iii) a n d ( v i ) . For p = 3 ( m o d 4), we use 2~ s = (3p + l ) / 2 . T h i s s u b s t i t u t i o n i n (2.10) gives l r-2( r-l 2 2 _ s r-2 _ r ! Z _ § E 1 = 2 ~ p 2r + 2 3 - 2~p S T 2 = 2 - (2 r = 1 0 = n/2 r 2 S + 2 - (3p ) r 2 S + l)p 2 s i f r = 2, b u t i f r > 2, p r o v i n g (v) a n d (vi), c o m p l e t i n g the l e m m a . I 2.2.3 Let V n Real Cyclotomic Units be the m u l t i p l i c a t i v e group generated by {±C„, (1 - C ) T h e c y c l o t o m i c units of Z [ £ ] , n C(Cn), are :K « < n - 1}. those units of Z [ £ ] contained i n n are interested i n the real cyclotomic units ft(Cn) = C(C„) n IR, 23 (see [14]). W e Chapter 2. Circular Units i.e., those units of C(Cn) which are possible images of units of £lC . n O(Cn) contains subgroups of the form V(Cn), where d is a proper divisor of n, but also contains subgroups of the form (1 - Q)^ where V \ a and p \ a. The group C(Cn) is then in the form s cf n ( / n z (i-c) )(n^)), z \l<a<n-\,l \a,p'\a \d\n J T J noting that —1 G V(( ). n In the following lemma, we give a more tractable description of f2(C ). n Lemma 2.9 fi(C„) = (Q^l - { ) )* * 2 VitfWtfWCn). /n n Proof. In the norm relation n (i-c»r=(i-cr ' ). p 1 6=m/(rip i)+i,o<i<np*i a all of the r 's are in G a (1 — Cn n provided r\ < r and s\ < s. These are the only cases for which ) is a unit in Z[£ ] and give l p 1 n ( 1 - C ) = (l-Cn) ^ 1 r II 1 (l-Cn)^" 1 b=in/(l ip i)+l r s which, modulo torsion, is in (1 — Cn) V^(Cn)- This simplifies the middle term in the above Z representation of C(Cn)Further, applying the operator r — 1 with T G G , we derive a a(Cn ) v = lpn n q ±V (Cn)^ { T b : b = i n / ( a - l r i p 3 1 ) + 1 ' 0 i < l n p € V(C„). For d = l p T Sl with 0 < s\ < s, we use n b=in/(l p i r Here the r^'s are all in s (i-c„ ) r T t )+l,0<i<p*i and we derive v(C) c 24 v(C). =(i-cr p a i )- S l } Chapter 2. Circular Units Similarly, for d = l p , Tl with 0 < r\ < r, we have s V(^) C V(C ). n Using these further simplifications, we derive C((n) = C„ " (1 - C n ) n C i > ( C r 2 / Z Z )V(( ). n The final three groups is this product are real. One of the final considerations is determining when a unit of the form —( ) b n is real. Note that C ( l - C n ) = C n - 2 + Cn 1 is real. For a unit of the form ^ ( 1 — ( ) = n of C n + 1 1 2 — Cn + 1 to be real, £n must be the mirror image the imaginary axis when we make the sustitution m = e ™/ . This occurs only if 2 71 2 | n and a + 1 = n/2 — a (mod n). Thus n/2 = 2a + 1 (modn) must be odd, i.e., n = 2p . This s gives, for n = 2p , s ft(Cn) = ( d pS 1)/2 (l-Cn)) V(C)V(Cr)^(Cn), Z and for all other cases Lemma 2.8 of the previous section reduces this representation further to the form stated in the lemma. I 25 Chapter 3 Setup 3.1 Derivation of Pull-backs Let C be a cyclic group of order n with generator x. The factorization n x -l = H$ (x) n d d\n leads to a decomposition where the component maps are induced by the character maps Xd '• x —• Cd, an d * root of 1 1 unity. This gives a representation of the group ring <QC„ as a product of fields and leaves little mystery about the group of units of Q C . n Using 7LC <->• Q C , the above isomorphism gives an injection n n ZCn^YlnCn}. d\n However, only in the elementary case n = 1 is this map surjective. Since a unit in Z C n must project to a unit in each of these cyclotomic rings, the knowledge of cyclotomic units gives a base for study. Our purpose is to outline an inductive method for the study of these units. The map C —» C , ni n sending a generator of C n onto a generator of C , ni where n\ \ n, relates units in 7LC to n units in Z C , which, we assume, are already known. To get the corresponding injection n i 26 Chapter 3. Setup for the ring Z C , we use the same product as for Z C , with the stipulation that the map n i Xd • Z C n —> Z[£d] is trivial where n\ is a proper divisor of d. Where there is little danger of n i confusion, we will use the same notation for character maps of these different groups, i.e., in our notation we will not distinguish between Xd and x'dThe only component of this product ring not accounted for in this way is the highest component, Z[Cn]- In this section, we develop a pull-back, enabling us to eventually eventually give a description of the units in Z C by intertwining knowledge of units in Z[( ] with knowledge of n n the units in the Z C ' s . We start from the following framework, devising a method for splitting n i off this highest component. Where I and J are ideals in a ring R, we have the following pull-back diagram R/{inJ) >R/I > R/(I + J). R/J The ring R we use in this setup is Z[x]. Since ZC ~ Z[x]/(x - 1), n n we place Z C in the upper left position by having ID J = (x — l). The factorization n n x l p - l = (x- l)$, (x)$,(a;)*p(x) p extends to x - 1 - {x ' n n lv since <&/ (x /' ) = $ (^) for n = l p . n p r p s n - l)§ {x)$ {x l )%{x l >), n lv n n lll l For I we take the ideal generated by Q (x), giving n Z[£ ] — R/I as the ring in the upper right of the diagram. For J , then, we use the ideal n generated by (x /' - l ) $ , ( x ^ ) $ ( x / ' P ) . For R/J we use the notation Z C n p n n p which we will clarify in the lemma following the main proposition below. The following lemma gives the calculation for the ideal I + J. 27 n / , x n/lp ZC n / p , Chapter 3. Setup Lemma 3.1 ($ (s), (x ^ n n - l)^(x ^ (x ^)) n = fZ, n p P r o o f : For simplification, we start with the case where n = Ip. Since ( $ / ( x ) , $ ( x ) ) = ( 1 ) , we also have ( $ / ( x ) , $ ( x ) ) = ( 1 ) . Thus 2 2 p p (*, (x), (x - l ) $ , ( x ) $ ( x ) ) = ($, (s)*j(x) , $ ; ( x ) $ ( x ) , (x - l)$,(a;)^(x)) 2 2 p p p p = (^(x^Mx), (x - l ) S , t > ) * ( * ) ) %(x )$ (x), l P p = (^i(»).P* (»), - l)$j(z)$ (a:)). P P Next we note that G ( Z $ , ( x ) $ ( x ) , p $ , ( x ) $ ( x , p ) ) C (Z$,(x),p$ (x)), p p p and lp€(lp$l{x),lp$ {x))c(l$i(x),p$ (x)), p p so we have ($pq(x), (X - l ) $ , ( x ) $ ( x ) ) = (Z$;(x),p$ (x)) p p = (Zp,Z$,(x),p$ (x),$,(x)$ (x)) p p = (Z,$ (x))(p,$K^)), p finishing the proof for this case. However, if we replace x by n/lp x i n the above polynomials, these calculations are still valid in the ring 7L\x\ giving (<Mz),(x /'P - l)$i(x )$ (x )) n n/lp = (l,$p(x )) n/lp n/lp P To complete the proof, we rewrite <J> (x / ) and $ / ( x / ) as 3 y ( x n /p n ( p (p,*l(x )) n,lp *) and <&;>-(x r pS p tively. I We can further describe the structure of R/(I + J). Lemma 3.2 Z[x]/(7 + J ) ~ F < [ x ] / ( $ 3 ( x P r _ 1 )) x F [x]/($,, ( x ^ ) ) . 1 p 28 • *) respec- Chapter 3. Setup Proof: This follows from the Chinese remainder theorem since I and J are coprime. I We now have the main vehicle for our inductive analysis of units. Proposition 3.3 Z C is a pull-back n TLC n zc , x n/ in the following diagram: > Z[C„] : n//p zc •F,[t ]/($ .(*r )) x I F N / ( M < ) ) • 1 n/p 1 - 1 p P Proof: This ties together the above considerations. The ring in each position of this diagram is derived as the quotient of a polynomial ring in one variable over Z by an ideal. The maps I arise from maps from the variable in on case to the variable in another. The last step is to give a description for Z C / / x „ ^ n L e m m a 3.4 Z C / j n/lp ~^-C / is a pull-back n p ~%-C n x /i n n ZC / p n (( n/i x P __ i)$ ( n/jp)) p s a n d Jl = n diagram: > %-Cn/lp- R = Z[x] with new ideals I' = {x l n l — l) = - 1 ) $ , ( W P ) ) . That I' D J' = ((x /'P n x /p p n ( n/p _ i) = l)$/(a;"/'P)$p(x / )) is clear. I'+ J'= n > ~%-C n p ^Cn/p Proof: In this scheme we are again using ZC / . in the following x n p ( W ? - l) since ( $ , ( x / ' ) , $ ( x / ' P ) ) = (l). n p n p I We can apply the functor fi to each of these pull-backs. Applying it to the second defines a group which we label fiC // ~x /i n n fiC / : p n p x n/(p ftC / n p QC / • fiC„/{ >• ClC /i . n p n 29 p Chapter 3. Setup The images of £2C /j x /i n ^^n/l n nC / P n in fiC„/j, SlC / and ClC /t we will denote fiC^, fiC^j and p n v n p respectively. We rewrite this pull-back n/l nc:n/p which we label PB1. In the cases where £lC /i n the qualifier p = YC / , fiC„/j n ip = ^ C ^ , and we can remove / p from all of the groups in this pull-back. However, without this condition, we dare not be so bold. Later, we will be able to prove that for some other cases as well, this qualifier may be removed. For the first pull-back we use an observation to clarify the entries in the lower right. QC n consists of elements which are of the form u(9), where 6 = x + x~ . Except where p = 2, we l s can rewrite the polynomial in the form x-^l %s{x) 2 %s{0). We then use the notation T \e ] linll for p which contains the image of VtC . ^/(^(c^ )), 1 Where p = 2 we will simply use the notation Fi /i[0] to s n %n denote the image of the inversion invariant elements of Z C / r - i in IF/[t]/ (<H> s{t lr p 2 The first pull-back, applying the functor O, becomes • fi(Cn) SlC n ^Cn/J X /i n p QC / > Fl,n/l[0p] n p x ^p,n/p[^l], which we label PBO. The map across the top maps the generator x of C to Cn- The map on n the right is derived from group maps sending a generator to a generator. Every element in the group in the upper right can be written a a polynomial in ( + (~ . The map on the right sends 1 n Cn + Cn to 9 on one side of the product and 0/ on the other. 1 P 30 Chapter 3. Setup T h e group of constructible units is of finite index i n QC a n d using L e m m a 2.2 we are able to n w r i t e d o w n a m u l t i p l i c a t i v e basis for this group. I n our investigation we want get a d e s c r i p t i o n of u n i t s outside this group. W e use the m a p across the top to split this investigation into two parts: (1) W h a t we t e r m the question of liftability looks at the image of this m a p , w h i c h we l a b e l L((n)- W h i c h units i n Z[£ ] may be lifted to units i n 0 C ? T h e group YC n is its image, Y(( ), n in Z[£„]? n n being known, what W h a t conditions determine w h e n the factor group L(( )/Y(( ) n n is n o n - t r i v i a l ? (2) T h e investigation of the kernel of this map is more involved. Because we are w o r k i n g i n a pull-back, this kernel injects into the group o n the lower right. C a r e f u l consideration of n o r m relations i n Z[£„] allow us to describe the kernel of YC . n However, we need to get a d e s c r i p t i o n of image of this kernel i n the group i n the lower right. I n the cases n — Ip, 4p, 8p, 9p or 36, the u n i t group U2C /[ n p u l l - b a c k P B 1 . T h e u n i t group QC /i n x /; S7C / n p n p p is t r i v i a l a n d we do not need the is s i m p l y the direct p r o d u c t 0,C p n x fiC / . n p W e c a n t h e n use the m a p S described i n [7] for the analysis of units. However, for other values of n we need a more general m e t h o d . T h e e n d game i n this p r o g r a m looks at the c o o r d i n a t i o n of images from the upper right a n d lower left i n the finite rings i n the lower right. (1) a n d (2) tell us w h a t to look for, b u t a final d e t e r m i n a t i o n for possible non-constructible units requires c a l c u l a t i o n of these images. W h e r e n = lp these rings are semi-simple. W h e n the powers of these primes increase, we get n o n - t r i v i a l radicals. 31 Chapter 3. Setup 3.2 Description of the Finite Rings O b t a i n i n g precise knowledge of non-constructible units requires a n analysis of interaction between images of Cl[( ] a n d £tC /i n n xp n p ClC / i n the finite r i n g n p Fi[<i] IF N P x (v(*r )) ( M O ) ' -1 K n o w l e d g e of the structure of these rings w i l l not give us such precise i n f o r m a t i o n , b u t w i l l enable us to compare group orders a n d determine, is some cases, the existence of non-constructible units. T h e i n f o r m a t i o n w h i c h is of interest to us is the structure of the group of units i n m (Me'' )) -1 w h i c h have i J / n o r m 1, i.e., those w h i c h are the possible images o f units i n Q,C /i. B y s y m n r n metry, we w i l l then have the corresponding i n f o r m a t i o n about the other r i n g i n the p r o d u c t . A u n i t u i n QC /i is i n the form u(y + y ). 1 n For p s ^ 2, we c a n w r i t e $ s(t) i n the form p t<t>(P y $* (t + i " ) . T h e image of 9,C /i w i l l lie i n the smaller r i n g 3 2 1 3 n HO] where 6 = t + t~ . l It is i n these cases a n d i n this restricted r i n g that we c a n determine the existence o f non-constructible units b y this comparison of group orders. Lemma 3.5 (1) Where n ^ 2l and f is the order of the subgroup < TI > of H s and g = h s j f, T p Furthermore, this group of units has structure r-1 i=2 In particular, the l-subgroup has (f>(n/l)/2 independent generators. (2) Where n = 2l , r \T ^[er\ = l 32 (i-i)i^ - - \ r l i p Chapter 3. Setup Proof. W h e n r = 1, the r i n g is semi-simple. I n IF/[0], the p o l y n o m i a l $ ^ ( 0 ) has g factors, each of degree / (see [7], page 115). Consequently, ¥i[0]/ ($* (f?)) ~ (IFj/) a n d 9 s HQ] x ( ("-i))" C w i t h order {l - 1)». f W h e n r > 1, the r i n g is no longer semi-simple. W h e r e $^{0) t o r i z a t i o n o f Q*(0) i n Fj[0], the p o l y n o m i a l s $*^(0 nf=i^(j)W * * s n e f " a c *) are s t i l l relatively p r i m e , so lr (*j.(» = )) M(v " 0' r _ 1 , l and we w i l l analyse the j t h component ring. / r—1 r _ 1 Since i n Fj[0], $ y ) ( ' ' *) = > 0 of degree less t h a n Z r t h e / ( r _ 1 _ 1 ) / p o l y n o m i a l s of the form f(0)$* {0) (j) ^ m a p injectively into the radical, "R-(j), o f this r i n g . T h e set 1 + consists o f units of / power order. T h e u n i t group o f the s u b r i n g has order if — 1 p r i m e to I. Since ^- -D/ /_ a /(r- -i)/ 1 1 1 ) + r- / 1 = z ) the order of this component ring, we conclude that it has a unit group of order l^ ~ W(lf 1 and a r a d i c a l of order l^ 1 — 1) T h e full group of units, therefore, has order r ( Z C ' - - ! ) / ^ / - l ) ) = / ( r - - W ) / 2 / _ 1)9, 1 9 1 (/ as stated. For the second part o f the l e m m a , we have already given the structure for the subgroup o f the u n i t group w h i c h has order p r i m e to I. W e now proceed w i t h a n i n d u c t i v e p r o o f o f the statement for the /-subgroup. 33 Chapter 3. Setup If r = 1, this /-subgroup is an elementary /-group of order /«K'P )/ and consequently is isos 2 morphic to (^(I-IMP')^ ^ establishing the result for this case. If r > 2, the elements of order / are represented by Z3(' ~ *(p')-' *G'")) = J5*C'"V) polynor 1 r-a mials of the form l+f(9)% {9 ~ ). These are the elements in the kernel, K _ i , of the map ir 3 r Fi[*] (*p.(* Ht] x ,r_1 )) x (*p.(t''- )) a From our induction hypothesis the /-subgroup of the group on the right has structure r-2 i=2 and <j)(n/l )/2 independent generators. Let gi(9),g2(9),... g,p( /i^(9) be polynomials in Fj[0] 2 n which are independent and generate this / subgroup in Fj[i]/ ( $ » (£' ) J . We will prove that p they are independent in Fi[t]/ ($ p s(t as well. l There are Z s C ' " ' ^ ) - ' ' " ^ * ) ) = l\4>{l ~V) polynomials of the form h(0') = 1 + f(0')*$« (^'~ ) 2 r 3 • r _2 \ X with the degree h(6 ) less that V~ (j){p ). h(9) is of order / in Fj[t]/ ($ »(t.' l l )J , so, for some a p integers a\, a , . . . 2 V)/2 h(#) = II (mod^(0 - )), r 2 i=l in ¥i[9]. But then «(J—V)/2 M**) = II (&W) lai (mod^^- )), 1 so that cosets of the gj's generate a subgroup, K , of order lW 2 p S 2 the above map. Hence, the gj's are independent in order l ki+l )/2 j (/pS) K i=2 34 r ki x f[(^-0 ^ ^ " r K _ i , the kernel of If g% has order l in Fi^/i, so the subgroup generated by the gj's isomorphic to (O-0^ n v) (/i lpS)) • in F^n/p, it has Chapter 3. Setup Now, since K _ i is an elementary /-group, we can write it as a direct product r K _! = Ki x K r 2 for some group K i of order j H W V J - W - V ) ) , We then have that this /-subgroup of is isomorphic to i=2 i=2 as stated. As a further check, the order of this group is (r-i)^(<p*.)+Er; ('- )(*(' P*)-*(' - p')) 1 i (2) Fi,2l - [6\ T l i s i i i iT.lZlWv*) 1 2 = /» -i. r_1 = the image in F/[i]/(l +t) ~ of inversion invariant elements of r 1 ZC jr-i. 2 Note that the element c 0 ciy + c y + ... + q r - i . x y ' ' " - + c j r - i j / ' ' 2 + of ZC /r-i c - 0 2 1 - 1 cjr-i + (ci - r 1 _ f c + c,r-i y ~ r + 1 + ... + c y ~ ' 2r 1 + 1 1 2 2 + ciy '" 2 - 1 - 1 maps to the element of Fj[t]/(1 + of £ 1 2 C i r - i . i ) * + (c 2 - qr-i_ )i 2 2 + ... - The coefficient of i for 1 <fc< fc (c 2 - q r - i _ 2 ) t r - 1 ~ 2 - (ci - qr-i.!)*'"" - 1 - 1 -1)/2 is the negative of the coefficient . Taking into account the constant coefficient as well, we have that \^2l^[0}\=l ^~ - . 1 1 1)+1 Elements in the radical, 1Z, of Fipi,T-i[0\ are those divisible by 1 +1. For these, the constant coefficient is determined by the other coefficients, so that this radical has elements. Thus \{F?)[l + 1l)\ = {l-l)lk - - ). r 1 1 Since then \K\ + |(F )(1 + TZ)\ = |^i jr-i[0]|, ( F f )(1 + TZ) is in fact the full unit group of X )2 ^l,2l - [0] and the proof is complete. r l I Of particular interest to us in the group ^ i , / : is the subgroup n norm 1. The following lemma describes the factor group. 35 the elements of H /i n Chapter 3. Setup L e m m a 3.6 The group Fl,n/l •r-U) is cyclic of order I — I if r = 1 but is an elementary l-group of maximum order l r Proof. W h e r e r = 1, F^n/i m o d u l o Z, where 9 s = ( p a Fj[0 «], the finite r i n g o b t a i n e d the r i n g p + 1 ^ . pS by reduction T h e galois-norm m a p 1 pS Z[# ] if r > 1. F|[0 .] - • F * X P is surjective (see, for example, [7], page 127). T h i s establishes the result for this case. Where > 1 we use a different consideration. T h e r i n g r ¥i[t]j ^$ p a(i is o b t a i n e d form r the r i n g by r e d u c t i o n m o d u l o /. W e w i l l calculate norms first i n this r i n g t h r o u g h a process described i n the next l e m m a . L e m m a 3.7 The H -i s-norm lr of a symmetric element in Z[y]y/ (^> s(y a r _ 1 p + g K-*-l-°r-i i=l y where a _j is the H^-ips-norm r a -i r g j=r-i j p r > J of the image of this element in Z[<^r-; ]. Furthermore, each p3 = 1 mod I and each coefficient in this sum is an integer. Proof. Since $ « ( y ' *)j is lr p *) = Ilj=o ^lip {v)i s w e have an injection Z[y]/(v(/ ))n2[Cpy]. _1 j=0 36 Chapter 3. Setup Thus, the norm in the ring on the left is completely determined by its images in the component rings on the right. Notice that = * l ( / " " v 1 ) * l , ( / " ) v = *j(/" " )*i,-p.(y). v Thus the product flj^r-i for k > r — i. In the ring 1 ®V(y ') takes the value l in ~2.[Q ] for A: < r - i, but 0 in ~Z.[Qk a] Z[£jfe»], 0 < k < r — 1, the above sum evaluates to p % kpS p cir-i + (a _2 - a _i) + ... + (a r r k p — a +i) = a , k k and we have proved the first part. For j > 0 we have = (^(/ P S ),^(/ = (/,$p(/ since + l p S + V _ 1 ),^ 3(y)) P - ),$ (y)) 1 p y = Mpy(y)), ^$ (/- ^- )^p(/-v1 1 p )$;p(y ^- if j > 0, and $ (y'P- ) = 1 p $ .(y)* p J p . This gives us the pull-back ->IF([CP ^] P 37 i ) Chapter 3. Setup Consider an element i(y+y )ElL\y\j ($ *(y x = H[r-i s p Gp-i (^ i {y)^i s(y)). and its image in r p lJ+ pS Jp x H s. The iJ ^-norm of f(£<> + C ) is an integer which is invariant under l p p pa p the r - ^ - 1) elements of G , - i . The ff,,_i .-nonn of f ( C . + fa ) in Z[s>], ao, is therefore 1 1 r P p which is 1 mod /. This pull-back for 0 < j < r — 2 then tells us that each of the Oj's is = 1 mod I. This tells us that I | (a _2 — a _i). r r For 2 < i < r — 1, the element invariant under the actions of a subgroup of H -\ s lr p f(Qr-i , + p Cj7-i a) iis p 5 of order Z* , so the integer a _j is an -1 r Z* th power. This makes both a _j_i and a _j Z t h powers. Since they are as well congruent _1 l-1 r r mod Z, we have Z* | (a _j_i — a _j). r r I We now continue with the proof of the previous lemma. To compute the norm in F^n/i, we reduce the norm of this second lemma mod Z. Thus, norms in F^ /i are of the form n r—1 / i + E i=l ibi V r—1 n j=r—i Since <I>/r-i(i *) = $[(t '~ *) is a factor in each of these products, and (^>i(t '~ ) J p ^l,n/u l 2p l 2pS S = 0 in each norm value has order Z. Thus the group of norm values is an elementary /-group of maximal order Z r _ 1 , and the lemma follows. I There seem to be good reason to believe that these norm values form a group of order Z. Norm values in these finite rings was used in [7] to determine conditions for liftability. In Chapter 5, we use a different method which gives more precise information. Thus, information on these norms is not as crucial in this treatment. 38 Chapter 4 The Case n = 4p I n this chapter we make a n interlude i n the development of the general theory to discuss the cases n = 4p. T h i s is a m o n g the cases where the pull-back i n the lower right of P B O is a c t u a l l y a direct p r o d u c t , a n d serves as a n i n t r o d u c t i o n to the later considerations. T h e r e are also peculiarities w h i c h arise w h e n n is even w h i c h must be treated separately; the o p p o r t u n i t y is taken now to p o i n t out some of these. I n the case n = 4p, the groups fiCj a n d Q,C are t r i v i a l leading to the following simplifica2 t i o n of P B O : QC4 > ^(Cip) P £lC 2p x1 • F [x}/($ (x )) 2 2 p x F [t]. p I n r e l a t i n g the different components o c c u r i n g here, we focus o n the character m a p Xn w h i c h runs across the top. T h e discussion is separated into the following three parts: (1) L i f t a b i l i t y . T h i s concerns the image of the m a p Xn- F ° rn = 4p, we prove t h a t the u n i t s i n fi(Cn) w h i c h are liftable to units i n Q,C are a l l i n Y(Cn)n (2) T h e kernel. Here we characterize Y/c, w h i c h i n the intersection o f the kernel of Xn w i t h YC± P a n d track its image, i m ^ / . j Y x , i n the lower left of the d i a g r a m . T h i s enables us to calculate its index i n Y C 2 . P (3) C o m p a r i s o n of images. Here we establish conditions for the existence of non-constructible u n i t s i n 7LC± . V 39 Chapter 4. 4.1 The Case n — 4p Liftability F r o m C h a p t e r 2, the real cyclotomic units fi(C4 ) have the characterization P (C^(i - C 4 P ) ) 2 ^(C4 )^(C Z 4 P P 4 P )^(C4 ). P T h e group V(£l) ~ V(i) contains only the torsion units ± 1 , a n d so m a y be d r o p p e d out. A s well, the n o r m r e l a t i o n (i-C4 ) P ^ 1 + T = (I-T4) - » i t _1 or (i - u y (i - c P 4 ) p T ^ - = ( i - ^4p) i 2 shows that C - (l-C4 ) 6Y(C p)^(C4 ). 1 (4.1) 2 4 4 P P T h i s leaves us w i t h the simplification tt(C ) = v(tf )v(( ). 4p p Ap For n = ip we are able t o get the simple characterization of liftability shown i n the following lemma. L e m m a 4.1 in fiC 4p (i) L{(,i ) = Y(£ p 4 P ), i.e., the only units in fi(C ) which 4p are liftable to units are constructible units. (") A(C4 ) P Y(CA ) = 2(p-l). P Proof: (i) T h e first step i n the proof is to o b t a i n a n upper b o u n d for the order of f 2 ( C ) / Y ( £ ) . 4 P For r E G , a n Ta-l ^(C4„) 2 = ( c r / a - c 4 p ) 2 ) T a e (v(c )v(( )) 4 40 Ta p Ap 1 c Y ( C 4 P ) 4 P Chapter 4. The Case n = 4p from (4.1). T h u s , V(( ) C Y(( ). 2 4p The map 4p v(Ci ) x v(( ) -»sue*), P 4p where (u, v) H-> UV, thus gives us a m a p H n(C ) f(C ) 4p w(c? ) 4p (4.2) 4p v(( )*w({ ) p Y(c y ip 4p T h e order of the first group i n the p r o d u c t o n the left is 4, while the order of the second group is p — 1. T h u s , the order of Cl(^4 )/Y(( ) p F r o m P B O , any element of £lC must have image 1 i n F [ i ] . 4p ^ V(Cip) W{(,4 ), is 4(p — 1) d i v i d e d b y the order o f the kernel o f (4.2). 4p p W(C4>) Thus, — 1 £ a n d a l s o so that (—1, —1) is a n o n - t r i v i a l element i n the kernel o f (4.2), a n d we have 2 P Y((4 ) <2(p-l). P W e now o b t a i n a lower b o u n d for the order of Q(( ) / L(( ) 4p 4p by l o o k i n g at the image o f f)(C4p) i n the lower right o f P B O . T h e image o f V((,4 ) i n Fj[t]/ ( $ ( * ) ) 2 p P image o f —1 G V(£ ) is V(t), while its image i n F [i] is { ± 1 } . Since the p is (1, —1), we see that the image o f V{C, ) i n the lower right o f P B O is 4p 4p precisely V(t) x { ± 1 } . T h e image of V{(£ ) i n Fj[t]/ ( * ( * ) ) is V{t) 2 p C V{t), while i t s image i n F [ t ] is F * . T h u s , 4 p p the image o f Q(C4p) i n the lower right of P B O is precisely V(t) x F * . L(( ) 4p has t r i v i a l image i n F [ i ] . T h i s gives that (p — 1) | the order o f Q((4 )/L(£ ). p P 4p We will now show that V(t) contains elements w h i c h do not have i?2p-norm equal to 1. These cannot be t h e images o f elements of O C 2 p . Hence, the image of L(( ) 4p giving ^ > 26,-1). Since, Y{C,4 ) C L(Q , we have P P ) tt(C4 ) ^(C4p) L(U ) o(c 4 p P P c o m p l e t i n g the p r o o f of the l e m m a . 41 = 2(p-l),- is a proper subset o f V(t) x 1 Chapter 4. The Case n = 4p Let v ((4p) Q G V((4 ). T a k i n g the preimage v (x) € 7LC P ZC We first calculate the H 2 2 2 p we have a n injection p Z[x]/$ (x ) Z[( ] x Z[C ]. 2 p v (x) 2 p a — Q p{x)§ (x), 2 F [*]/* (t ). -> Z[x]/$ (x ) 4 p we look at the maps n o r m of v (x) i n the m i d d l e r i n g . 2P Since $p{x ) Av a 2p P maps to a real unit i n each r i n g o n the right. T h e H a n o r m i n each is therefore ± 1 , 2P d e p e n d i n g o n a. F o r v ((4 ) to be liftable, we must have this n o r m of its image i n IF2[a;]/<I>p(x ) 2 a P to be 1. I f the norms i n Z[£ 2 p ] a n d Z[£p], are of opposite sign, then the n o r m i n Z[x]/($ (x )) 2 p is ±.x w h i c h does not m a p to 1 i n IF [x]/ ($ (x )). p 2 2 p S t a r t i n g from the augmentation m a p Z[y] -> Z where y H-> 1, we get a m a p Z[C ] ^ Ztj,]/(* (j,)) -> IF . p P p Thus Ua(C ) = afr-W Ei/2p P (mod p). 2 Z[£ ] is a c t u a l l y the same r i n g as Z[£ ] w i t h £ N o t e now that 2p p of the second r i n g . I n the augmentation m a p above v {—y) E i f 2 p P = (-l)^- )^- )/ T h u s , i f ( _ i ) ( a - i ) ( p - i ) / 4 ( - i ) / 2 = _j_ a mod p, then For r a i> (£) a P does not have H 1 m o d p ) o r 1 = — £ for some generator £ p (—1)(° )/ - 1 a ^a(C2 ) 2 p 2 e Z . Thus ( m o d p). 4 equivalent^ (_i)(o-i)/2 > p a j sa non-square n o r m 1 i n IF [t]/ ( $ ( i ) ) . 2 2P 2 p to be i n ( ? 4 , we need a to be o d d a n d (a,p) = 1. W e w i l l further choose a = 1 m o d 4, P w i t h a a non-square m o d p . For p = 1 m o d 4, we take 6 to be a n o d d non-square m o d p . I f b = 1 m o d 4, we take a = b; otherwise, we take a = 2p + b. For p = 3 m o d 4, we take 6 to be a non-square m o d p . I f 6 is o d d , t h e n either a = frora = 2p + 6 w i l l give the required a. I f b is even, either a = p + b or a = 3p + b w i l l work. 42 I Chapter 4. 4.2 The Case n = Ap The Kernel W e first introduce some n o t a t i o n : Y(C 4p | p, 2p) w i l l denote the kernel i n the restriction %ip '• YC —>• Q{( ). F o r this kernel, the 4p character maps Xi>X2>X4> a n d X4 a r 4p YC t r i v i a l . It is composed, then, of those elements of e P w h i c h m a y have a n o n - t r i v i a l image only i n the characters \p a n 4p d X2p- i m ^ . j X w i l l be used t o denote the image i n Q C p i n the lower left of P B O of a group X i n QC . 2 a; is a generator o f the group C . YC 4p y is a generator o f the group C 2p = W{x) x W(x ) x W(x ) ~ WC 2 4p 4p = {1,T 2 P + 1 4 x WC2 4p i n the lower left of P B O . YC2 P x WC . p = W(y) x W(y ) a n d the m a p 2 P d o w n the left of P B O is characterized by x G 4p y. }. T h e two lemmas o f this section describe impj]Y(C \ p,2p). T h e first gives a c h a r a c t e r i z a t i o n 4p of this group, w h i l e the second calculates the order of Y C 2 / i m [ , i ] Y ( C 4 | p, 2p). P Lemma 4.2 im Y(C [LL] Proof. I n 0 ( ^ 4 P \ p, 2p) = ( Y C ) ~ 2 4p T 2 2 p - - P . ) we have the n o r m relations Wa(C4p) = ™ (Cp) E ( ? 4 Q 4 and M(i ) = M(ip) ~ ~\ 1 T2 P for a € H 4p a n d (3 € H p . T h u s , we may w r i t e 2 W(S)£G4-T r 2 W ( 2)1-T (I-T-I) x 2 (C \p,2p). Y C 4p is a h o m o m o r p h i s m o f W(x ), b u t o f neither W(x) n o r W(x ). 4 2 2 h o m o m o r p h i s m of W(y ), b u t not of W(y). A l s o , r 2 ) 1 - - ^ 1 - ^ 1 2 2 T ) ) = W(y) - 2^(y ) - 2, imp.,.]Y(C 43 , however,r = 1 in i 7 . Thus, 2 p 2 and we have (YC p) - 2 C 2 p 2 p + 1 2 imp.,.] ( ^ ( x ^ - W t x In,Y C 4p | p,2p). T 2 2 T 2 is a Chapter 4. The Case n = 4p We finish the proof by establishing the reverse inclusion. Let u(x) = w (x) wp(x ) u) (x ) G Y{C 2 \ p, 2p), where a G A H± , 4 a 7 2 Ap P (3 G A H2 2 P and 7 € AH. 2 p Since T2 +i does not affect images in the character maps %p and \2p, « ( X ) P + I = u(x). Since T2 P YCi is the direct product of the compoments W(x), W(x ) and ff^x ), we have as well 2 P 4 w {x) p+i = w (x), T2 a a meaning T2 +i a = a in H . Hence Ap P a = (1 + T i)a' 2p+ for some a' G Hi . We must then have a' G A i ? 4 with 2a' G P p A .ff4p. 2 The identity map r - 4 r induces a map a a i ? 4 p —> i72p. i7 4p has order p — 1. Since G is cyclic of order p — 1 and has an isomorphic image in Hi , we p see that H\ p is cyclic. H 2p p is then cyclic of order (p — l)/2. We now use the isomorphisms AH^p AH2 P The further induced map AH^p AH 2p A tf p ~* A i? p 2 2 4 2 maps a cyclic group of order p — 1 onto a cyclic group of order (p — l)/2. Since 2a' G A i ? 4 p , 2 the coset containing a' must map to 0 in AH2 /A H2 . 2 P 7LH.2 , this says that a' G A H . Considering a' now as a member of The element 2 P P 2p In fact, its image must be 1, since —1 ^ Y(( ). Ap w >{y) w ,{y )2 a 2 1 a 44 Its image in YC2 , P e{YC ) - *. 2 2v r in the lower left of PBO, is Chapter 4. The Case n = 4p W e now rewrite o u r element u(x) — (w (x)w i(x ) )u'{x ) 2 a l a 2 where u (x ) = w t(x )wp(x )wy(x ) a W e are now operating i n the pull-back nc • 2p o(c 2 p ) >F [C ] x F . YC x 1 P 2 I n [6], it is proved that im^Ll.](u'(x )) € (YC2p) ~ 2. 2 2 p P T T h i s is a simple extension of the m e t h o d I of the last few lines a n d completes the proof of the l e m m a . Lemma 4.3 YC (YC ) 2p T,-2 = 2^ ^ (2^ — l ) , where f is the order of r in H and g = p 3 2 9 2 p 2p [Hp :< T >]. 2 P r o o f . F r o m [6] we have W(y ) 2 = {2 - iy f \T,-2 W(y ) 22 W e w i s h t o calculate the order of YC {YC ) ~ 2 W(y) x W(y ) W{y) - 2 x W(y ) ~ 2' 2 2p 2 T 2 2p Since m u l t i p l i c a t i o n b y 2 — r W(y) a n d W(y ) 2 T 2 is injective i n b o t h A i ? 4 T a n d A H2p, 2 2 2 2 P T h e order is finite. B o t h have rank (p - 3)/2; we choose m u l t i p l i c a t i v e basis elements U{, 1 < i < (p - 3 ) / 2 , for W(y) a n d ttj, 1 < i < (p - 3 ) / 2 , for W ( y ) . { u 2 2 T 2 , vf T 2 } then form a basis for (YC ) - 2. 2 T 2p W e form the m a t r i x whose row entries are the exponents of u\, u2,..., v\,... i n the expansions 2 —T of each o f the basis elements u 2 1 2 —T , u 2 2 , . . . T h e determinant of this (p — 3) x (p — 3) m a t r i x gives us the order of the factor group. A s this m a t r i x has been formulated, however, the first (p — 3 ) / 2 columns each have the single 45 Chapter 4. The Case n = 4p non-zero entry 2 i n the diagonal p o s i t i o n . T h e determinant is therefore 2^ ~ ^ times t h e dep 3 2 t e r m i n a n t of the smaller (p — 3 ) / 2 x (p — 3)/2 m a t r i x whose entries give the e x p a n s i o n for basis elements of W(y) ~ 2 i n terms of basis elements of W(y). 2 T T h i s smaller determinant gives the value ( 2 ' - l ) , the order of W(y )/W(y) - 2, 9 done. 2 2 T a n d we are I T h e i n d e x [S7C 2p : YC ] = d (( ) is the defect as defined i n [7]. W h e r e im WC 2p 2 of WCP i n F [ C ] , <h((p) 2 h a p s t h 2 e v a l u ( e P 2 / - is t h e image p I i m WC |. F r o m this definition the following 2 P corollary becomes immediate. Corollary 4.4 $lC 2p (Yc r^ 2 = d (C )2( - )/ ( 2 > - l ) » . p 2 3 2 p 2P 4.3 C o m p a r i s o n of Images We are now able to give a n account of the circular index c ( C ) = [ O C 4 p 4 p : YC 4 p ]. Lemma 4.5 Where r is the following map across the bottom of the pull-back PBO, 2 r :Z C 2 -»• 2 p F [t]/($ (t )), 2 2 p we have • c(C d 4 p ) = 2 ^ 2 ( P " 3 ) / 2 ( \r (nc )\ 2 2 / - 1 ) 9 4p Proof. F r o m P B O we get the sequence Cl(C p\p,2p) Y(C \p,2p) A 4p nC^p L(( ) YC ^Y(( y 4p 4p where Q(C4p\p, 2p) is the kernel i n the m a p x±p '• QC4p Cl(C \p, 2p) f l Y C 4p 4 p 4p —> 0 ( C 4 p)t . Since the group o n the right is t r i v i a l , we have ftC YC 4p _ n{C \p,2p) ~ Y(C p\p,2pY 4p Ap 4 46 g i v i n g Y(C4p\p, 2p) Chapter 4. The Case n = 4p From the pull-back, Vt{C \p, 2p) and Y(C \p, 2p) are isomorphic with their images in QC2 , 4p 4p im[,./]0(C4 |p, 2p) and (YC2 ) ~ , 2 P P respectively. From the pull-back as well, we see that imp.,]fi(C4 |p, 2p) T2 P P is precisely the kernel in O.C of r'2, which we write k e r Q C 2 . Thus, r2 2p OC p _ im[ nSl(C \p,2p) 4 p L YC ~ Ap Ap (YC ) - * 2 ' T 2p The corollary then follows from the sequence ker fiC2 7^2 ^ (YC ) 2ri QC ^ - 2 "» r ( f t C p ) , (YC2 ) * P T 2p 2 2 r 2p 4 P I using the corollary above. We next give a description of r2(0,C2 )P Lemma 4.6 Let a = (r — Y)[r\, — 1), where r and rj, eac/i generate H2 . Then r ( O C 2 ) is a a 2 P P £/je product of the group r (VF(y )), which has odd order, and an elementary 2-group generated 2 2 by (p - 3)/2 H -conjugates of r {w (y)w (y )~ 2 2 2p 2 a T a Proof. Note that (r (flC )) = r (ft(C p) ) = r (r (WC )) 2 2 (F [i ]/ ($p(t ))) 2 2 2 x 2 2 2p 2 2 2 p = r {W{y )). But, r {W{y )) C 2 2 2 ~ IF2[Cp] ) which has odd order. In fact, r (W(y )) x 2 must contain all the 2 2 elements of odd order in r ( f J C 4 ) . 2 P Simce | f l C p / Y C 2 p | = d ((p) is of odd order, we must then have that r (fiC4 ) = r 2 ( y C ) . An 2 2 2 P 2p element u(y) = w\(y)w2{y ) € W(y)W(y ) = YC2 , we rewrite in the form u(x) = iii(y)u (y ), 2 2 2 P 2 where ui{y) = u} {y)m{y y 2 and T21 1 Note that (r (ui(y))) = wi(t )wi{t )~ * 2 2 2 T u (y ) = wi(y ) 2 r (ft<? p) = r (Wiy) -^* )) 1 2 T21 w (y ). 2 2 = 1. Thus we may write lT2 2 2 2 2 1 2 r {W(y )) . 2 2 T2 {w{y) ~ ^ 2*) j is an elementary two group. The proof is finished by observing that W{y) l T T is generated by (p — 3)/2 conjugates of w (y). a 47 I Chapter 4. The Case n = 4p I n YC , W(y) ^2 1 2p T ) is a free group of rank (p - 3 ) / 2 . r Its image i n f [t]/($ (t )) 2 2 p may fail to have the same rank. W e define a new defect t e r m 2 <k(p) = (p-3)/2 W e t h e n have the following theorem. Theorem 4.7 c(C p) = d (( ) di(p). 2 4 2 p Proof. U s i n g the lemmas of this subsection, we have d (Cp)2fr- >/ 3 c(C4p) = (2/-1) 2 2 9 h(ftc p)| 4 (2^-1)5 2(P- )/ r (W(y ))\ | (VCy) -^" )) 2 2 = 3 2 1 1 r2 d {( ) d (p), 2 2 p 4 according to o u r definitions, m T h e first t w o primes p for w h i c h the defect d ( p ) is n o n - t r i v i a l are 29 a n d 113. I n each o f 4 these cases this defect a n d also the circular index is 8. For c{Cus) = 9. I n the same manner giving c ( C n 3 p = 37, we have c ( C 7 3 ) = 49 a n d 0(6*4x97) = 49. F o r 4 X (^(£37) p — 113, d (Cii3) = 9, 2 ) = 8 • 81 = 648. I n the following, we outline a m e t h o d for the calculation of d (p). 4 W e start w i t h the observation that IF C p 2 l)$ (t )) 2 F [i]/ ((t 2 2 p 2 ~ IF C x 2 2 ¥ [t}/ ( $ ( i ) ) 2 2 p since ((t - l ) , $ ( t ) ) = 1 i n ¥ [t). 2 2 p 2 Because the image o f ClC 2p i n 7L.C is t r i v i a l , we are able to work w i t h the m a p 2 r' : fiC 2 2p —)• I F C 2 48 2 p = 3, so Chapter 4. The Case n = 4p in place of r . 2 Let a = ( r — 1)(TJ, — 1), as above, where r a and Tf, each generate i ^ p , and let u(y) = a w {y)wa{y y 2 T21 a Thus, H 2p = {l,T ,T \...T^-W} A A and in IF2C2P V U (p-3)/2 T ; (i)2^i=0 * = 1. We have, then, the following lemma. Lemma 4.8 Let h(r ) be a minimal polynomial such that u(t) ( ) = 1 in f C h a h(r ) I E £ O a 3 ) / 2 Ta 2 2p Then, T J (mod 2), and d ( p ) = 2 ( P - ) / - W r - ) ) . 3 2 d 4 Proof. This follows since in exponents of u{i). is an elementary 2-group, so we can work mod 2 I The procedure, then, is to factor ^fS^^ T* mod 2, and find the minimum number of these 2 factors which annihilate u(t). We can simplify this approach using logarithms. InlF C p, (u(f)-l) 2 2 2 = 0. Thusu(t)-1 is in the radical of IF C p, so we have ^ - l | (u(i)-l) 2 =0 2 2 mod 2. Thus, u(t) has an expression u(t) = 1 + f(t)(t - 1) = p exp{/(*)(<* - 1)}, where f(t) G IF2Cp. (t — 1) is invariant under the actions of H , p 2p l + f(t) (t T p - 1). Since t {t d+p p + 1) = t d + 2 p + t 49 d + p = t (t + 1) d p so for T G iJ2p, u(t) T = Chapter 4. The Case n = 4p in we have the group H operating on f(t) to obtain conjugates of u(t). In particular, \F2C2p, p /(£) = fit* ) in f C , and we may write 1 2 p (p-3)/2 f(t) = c + ]T 0 cit + t- )^. 1 i=0 By the first lemma in the section on constructible units, the coefficient of y in the expansion p of u(y) is even. This implies that Co = 0, so that in fact, we have u(t)=exp{((i + i - ) / i ( r ) ) 1 where hx(r ) = £ £ o " 3 ) / 2 a 1 (t -l)} p a ar*. We are looking for the minimal polynomial h(r ) so that a u(t) = exp{((t +1- )/!!^)/!^)) (t - 1)} - 1. 1 For this, we must have hi(T )h(T ) a p = 0 mod 2 in H , i.e., a p T^- )/ -l|/n(T )Mr ) 3 (mod 2 a a 2). Since h(r ) \ 2~jl=7j ^ a ° d 2, we have r + 1 | hi(r ) mod 2. Thus, 3 2T m a a a V-(P- )/ T-J 3 /i(r ) = ^ a ffcl(Ta) VT 0 2 « ( -3)/2 0 V T A ' P +1 ' ^ i = 0 'ay Example. For p = 29 and a = 3, 6 = 11, M * ) = (« + ! ) ( ^ + ^ + * + +1)(^ + * + 1) 9 4 z 3 and (p-3)/2 £ ^ = (z + l ) ( 2 + z + l) (2 3 2 3 + ^ + l) 2 2 (mod 2). i=0 In this case, = (z + l)(z + z + l)(z + z + l ) and d (p) = 2 . 3 3 1 2 3 4 We obtain a non-constructible unit by first calculating the value of u(y)( Ta+1 ^ <T 3+Ta+1 ^ <' <' ) T 3+T 2+1 in HLC . We are then able to lift this to a unit u*{x) € Z C , where u*(d ) = 1 and u*(x) i-> 1 2p 4 p in Z C . 4 50 p Chapter 4. The Case n = 4p u*(x) G YC4 , a n d we can as well calculate its precise preimage i n iif x H 2 P 4p u{y? = w (y) w (y y ^ 2 2 a Take a = ( r — 1){T — 1) i n i 7 a = 2T a 0 G Ti 2 is i n Tlt( Tail T 1) p w {y) -^w {y Y^- \ 2 2 a 2 a as well. T h e n 4 p w (x)^ *- w {x ) ' ~ ' ~ a H. x 2p 1) ( )^ ^ G a = W a x W a ( )-^i 2 x i W a ( ^y x l s ^-r 1 6 ) Y{Cip\p.2p) and, i f fact, equals u*(x) . T h u s , 2 u*(x) = tu ( a ^ G , a ( - l - r ) , a r ( l - r )) . 2 4 1 5 1 5 15 is a n elementary 2-group of order 8. T w o other independent generators are given u*(x) 3 a n d T by «*(x) 9. T I n the first o d d 100 primes we have the following cases where d ( p ) is n o n - t r i v i a l : 4 p 29 113 163 197 239 277 311 337 349 373 397 421 (k(p) 8 8 4 8 8 16 1024 64 16 32 16 16 a n d where d (£ ) 2 p 37 d (C ) 2 P p 8 64, is n o n - t r i v i a l : 73 97 101 7 7 3 3 463 491 113 197 199 3 7 9 229 269 313 3 3 7 349 3 353 9 373 389 541 3 7. 3 F o r p = 113, 197, 349, a n d 373, b o t h defects are n o n - t r i v i a l . F o r 25 of these 99 o d d primes, the circular index c ( C ) is n o n - t r i v i a l . 4 p Example: F o r n = 12 these defects are t r i v i a l a n d a l l units are constructible. T h e rank of the the group of constructible units is 1; this group is generated b y $35(0;), o r its inverse $35(% )- T h e u n i t group is thus given by: 5 < -1, x, - 2 x + 2 ( x + x ~ ) - ( x + x 6 5 5 4 - 4 51 ) + ( x + x ~ ) - 2(x + x " ) + 3 > . 2 2 1 Chapter 5 Liftability In this chapter we concern ourselves with the map Xn • ttC - r fi(C„) n across the top of of PBO. The first step is to identify Y(( ), the image of YC , and calculate n n its index in fi(Cn)- This is the subject of the first section of this chapter. The second section deals with the broader question of the liftable units, L(Cn)> the image of SlC . We will establish necessary conditions for liftability, and identify some cases where the n index [L(( ) : Y(Cn)] is non-trivial. n Some definitions which we use in this chapter: m = (4>(r),<£(p ))/2 if 2p . m = 1 if n = 2p . s fi - I < n > I in H s. gi = \H s\/fi. p p s (f = p (g = \Hir\lf .) p p s | < T > | in H r. P t m" = {gi,g ). G , where di \ d, is the subgroup {r p dud a :a = i-^ + 1,0 < i < d\, (a, d) = 1} of Gd, the galois group of Z[Cd]- Where d = n we may write this as G' or even G^ where there is unlikely to be confusion. di 5.1 Constructible units in fi(£n) In the analysis of the map YC n fi(Cn), 52 Chapter 5. Liftability the norm relations in Z[£ ] allow us to narrow our focus to simpler groups. In Chapter 2, we n established 0 ( £ ) as the product n In the following lemma, we show that we need only work with a smaller group on the left of this map. Lemma 5.1 W (x)W (x )W (x ') -» Y ( £ ) . r p n Proof. Consider a map W{x ) O(Cn), d where d = l p> \ n. If 1 < i < r, and a E A H / l then the norm relation 2 n d Wa(Cn )=t«a(C„ '') ' -^ d P 2:G i gives that Wa(C') e Xn (W^j) C Xn (W(x)W(x °)) p , and ^a(C„ ) € Xn (w(x)W(x )W(x* )) lr d , S using the similar argument in case 1 < j < s. Thus X(YC ) = Xn ( II W(x ) J - Xn (w[x)W(x *)W{x^)) , d l n as claimed. I Using the isomorphism w : A H j ~ W(x ) we have the following corollary, which enables us 2 d n d to work more easily with norm relations. Corollary 5.2 There is a map :AH 2 X w x AH x A H r -» Y(C„). 2 n 2 pS t 53 Chapter 5. Liftability Proof. Xw is the composition Xn°w : ( a i , a , a ) >->• w ( C n J ^ f C D ^ a s C C n " ) - 1 2 3 ai Our problem, then, is to determine the order w(C )w(C)w(cf) n in view of the norm relations ( l_ C n ) EG r = _ r i-r,_ ( 1 c ) 1 and (i-Cn) » E G -(l-cr) - '"- . 1 s 7 1 We thus concern ourselves with the ideals (1 - T- )7LH r (1 - r f ^Zflp. C AH , x r C Atf,-, { and (E GJO Zi?n + ( S G 3 ) 7LH C Ztf„. P Note. The map x w n does not extend to a map A 7 J n x AH » x Ai//r -» fi(£ ). In H , for n P n example, we have identified T\ and r _ i which have different roles infi(Cn)-Thus, for a ( r j - i — 1) in the first ideal, we have in fi(Cn) (1 _ r a(l-r,_ ) c ) 1 = ( 1 _ C n ) aEG,. > Since (1 — Tj )Z.Hp* <jt A H <>, we can only study the effects of this conversion in conjunction _1 2 P with x-w o n elements of the form OJ(T;-I — 1) + p 6 A H s. 2 In the following lemma, we work with the first two ideals. Lemma 5.3 V(( ) 2m n C Y(C„). 54 p Chapter 5. Liftability P r o o f . C o n s i d e r a n element v {C, ) G V(£ ). a n T h e n ( r - 1)(1 - r n 0 X. ((0, (r - 1)(1 - rf'),0) = VaiO^r = 1 fl = "a(Cn) 0(r) _ 1 ; ) G A H s, 2 p and v (( )Z " G a n ^(T -l)(i:G r-^'-))(Cn), o i a c c o r d i n g to a r i t h m e t i c developed i n C h a p t e r 2. T h i s gives l)(<A(r) - Xw ( ( r « - E G , . ) , ( r - 1)(1 - r f ^ , 0 ) = « ( C „ ) * . (5.1) (r) a a Similarly, Xw ( ( r - l)(0(p ) - S G , ) , 0, ( r - 1)(1 - r " ) ) = <; (C„)* . s 1 a P (5.2) (pS) a a So t h a t , Va(Cn) = V (Cn) WlrU(pS)) G Y(C„). 2m a • R e m a r k I . T h i s takes care of the cases i n v o l v i n g the first two ideals, as we show i n the following. Suppose we have a = a i ( l — T, ) + a G A H *, where we use the n o r m r e l a t i o n t o convert 2 -1 2 P the first t e r m o n the right. (1) I f cti G AHps, t h e n a\ = ( r — 1) + a[ for some r G G a have « 2 S A H s. 2 p a n d a[ G A H . 2 n a n A l s o , we must Then X t l > (0,a,0) = v ^ y - ^ w ^ C Y - ^ w ^ ) = V (C„) E a G " W (C„) a[ EG " (C)• Since ^(Cn) E G ' G W(Cn), r this gives no new i n f o r m a t i o n b e y o n d (5.1). (2) I f « i ^ AH a, p + ( a i — fc). T h e n then for some integer A;, a\ — a = fc(l - T , ) + ( a i - fc)(l - T , ) + a . _ 1 - 1 2 T h e second t e r m o n the right was dealt w i t h i n (1) a n d we drop i t out a n d now rewrite a = fc(l - r _ 1 z ) - fc(l - rf ) 1 55 + (a 2 + fc(l - T," )). 1 Chapter 5. Liftability For the last term on the right, now in A H *, and we take its image in W(C,T)- Dropping this 2 P term leaves us with the trivial element in A H s. The possibility arising in this case concerns 2 p the third ideal, and is considered below. Because of differences in the arithmetic depending on when n is even or odd in considering the third ideal, we separate the further discussion accordingly. 5.1.1 n odd With n odd, the groups G/r and G * are both cyclic of even order. Let r p a2 and T{, , respectively, 2 be the elements of order 2 in these groups. Then a = —62 mod n so that 2 Va {(n) = 2 -V {Cn)b2 We start with a lemma which identifies further elements of Y(Cn) in the cases where m is non-trivial. Lemma 5.4 (i) (Cf) v -„ p and &2 are integers such that (ii) ( C n > ( C ^ l - C n ) ) ) " ^ ( C n ) G Y(( ), P 2 2 where h m 2 n + & ^ ^ = 1. 2 ( - i ) ^ ^ v ^ ( C ) v ^((X I 2m )er(Cn). 2m p Proof. Setting «[,,] = ( E G « r - 0 ( Z ) ) - ( T R a a -l), (5.3) we see that aprj is the sum of elements of the form (r — 1) — ( T " — 1), since T 1 a element in Gjv which does not have a distinct inverse. Thus J] X«,((a[J'],0,0))= I V (Cn)\ a \T eGir £ AH, 2 n A 2 is the only and we then have Va.iCn)- 1 J a = «|-i(Ci )(C (l-Cn) ))-* ' r 1 56 2 ( , ) / 2 |;a (Cn)- . 1 a (5-4) Chapter 5. Liftability Similarly, for (5.5) we have X»((a ,0,0))== i(Cr)(C (l-Cn) ))"^ 1 M 2 ) / 2 V The elements in (5.4) and (5.6) generate a subgroup ofQ(( ). (5.6) »6 (Cn)- . 1 a We can find equivalent generators n in a more convenient form in the following way. Since 2m = (4>(l ),<f>(p )) we can find integers r s ki and k so that 2 With fc = and k = 3 we have 4 and the determinant h k kz ki = 1. 2 Thus, with a[/r] and a^j as in (5.3) and (5.5), Xw {(kia + ^ 2 0 ^ , 0 , 0 ) ) = «l-x(C4') ^-i(Cn ) P3 fcl m fc2 ( C H l - Cn) ))" ^a (Cn)2 m 2 fcl ^(Cn)"* 2 (5.7) and ^ ( ( - 2^T ['1 + 2 ^ r M ' ° ' ° ) J = ^- (Cn ) a a x V^Cn ) 2 m 2 m u a (Cn) ™ «6 (Cn) 2 2 2 *(') r 2m (5.8) give an equivalent pair of generators in fi(Cn)- We note that <UCn)- fel M C n ) - * = ^ (Cn)" 2 2 fcl (~V (Cn))"" a2 = (-l) ^ (Cn)- " fc a2 = (-l) « (Cn) fc2 57 a2 fcl 2 fc2 (modW(C„)), Chapter 5. Liftability since k\ + k must be o d d , g i v i n g a^ " "* 1 1 12 2 = a m o d n. 2 Also uj-iK,!')* (CD (modW(CT)) 1 and (mod = Vp-xACf) v -i(tf)" 2 p W(Ci')). U s i n g these equivalences i n (5.7) gives (i). For ( i i ) , we first note that «<-i(C,rr^ V ^ C f ) ^ I 2m W e t h e n need t o consider the parities of (1) I f they are b o t h o d d , t h e n ( - 1 ) 4>( ) S «a (Cn) V 2 m 2 (2) Suppose that Gir, _WCf) p 2m =v,ltEn(C)v 2 (mod W(C)W(C[)). and w = - 1 and 4>(l ) r «6 (Cn) 2 = V (( )v 2 m a2 is even, g i v i n g (—1) n 4m b2 2 (C„) = - 1 (mod W(Cn)) = 1 . T h e n , since r a 2 is a ^ ^ t h power i n | ^ p , u (Cn) S ^"(Cn) b y the preceding l e m m a . A l s o v& (Cn) ^"^ € W(C ), so a n d 2m 02 - 2 n that ^(Cn)^^^)"^ =1 (modY(Cn))(3) W i t h (5-9) even, we get the equivalence (5.9) as well by the corresponding argument. U s i n g these equivalences i n equation (5.8) completes (ii). I R e m a r k I I . W e have, i n fact, covered the cases for t h e t h i r d ideal. a ^ E G p s + 0J3 € A H . W e then have the following considerations for a\, a n d s i m i l a r l y for a : 2 n (1) I f a e A H , 2 then 2 x L e t a = a\TiGir + n X w ((aiEG ,0,0)) = ; (Cn) ' - ^ ( C D " ^ r U E G Q l r 1 1 eW^). (2) I f a\ £ AH , t h e n we c a n find r so that a i = r — 1 + ct[ w i t h a\ £ A H . 2 n a a n rewrite a = (aiEG,r - <f>(l )) + (0(P>i + a T,G * + a ) • r 2 58 p 3 T h e n we Chapter 5. Liftability T h e resolution of this first t e r m on the right is given i n (5.7 above. (3) F o r oti = k + ( a i — fc), where a\ — k € AH , we rewrite n a = ka r + ( a i - & ) E G j r + ((0(p ) + ( r s {l } a 2 - l))a + a ). x 3 E q u a t i o n (5.3) gives the resolution for this first t e r m o n the right. W e are now i n a p o s i t i o n to calculate the order \Sl(£ )/Y(( )\. n F i r s t , however, we introduce n the new invariant ra', for n o d d : 4>(i )4>{p") / i \ / - \ \ ra' = ra" d i v i d e d by the order of ( - 1 ) 4mm" ( j / -^— j i n r p T h i s element is ± 1 , ra' is either ra" or ra"/2. Proposition 5.5 fi(Cn) = ^(Cn) m m'<t>{r)<f>{p ). 2 s Proof. F i r s t we note that Y ( £ ) C ft°(C) = V"(Cn)V(Cn')^(Cn"), so that n = m fi°(Cn) leaving us to calculate the order of the group on the right. U s i n g the inclusion m a p o n each component, we have a m a p v(Cn)x v ( c i ) x v ( c ^ ) - > n ° ( c „ ) r from w h i c h we derive the further m a p V(Cn) W(C„) W(CT) 2m W(tf) y(Cn)' T h e orders of the groups i n the product are 4m , (f>{p ) a n d <^(r), respectively. W e w o u l d like 2 s now to determine the order of the kernel i n this m a p . = (—1)(—1) = 1 a n d u-i(Cn)w-i(Cn ) = 1, so T h e first consideration is that v-i(£ )v-i((£) n the kernel has order at least 4. F r o m preceding lemmas a n d the remarks following, we are left w i t h d e t e r m i n i n g the order of the p r o d u c t of cosets corresponding to the element , (-1) WW) 4m» X I 2m 59 p 2m Chapter 5. Liftability The least power k so that both v {^) I 2m i m k i = ±l (modlf(([)), and v _*<n((f) k = ±i (modW(cr)), 2m p is k = m/m". Then If the image of this element is 1 in 0 ( d ) , then —1 ^ Y(( ) and the order of the kernel n is Am/m" = Am/m'. If its image is -1 , then —1 G Y{(, ) and the order of the kernel is n 8 m / m " = Am/m'. Thus = 4m <j>(l )<f>{p )/{4m/m') = m m'<j)(l )<j){p ). I 3 Y(Cn) r s 2 r s Remark. In section 5 of [7], an element of YC in the kernel of Xn is constructed starting from n the element do = (h /m) p AH. 2 n G\ — {hi/m) This is essentially the element G' € AH which is adapted to give an element of p n i l ~ ^2m~ \p ] § i a lr a a v e n above. The conditions given are not quite sufficient for the element to be in the kernel, and fall short precisely when —1 € Y{C, ). n The order for the group of Proposition 5.2 is m! as defined above, which is occasionally 1/2 of the order m" stated. 5.1.2 n = 2p r a Where n is even, the group G * is cyclic of even order as before, and hence has the single p element r; of order 2; however, the group GV contains en elementary 2-subgroup of order 8. p Since then n^=i, the identity element, we have that The further elements of Y(( ) for this case, we identify in the following lemma. n 60 Chapter 5. Liftability Lemma 5.6 (-l)*^C^Ml " Cn) ) « *i (C) V» (^) «*(Cn) G ^Cn), where k (i) 2 m (p i) 2 2~ and k are integers satisfying k\ m r 4>{p ) \- k 2m 2 2 (ii) {-lffiv x = 1. 2 _iM{C?)v 2 2 s ^(Cn)GY(C„). XzlitfU, (j ) m m p 2m p Proof. Some part of the following discussion is concerned w i t h d e t e r m i n i n g whether or not is constructible. W e note first that 2 + 1 is a 2 r t h power m o d 2 r - 2 r + 1 . If p ^ 2 r _ 1 + 1 m o d 2' 2 m | 2 " , so that r 2 -l = With Gr 2 v r (( )€V(0 cY(( ). 2m {2 +l) n n = {r :a = 2ip + 1,0 < i < 2 }, s r a Xw{(<x r],0,0)) =v [2 a)ttn) JJ {Ura€G2T V ^ - a T eG r a W e prove first that for p = 1 m o d 4 a n d r > 2, n „eG 7s T Xio ( ( a r ] , 0 , 0 ) ) [2 2 =^ P +* a r 2 = - JJ S m o a -2 r + 1 p ) giving S «(Cn)0 Ta&G r 2 T h a t this is the case for r = 3 is proved for the case n = 8p. N o t e that 2 r-l_l 2 " -l r JJ (2ip + l)= s i=0 2 ~ -l r 2 2 JJ (2ip + l) JJ ( 2 - y + 2 ^ + l) t=0 i=0 2 -l ( ~ -\ s r R r _ 2 s T~ -\ 2 2 2 • - = JJ (2*p s JJ ( + 1) r - 2 _ 1 = JJ (2ip* + l) (T~ -\ r + 1 II ( \ 2 a n d hence m o d 2 )+ 2 r _ 1 P T 2 S £ ( *P* 2 + 1 S + ) + "V (- • • X 22r s i=0 JJ ( 2 ^ + l) + 2 2 r -V((2'- -l)p 2 )J (mod2 - p ), 2 r 2 s s r - 2 _ 2i 2 p , for r > 3. A s s u m i n g that 2 ( P ~ -\ y i=0 t=0 = ( ^ y i=0 i=0 2 2 i + X 1 JJ {2ip + 1) = fc2V + 2 s i=0 61 r _ V + 1 2 s + l) + 2 '-V ( 2 s Chapter 5. Liftability for some k, we have 2 ~ -l r 1 l[ (2ip + 1) = (k2 p + 2 ' p s r s r l + l ) = 2p + 1 s 2 r (mod s 2 p ), r+1 s i=0 a n d (5.1.2) follows by induction. U s i n g results i n C h a p t e r 2, we then have X For Gps ={Ta:a - C n ) ) " " " «2-i(C„ ')- - (H>-],0,0)) = - {Q {\ l W =2 i 2 2 + 1, {a,p) = 1,0 < * < p } , and a r+1 2 2 = £ G > - <£(p ) - ( r s Xw{(a\ps], 0,0)) = I - s M %(Cn) MCn) Yl (5-10) 1 ip 1), _ 1 •HP") = (C (1-Cn) ) 1 2 ^-i^')- ^^)- . 1 2 (5.H) 1 Thus, X - ( (fc[2] «[2^] + fc W (C-^l - Cn) )"" ^-.(C ')-*™ V ^ ) " * ' a ,0,0)) = 2 M 2 1 1 1 «ip(Cn)-' (5.12) EE ( - l ) M ( C ^ l - Cn) 2 )"" 1 V m (Cn~2r) f c V „ (CnO « (<p) > (Cn) W m o d y(Cn), proving (i). Also, Xw {- V [2 ] a r + ^Sr M' '°)j = a 0 2 m U 2-i(Cn ) 2 m V>(Cn ) P m MC") ™ (5.13) = (-1) 2m 2 mod Y(C ), n proving (ii). 2Lzl(Cn )V v 2m p ) ,. ,_2Tzi v (j ) m p (Cn) m I R e m a r k I I I . A g a i n , this covers the cases for the t h i r d ideal as outlined i n R e m a r k I I . W e now define the invariant m' for even n: 4>{2 )(j>(p ) / i \ T m' = m"/2 m'.= if ( - 1 ) 8mm" s 2=_ =-1, r > 3, and p = 1 (mod m" otherwise. 62 JT" ). 1 Chapter 5. Liftability Proposition 5.7 = m m'<j>{2 )(f>{p ). 2 r s Proof. F o l l o w i n g the same analysis as i n the proof of the p r o p o s i t i o n i n the previous subsection, we are left the consideration of the element « r) (C ) X V s (-l)^jr V ra 3 (i ) Hp ^ ,) (Cf )• ) Xy (C } iT^r p (a 2^jr p If 2m" = <t>{2 ), we must have that p = 1 m o d 2 i n order that v _^2n (Cn ) = i l > a n d also r r p 2m" (Cn) £ that m = m " . T h e m 4 m | 4>(p ), i m p l y i n g that T j is a 2 m t h power i n G * a n d v s p p (jp) 2m" V(Cn) ^(Cn). 2m If 4 m " I <£(2 ) then « (Cn) € W(Cn). r (ip) 2rP^ I n any case, we have reduced our consideration to the element <KP ) S ( - l ) l ^ r, x « ^ ( ^ ) x « , ,HP11 (2 (C ) = ( - l ) l ^ x / > 9 p / 2m" 1 2m" = m m"4>{2 )<f>(p ). 2 l P so n Y((n) r X v _^n(Cn )• p 2m n If p = 2 ~ " p a 2m" If p ^ 1 m o d 2 - , then - 1 G V{( ) W{( ), 7 - 1 / m p r s + 1 m o d 2 , T h e n m = 0 ( 2 ) / 2 a n d </>(p )/2m is o d d . T o have v _ ± r r s m p we must have 4 m " | 4>{2 ) a n d hence also | (j){p ). T h e n r (-1)<2& s 4 p V (2r) ((f) = (-1) = ( - 1 ) sJ"m 2m" If p = 1 m o d 2 , then 4 m | 4>{p ) and fp(Cn ) = ( - l ) * " ^ , g i v i n g s r 4 S . MP ) 3 M?^(Cn = P •n 9™" . , ^(2 )<t( ) r "I Accordingly, when «2»W) (-1) / 2 /m"' I —p a— | = 1, 1 8-"- we have m m"(j>{2 )<p{p ), 2 r s and when ,» ( 2 W ) ( - 1 ) 8m"m / / 2 /" " 1 I 63 P 1 s 3 P = -1, 2m" = ±1 Chapter 5. Liftability we have m m"4){2 )4>{p ) 2 r s The proposition follows from our definition of m'. B 5.1.3 T h e i n d e x for c o n s t r u c t i b l e u n i t s Though the methods of proof differ for n even and n odd, we arrive at the same result. T h e o r e m 5.8 = Y(U) mm <f>(n) P r o o f : This is the restatement of both Propositions 5.5 and 5.7. Note the m' = m" or m"/2, but there is a slight difference between the definitions for the even and odd cases. 5.2 5.2.1 I Liftability and non-constructible units n = Ip In this subsection, we wish to review the topic of liftability for the case n = Ip and provide an exact determination of the order of L((i )/Y((i ). p As well, we shall look at the evaluation of p some non-constructible units. Consider the pull-back PBO: TLC • Z[Ci ] lp ZC P X l p TLCi • F,[C ] x Fpfc]. P A unit infi(Czp)is liftable to a unit in OC; if and only if its image in the lower right is in p W(£p) x W(Q). This, of course, is both the image of YC P x YCi across the bottom of PBO, and the image of Y(Q ) down the right side. p The subgroup V(£ ) x V(Q) in the lower right is of interest. The invariants p V%) ^(Cp) W(CP) ncp) w%) 2 2 ei(Cp) = and e/(Cp) = are defined in [7], and show up in the following proposition giving the order of 64 L(Q )/Y(Q ). p p Chapter 5. Liftability Theorem 5.9 V e « « P ) ' «P(C/)y n — 2p, we have L(& ) = Y(& )- T h e p r o p o s i t i o n is therefore true for these cases, Proof. For P P so for t h e following, we assume that n is o d d . W h e r e i m [ / ] designates the image i n the lower right of P B O of subgroups of 0(<^ ), we have r p im[(. .]fi(0 ) r x W(C ) L(Ci ) P P P (5.14) W(Ci) W h e r e a a n d p are generators of the groups G a n d G / , respectively, we consider t h e a c t i o n of p the operator (a — 1) x (p — 1) on subgroups i n the lower right of P B O . Elements i n t h e kernel of this operator are i n IF* x IF*. Since IF* x F * C i m [ ( ] r ^(Cjp^CCip)) F f x F * ^ imp^nccp) - » w e n a v * e sequence n e {im^nic^f- ^^. 1 Since these groups are finite, we derive im .]fi(C(p) [Lr = (/-l)(p-l). ( [i.r.] '(Qp)j lm S F i n d i n g the order of ( i m ^ n ^ ) ) ^ 1 ^ - 1 * / (W{( ) x W(Ci)) w i l l now enable us t o establish p the order for t h e groups i n (5.14). (im^V^V^ViQ,)) We note that C i p ) 2 ) z / m z v ( C P ) r ( C £ ) F ( C ( p ) ) ( < T t h e g r o u p - 1 ) x ( p " W { C p ) = ( i m [ ; r ] f i ( 0 p ) ) x W { ( l ) (.-Dx(p-i) a n d its order is given b y the l.c.m. o f componentwise orders. of i m ^ r . j Q ^ p ) ^ ) * ^ ) i n F/[( ] is precisely V(( ) W(C ) -1 -1 V(Q) W((i). 2 P P ; p ( w ( C p ) x W { Q ) ) i g c y c l i c > W e next note that t h e image a n d its image i n F [(j] is precisely 2 p . Since Sl(Q ) = ( ^ ( l - 1 ) p F r o m the definitions given at the beginning of this subsection, i t follows that L(Cn) = a-l)(p-l)l.c.m.{e,(C ),e (Ct)}. P p We now use the result of P r o p o s i t i o n s 5.5 a n d 5.7 o f the previous section, namely, = m m'{l-l){p-l). 2 Y(Cn) 65 Chapter 5. Liftability Then / L(Ci ) / P Y(Cn) Y(CIP). mm HCn) l.C.m.{eKCp),e (Cl)} P mm' (-^r, , \e,(C )' e (Ci)/ ' P since b o t h e , ( £ ) a n d e (Q) p I d i v i d e m. p p W i t h this theorem we can extend the results of [7] to give a precise c a l c u l a t i o n of T h e factor i n P r o p o s i t i o n 5.9 is a divisor of c ( C , ) . ered b y considering the kernel of the m a p xip- W h e r e Y(Ci \p) p w h i c h have n o n - t r i v i a l image o n l y i n the character x p is the lower left of P B O . W e define the defect _ 1)91 (lfl / — di(p)= - \vmiYC \, } K m P p fl ™[u.]Y{Ci \p) where i m , : YC YCi (l ~ 1) 9l m{l - 1) V p denotes the elements of ( cf. S c h o l i u m 4.3 a n d L e m m a 6.1 of [7]), P where i m p , ] denotes the image i n £lC p Possible a d d i t i o n a l divisors are uncov- p YC c(Ci ). p j —)• F j [ £ ] . p T h i s differs from the definition of di(( ), w h i c h relates to |im,V(Cp)|- U s i n g di(p) allows a clearer p d i s t i n c t i o n between factors of c ( C , ) arising from the kernel of xip a n d those from the co-kernel. p These separate definitions are related through the formula (cf T h e o r e m 0.3 of [7]), mdi(p) ~ d((C )ej(C ). p p d (l) p is defined i n the corresponding way. W e now have the following theorem. Theorem 5.10 c{C )=md (p)d {l) lp l p ( s g y . ^ y ) . C o n s i d e r the exact sequence 1 -> n(Ci \ ) Y{C \p,l) p lp Y(C \p)Y(C \l) lp lp ^ n(c \i) P Y(C \p) lp X lp Y{C \l) lp 66 ~* nc ip L(Ci ) YC lp Y(Cip) P Chapter 5. Liftability from [7]. T h e first group has order m'. (See remark at the end of the subsection n odd above.) T h e components i n the p r o d u c t i n the next group have orders d ; ( £ ) a n d d ; ( £ ) respectively. p p T h u s , we have m'c(C ) = (d«(C )d/(C )) (mm' ( ^ , lp a n d the theorem follows. P I d[(p) a n d d (l). T h u s , as a m i n i m u m , m(m') 2 T h e factor m ' is a divisor of b o t h sor of ^ ) ) P p c(C[ ). T h e factor ( ^ £ - y > e~^T)) & yes p * n e is a d i v i - order of the image i n the m a p vWY L [ C l p ) For n = 5 x 29 it has the value 2. It is a divisor of m " , a n d there are reasons to expect that, w h e n it is n o n - t r i v i a l , it has the value 2. Examples (for 4>{n) < 300). T h e r e are the following cases where at least one of the defects is non-trivial: I 3 3 5 5 5 7 11 11 11 13 23 47 61 p 61 97 41 61 73 43 13 17 19 17 11 5 5 di{p) 44 73 2 22 27 8 3 3 19 2 11 3 2. O n l y for n = 5 x 29 is the factor ( —TTT, —TTT ) n o n - t r i v i a l . T h e n o n - t r i v i a l c i r c u l a r indices for n = Ip i n this range are then: n 65 85 c(C„) 2 2 n 217 221 235 247 253 c(C„) 3 4 3 3 11 91 3 133 3 143 3 145 183 185 44 2 259 265 291 301 305 365 3 2 73 24 8 2. 4 67 187 3 205 4 209 19 Chapter 5. Liftability n = Fp* 5.2.2 T h e m e t h o d used to determine the order of for higher powers. T h e groups Cl(( )/Y(( ) n for n n = Ip does not d i r e c t l y generalize H^-ip, a n d Hiv ~\ applicable i n the lower right of P B O m a y no pS longer be cyclic. A l s o , the groups W ( * i ) a n d W(t ) i n the lower right of P B O u s u a l l y do not 2 YC p a n d comprise the whole of the images of YC / . n n p A t least for n = 8p, however, the m e t h o d does work. I n the statement of this extension, we use the definitions V(h) W(h) __ 2 ei{l ~ p ) = r l s _ ___ J nr and a V(t ) W(t ) W(t ) 2 u 2 e,(ZV~ ) = 2 2 where n = lp r s T h e o r e m 5.11 For n = 8p, = Y(Cn) m m \ Ve (4p)' e„(8)J ' I j mm 2 if 2 \ \Q,C4p/YCi \. In any case, the factor on the right divides the order on the left. P P r o o f . P B O i n this case is nctp x nc • 8 YC = W(y)W(y )W(y ) 2 ip 4 for the cyclic groups H - » W ( i i ) and x IF[C]. P 8 YC -» W{t ). A l s o , we c a n find generators a a n d p 8 2 a n d H%, respectively. Ap T h e difference i n the proof, i n this case, is i n d e t e r m i n i n g the kernel of the m a p (a — 1). T h e subgroups of s y m m e t r i c units i n J^ ^-. is a n elementary 2-group, K , of order 4 generated b y 2 A (*p(*i)) t 2p and t +t p 2p + tl . F r o m ref t p € V(h). A l s o , v ((£ ) ^ i f + t 2p +t . 2p 5 3p p sequence K x A IF X P imp.r.jfiCCsp) - » ^m _ a{^ )) ' ^ {a l) [hr ] p l) p a n d the p r o o f proceeds as before, g i v i n g ™[I.r.]Q(C8p) W(ti) x W(t ) = mm' 2 68 (I.e.m.{e2(4p),e (8)}). p , W e t h e n have the Chapter 5. Liftability A n element o f O(Csp) f i ( i i ) x Wfo), liftable i f a n d o n l y i f its image i n the lower right o f P B O is i n the image, across the b o t t o m of Q.C x YC%. T h i s order o n the right is a 2-power. I f 4p 2\\nC /YC p\, 4p I S then 4 im[;. .]f2(C p) r , 8 flfc) x W(t ) = 2 and the result follows as previously. I f 2 | \£lC /YCi \, 4p is smaller, so that mm' ( ^ ) > i"7Jj)) * e2 (Mej^),^)}). p sa there is the p o s s i b i l i t y that the order p proper divisor o f | i (Csp) / ^ (Csp) I - ' 1 For p = 3 ( m o d 4), this gives that Y(Csp) = £(C8p)> while for p == 1 ( m o d 4 ) , there is a n i n d e x o f at least 2. We are able to give a n exact answer for this index i n the cases where ( m , Ip) = 1 as w e l l . W e start w i t h the following p r o p o s i t i o n . Proposition 5.12 Consider the following diagram, where the map on the right sends v ( Q r ) £ V(Qr) tov((f): (fi C )£Gp._*»_>v'(C *) n jP nc —^ lr There is a map (HC )^-- p G s n Proof. L e t u(x) = YA=O of r . a —> fiCjr CJX 4 making the diagram commutative. G ( f i C ) ^ p ' . For r G n a G G s, p u ( x ) is unaffected b y the o p e r a t i o n Thus, (CjX ) — CjX = C[ .j]X^ ^ a for each i, where [a • i] = a • i m o d n, i.e. Ci = C\ .i\ . n In p a r t i c u l a r , since 69 Chapter 5. Liftability where a,j = j 7 p r c \ + 1, we have s _ S l _ 1 = C[ kpS for 0 < »ij a fcp j < p whenever r aj £ G *, i.e., p w h e n (a,j,n) = 1. N o t e that for (k,p) = 1, =x JV'V- kpS1 3=0 1 = x $ (x ). kpS1 r pS 3=0 W h e n 0 < si < s — 1, (a?,n) = 1 for 0 < j < p; i n fact, { a , } form a subgroup o f G =. p enables us t o sort the terms i n the expansion of u(x) w i t h p s _ 1 This \ i into groups of p terms w i t h the same coefficient, g i v i n g u(x) = £ CiX + g(x)$ '(x ), 1 lT P p"~ \i l for suitable g. W h e n si = s — 1, we have p \ jy.\ + 1 = a,j for some j^, i n w h i c h case we now w r i t e T k u ( ) = ~ [i+/- *-i]K x gi{x °~ ) %s(x )+g(x) p + c P 1 r %s(x ) r p \i s = v{x )+g {x) pS 2 %s(x ), lr for suitable v, g . T h e m a p p i n g 2 u(x) u(x) l-r gives t h e required h o m o m o r p h i s m , as we now show. A p p l y i n g the same procedure to the inverse of Since $ » ( x ) = n[=o ^ V ^ ) ' r a n c p * ( u x P u(x), we get a n element vi(x ) i n 7LC . pS n " ) have the same image i n Z[£,< *], 0 < % < r. p Since the p r o j e c t i o n 2[/]/($ ,(/ ))4Z[/]/($ i 1 l y ( )) Z is 1-1, m a p p i n g a copy of Z[£,r] onto another copy of Z [ £ , r ] , we have v{y )-v {y ) pS pB l in =l 7L[y ]l ($,t(y ))> 0 < i < r. T h e p r o d u c t of these rings gives a complete W e d d e r b u r n pS pS d e c o m p o s i t i o n for H\x \j pS gives the required m a p . (x " r p — l ) , so v(x ") is i n fact a u n i t i n Z C , a n d the c o n s t r u c t i o n p n I 70 Chapter 5. Liftability M&) C o r o l l a r y 5 . 1 3 An element «(£») = ( C ^ U - Cn) )" « (Cn) « (C£ ) 2 to a unit in flC C only if the element v -i(Cr) ° d{Cl )^"^ -2 n 6 p v r ? ^{Cl ) n *'* T € «(Cn) " liftable liftable to a unit in fiCjr. P r o o f . Suppose v(( ) is the image of the unit it (a;). T h e n n u(x)^ ^ (QP\I - cr ) T where a = (TJ, — 1)(1 — r - i ) . p- {Q )~ " a{C,i )«d(C/ )^ 1 r 2a w r r pS > i • MC?-) -*'1 i • v (czy^ • 1 d U s i n g the m a p described i n t h e p r o p o s i t i o n , we get that p v (1_V1) liftable t o a u n i t i n s ftCV. Since w (Qr) is constructible, a I the corollary follows. T h i s leads t o the next theorem. T h e o r e m 5 . 1 4 When (m,lp) = 1, = l I m m \ e (C,r)J mm {eatCr*)' P P r o o f . B y L e m m a 3.1 o f [9], |F(CiO/L(C«0| = 4>{l ) = ^ ( Z - 1). A l s o cj)(l ). A n element v {( ) v (( )v {( ) r T b n c d <f>(p ), we must have that l ~ s r E n°(( ) pS n l is liftable only i f v {C r)<fa') d n = V{tf)/W(tf) r l Since I \ e | |ft°(Cn)/(L(Cn) n fi°(C»))| • S i m i l a r l y , p * " | |«°(Cn)/(£(C«) n « (C„))| 1 0 U s i n g now that |^(Cn)/^ (Cn)| = m is relatively p r i m e to b o t h I a n d p , we have that Z 0 r - 1 p |fi(Cn)/^(Cn)| w h i c h has the value m m'4>{l )(t>{p ). T h e quotient m m V ( r ) ^ ) / I - V " 2 r s 2 s r s - 1 1 | is p r i m e t o /p. T h u s n(Cn) i(Cn) |fc1 ^ L(CnY P KL K where we choose k\ > r — 1 a n d k > s — 1. T h e value of this second i n d e x is t h e n p r i m e t o Ip. 2 We choose &i a n d k large enough as well so that i m [ ; j (Q{( ) ^) lklpk2 2 Let r i : £lC /i —>• Fj[£] j n r (<£y(i r n has order p r i m e t o Ip. be derived from the m a p across the b o t t o m o f P B O . W e t h e n have that ( n (nc ) n/l x i)nimp,., (fi(Cnf * ) lp 2 = (ri 71 ((nc„/,) ) r_1 x 1) n i m ^ (n(c„)' *). fcl1 Chapter 5. Liftability Since ri(u(y) SlC s p c Z[y r r _ 1 1 ) = ri(u(y ]/(y r _ l p S _ 1 *)), we are able to restrict ourselves to units in the subgroup r For regular p, ftC> = YCp.; for irregular p ) oinC . n/l \SlC */YC s\ p p is a p-group so that (n(oc ) x n/i 1) n i m ) = (ri(YC» x i ) n imp.,] (fl(C„)' (o(Cn)' fclpfc2 M = This last line follows from the proof of Lemma o f 171=0 W(x ~ ) lr lpi (r^oT )) 1 6.1. fclpfca ) x I) nimp.,] {n(Cnf ) • lpk2 YC ° = YliZo W(y ~ *) r lp P which is, the image from YC in the upper left of PBO. These map onto W{C,£~ ) via Xn and, l n W(t{ ). in turn, onto rl Similarly, ( n ( i x nc )) (fi(Cn)' ) fclpfc2 nimp.,] n/p The fibre product W(y r 1 ) x / , l^(y n p p = (r (i x ^ ( y f ) ) ) nimp. .] 1 r x (n(c„)'* l p * ) • 2 ) is actually a direct product, since the images of the separate components intersect trivially in flCi . p Our situation now is as it was in 5.9. An element of 0,(^ ) is liftable if and only if its lklpk2 n *) — fi[Cp ] hes in W(t[ ) image in /([if ]/{^p {t[ 1 a rl s and its image in the other side of the s—1 lower left lies in W(t p )• The isomorphism groups acting on the images are H s and Hp, for p which we find generators a and p. The kernel of the maps a — l x p — l i n imp. .]fi(Cn)'* r precisely IF x IF*. X Thus we get L((nY = klpk as before and the rest follows. (l-l)(p-l)Lc.m.{e (( ,),e (( r)}, l I 72 p p l lpfc2 is Chapter 6 Setup (2) First we introduce some notation used for designation of various groups: A script notation will be used to signify a set of characters of the group C . In this context we n have - {Xi '• i\n but i ^ n}, V - {Xi '• P I * s b u i^n}, t £ ~ {Xi '• l \ i but i r 7^ n}. For a group X, the notation (X\S) for those elements of X which become trivial in every character which is not in the set S. For example: (YC \K) denotes the elements of YC n n (YC \V) denotes the elements of YC n n which are in the kernel Xnwhich have trivial image in both Z[C„] and Z C / . n p {YC /i\P) denotes those elememts of YC /i which have trivial image in Z C / , . n n n p This is in line with notation used in [7]. The next step in our program is an analysis of the kernel of the map Xn across the top of PBO, in particular (YC \K). n Our attention will be focused on several kernels contained in £lC , for which we use the further abbreviation Y$ for (YC \S), where S may be K., V, C or n n V U C We will also use the abbreviation Y for YC . n For a group X, we use the notation im^jX to denote the image of X in Q,Ck, where this makes sense. For the image of a group X C 0,C in the group in the lower left of PBO we use the n designation imp, ]X. 73 Chapter 6. Setup (2) Our work in this chapter is to get a description of the group QCn/i X /i n fiC /p p n ™[i.i.}Y K by disentangling information obtained in parts of P B 1 . We wish to develop the counterpart to the exact sequence (10) of [7], which relates groups in the upper left of PBO. This approach uses the fact that P B 1 is trivial, i.e., the group in the lower left of PBO is a direct product. To work with P B 1 more effectively, we switch our attention to sequences in the lower left of PBO. Consider the following diagram derived from P B 1 , with the squares being the pullbacks defining the groups fiC„/; x /ip ^C / n and YC /i x /i n p n n p YC / n which, for simplicity in these arguments, p we denote fi* and Y*: (The map i : Y* -> fi* is induced.) Y\ >• im Yic [n/l] fi* imr i/p]YlC ' [n/lp] K lm Y fiC^, for example, denotes the image if fi* in fiC /j. The " " designation indicates a subgroup L n defined to ensure surjectivity of the maps in the front pull-back. The following argument is strictly formal, and arises from this diagram. Because the maps into fiC^ p and im[ // ]Y£ are surjective, we get the commutative diagram n p which is exact in the rows: 1 • Y* ->fi* > im Y [n/l] K •* fic^ x /( x im[ p]YJc n/ nc n/p • '^[ /lp] ic Y + nc n n/lp > 1 -+ I For the maps from the product in the centre of these diagrams we take the multiplicative difference ((u,v) H-» uv~ ) of images from P B 1 . The maps going down are inclusions. l 74 Chapter 6. Setup (2) We then have a short exact sequence for the quotient groups. nc n/l n* 1— y— * > [n/;]^c Y The im nc.n/p nc.n/lp im Y [n/p] im / Y K [n lp] K group imp^jF/c is contained i n Y*, b u t m a y be p a r t i a l l y diagonal. B y extending this sequence we are able to isolate our group of interest: n* ua-[u.]Yic Y* i m [U.]Yt nc n/l nc n/p im ,]Y£ im Y)c nc n/lp im [n/p] [ n / [ n / i p ] Y*; -»• 1, giving the calculation nc.n/lp n* im [I.I.]YK im ~n/l]Y)C im im [U.]YK ' [n/lp]YK lm nc n/p nc n/l (6.1) n/p^K U p o n further analysis we find that the group im[n.]YJc has several aliases w h i c h we identify i n the following l e m m a . Lemma 6.1 ^ im[nit\Y-p c ^ im[ / ]Yp im[/.,.]Y/c im[ /i]Yp ~ im[ / Y u n p uC _ n n p] c ^ Y y ~ YY' Cu C V Proof. T h e p r o o f comes t h r o u g h a n analysis of the following d i a g r a m Y-pYc ' ' [n/l] V lm The Y n/lp ' [n/p]Yc ' x lm • Yp uC +Y ' K > ^[n/l]YpuC X-n/lp' [n/p]YpuC' lm > Y*. maps across the t o p as well as those across the b o t t o m are inclusions. T h e maps t o the b o t t o m are restrictions of the m a p o n the right side of P B O . The first two groups o n the b o t t o m have t r i v i a l images i n nCn/ip, a n d so are a c t u a l direct p r o d u c t s i n Y*. Since a l l o f the groups along the t o p are i n the kernel o f the t o p m a p i n P B O , they are isom o r p h i c w i t h their images i n the lower right o f P B O . T h e first group, YpYc, is a c t u a l l y t h e direct p r o d u c t Y-p x Yc a n d is isomorphic w i t h its image below. T h e images of the second a n d t h i r d i n the groups below m a y be p a r t i a l l y diagonal a n d this l e m m a gives several versions o f 75 Chapter 6. Setup (2) the c o n d i t i o n g i v i n g rise to this possibility. T h e right square induces a m a p im[n/i] vuc x i m [ / p ] F y u £ Y* Y n im[U.]YvuC im .]Y/c' [u I n w h a t follows, we show that this m a p is actually a n i s o m o r p h i s m . F i r s t , let u be any element i n Y*. W e pick an element v! € Yjt w h i c h has the same image as u in YC /i . n p T h e coset corresponding to u(im[{.z.]t/) maps to the coset corresponding to -1 u, so the m a p is surjective. N e x t , we note that any element i n imj/xjYc H i m [ / j ] } p u £ x i m [ / ] Y p u £ n has t r i v i a l image i n YC /i , n p a n d so must be i n im^^Ypuc- n p Hence, the m a p is injective a n d is an isomorphism. T h e p r o o f is completed by l o o k i n g at the left square i n the d i a g r a m . T h e kernel i n the m a p Ypuc -» im[ /(]lpu£ n is precisely Yc, so we have Ypuc • -y— ~ im Y-p , v [n/l] uC a n d hence YpY ~ im Yp c [n/l] W e now look at the maps imr //]Ypu£ n Ypuc ^ im[ mYp n uC x im Yp [n/p] TT~ X 1. -» -r 1 uC T h i s is a c t u a l l y exact as we now prove. Suppose the element i n the m i d d l e group, u — u\ x u 2 0. W e can find a n element (ui i m [ / ; ] U ) ra 2 _1 u' 2 € Ypuc s u c that i m [ / ] U n N P 2 x 1 is the image of a n element u[ € Yp C Y P U £ u' !->• 2 76 u. u- = a 2 n Then d we have u (imp.jju^) -1 Chapter 6. Setup (2) Any element of Yp c either is in Yp or has non-trivial image in nC[n/p] and its image is in the u kernel of the second map. Thus im[n/i]Ypu£ im[n/ ]Ypu£ x im[ /nYp P n uC ^[n/i]Yp [i.i.]Ypuc im By symmetry we must have Y-p c ^ i [n/p]Yyu£ Y Yc ~ im p]Y ' m u v [n/ £ as well and the proof is completed. I Further analysis of Yp c/Y-pYc will be left to the next chapter. We now interpret the other three groups in the calculation. For k = n/l, n/p and n/lp im Y [k] C YC C fiCfc K k which allows us to rewrite: net net YC (6.2) k YC im [k]Y)C k The first group on the left side of the equation we know inductively. For k = n/lp, we analyse the second group in the next chapter. For k = n/l and k = n/p we work out a further breakdown of this second group in the following. This will be done for k = n/l, the procedure for k = n/p being identical. From the right side of PB1, we have the commutative diagram 1 > (im[ /qYJC | V) > im j]Y,c n 1 > Y(C | V) n/l • im , Y,c [n/ [n/ p] •1 ->YC, n/lp • YQ n/l The maps going down are inclusions; the rows are exact. Upon division, then, we get the short exact sequence i } Y(C n/l | V) {im[n/l} ic | V) Y ) YC n/l im [n/i] 77 ? Y/c YC n/lp im > Y [n/lp] K i Chapter 6. Setup (2) The elements in (im[ /j]V5c I *P) are the images of elements in YJC and survive only in the n characters V U £ , i.e., (im[ /j]rx: | V) = im[ ]Yp . n n// u£ This enables us to extend the sequence of quotients to _^ imfn/qY^ur _^ Y(C \ V) _^ YC im[n/i]Yp im /qYp im^jYc n/l YC im ] Y/c n/l n/lp [n [n/ip Since the map Yc —>•fi*is injective, the first group is isomorphic to Ypuc/YpYc- This sequence then gives the calculation YC,n/l i m i n/l]YK Y(C Yjvuc Y Yc | V) n/l '^{n/i}Yr v YC,n/lp (6.3) ' -[n/lp] >C lm Y The second group on each side of this equation, we have already encountered. These are topics for the next chapter. The calculations for the first group on the left give the remaining topic for the next chapter. Putting these pieces together we get the following proposition: Proposition 6.2 Y(C \V) n/p n/l YC / {n/l] V im n p Y YCn/lp [n/lp] K. n/p [n/p] C im Y im Y n/lp Yp,C i m [U.]YK Y(C \C) YvYc YC /[ n p Proof: Using substitutions given by (6.2) for first k = n/lp, then k = n/l and finally k = n/p in (6.1), and, as well, a substitution from Lemma 6.1, we derive fi* fiC'n/lp fiC'n/lp im[u.]Y/c YG n/lp YG n/lp Ycuv Y Yp c fiCn/l fiCn/p YC n/l YC,n/p YC n/l im[„/*]Y/c YC,n/p imr x/p\YK Substitution for these last two groups on the right using (6.3) finishes the proof. I Note. In applying the proposition for n = Ip, we have that the last group on the bottom 78 Chapter 6. Setup (2) and the first, second and last groups on the top in the quotient on the right are trivial. The second group is the kernel of the map E of [7] and has order ra'. The groups U* and imy.qF/c are WC x WQ and imp.]Y(Q \p, I). Y(C p P p \ V) and Y(C / n n p \ C) are simply the groups WCp and WCi while rm^/qY-p and im.[ / ]Yc have the final designation n p W~ i l T and W~p p T in [7]. The proposition applied in this case becomes WC x WQ im[u.]Y{Ci \p,l) YCr, P p ra' YCi W~P P T The study of the the groups of this proposition is left to the next chapter. 79 (6.4) Chapter 7 Calculation of Group Orders 7.1 Images of constructible kernels in and O C / VtC /i n n p Our analysis of the group Y(C | V) n/l iva[n/l]Yv will proceed in reference to a filtration including the group Y(C /i n \ V) '~ , which has an easier T l description. Lemma 7.1 im Y [n/l] Proof. Since YC /i d T \ V) '~ . T n/l l is the direct product n rid|(n/0 W(y ) '- D Y{C v For each d \ {n/l), let a' e A H n n \ V) ~ n l C II d\{n/l) *Ay ) '- - w d T 1 with a' ->• a € A i 7 / ; . 2 d 1} C H . any element u(y) € Y(C /i d has a breakdown 1 «(y) = Since IV rid|(n/0 W{y ) 2 n d d n £lC /i, it is invariant under the actions of the subgroup Gi = { T j ( n ) n +1 : (i(j),p) = There are two quite different considerations according to the power of I dividing n = l p . For r = 1 this group has order (I — 1) while for r > 1, it is of order L We divide the r s proof for this first part of the lemma into these two cases. Case 1 (n = lp ): Here TJ is an automorphism of OC /j. Note that s n u (x) d = w (x ) - "w , (x )-ZG ld ad 1 T d a d l ^ d p - i ^ y d y d - i ) 80 = W a d { y dyi-\ Chapter 7. Calculation of Group Orders w{x) = rid|(n/n d{x) —> 1 for Xn a n d a l l the characters C. A l s o w{x) —> 1 i n YC n , u n p w(x) 6 Yp a n d we have u(y) G im[ /qYp. n C a s e 2 (n = T p w i t h r > 1): Here s u (x) = w (x )w ,(x rZG ld d () w x = Ud\(n/l) d( ) u ( *)n-i. d ad a l Wad y G Yp a n d w(x) -> « ( y ) . l x T h i s gives us the filtration n C / / I VY'- 1 C i m n [ n / q y P C Y(C n / , | P). I n the following l e m m a we establish the index of the first group over the final group. L e m m a 7.2 Y(C n/l iW- -mv )(ii | V) l _i) s 9 W) ' Proof. T h e proof proceeds b y i n d u c t i o n o n r , the power of I i n n = l p . r s C a s e 1 (r = 1): F o r 0 < j < s we have the short exact sequence 1 ->• (wbfyv) -> 1 ^ ( / ) - » W ( ^ ) -> 1. N o t e that z is cyclic of order p s 1 while y has order p so we have the i s o m o r p h i s m s W e also note that for each j, W(y ) p3 Tl 1 C W{y ) a n d that this sequence remains exact w h e n pl we a p p l y the operator T\ — I to each of the groups. F r o m i 1 > (w^^vy- 1 • (WiifyV) > w^y- 1 • WiyP ) 3 81 > Wiy^Y 1' 1 •W ^ ) 1 >1 •1 so Chapter 7. Calculation of Group Orders we get the calculation {W{y^)\V) W{y^) {w^^vy- W{y* ) +1 wiypy'- 1 1 It follows then that -n 3=0 Y{c \vy^ n/l {W{y^)\V) (W{yP )\V) 'j T W(y) w( y- 1 1 y establishing this step in the induction Case 2 (r > 1): We note that Y(C \V) = n/l fliWiv^m xY(C \V), n/P 3=0 Also ( ^ = 0 ( ^ ( ^ ) 1 ^ ) ) ' = Y(C \V) whereY{C .\V) = Y(C \Vy. n/l n/l (modY(C \Vy- ). 1 n/P n/l Thus, we have nUoiw^m Y(C \V) n/l Y(C ,\V) n/l Y{c \vy-i Y{c .\vy- 1 nll n/l our inductive relation. Prom the diagram 1 > (Whf^V) > Wiy^) • Wiy^ ) 1 > (WiyP )^) > WiyP ) • WiyP 1 1 3 1 3 •1 1 ) 3 >1 we get the calculation Wiy?) WiyP ) (W(y*)\V) WiyP ^) W{yV ) 3 3 1 +1 1 leading to (W(y)\V) (115=0(^(1^)1^)) (w(y)\vy = i{\H \-l) n/l = j(</>(n/0/2-l)_ Using 4>{n/l) = l ~ (l — l)4>(p ) we have r 2 s <t>{l ~ ) r <t>(l ) l r 82 Chapter 7. Calculation of Group Orders I a n d the i n d u c t i o n is complete. E s t a b l i s h i n g the order of im[n/i]Yp/Y(Cn/i \ V) ' T 1 is a more delicate matter, a n d involves a careful consideration of the i s o m o r p h i s m H ~ A H d for the various i s o m o r p h i s m groups H where d \ n, a n d of maps between these a u t o m o p h i s m d groups. m i™[n/i]Yp Proposition 7.3 Y(C | pS-i if (p,l - 1) = 1. {I-I,P-I) ^ p ^ s Q ^ | / _ p i . n/l 2 " s 2 ifp = 2, s>2 and I = 1 ( m o d 4 ) . 2 " 3 ifp = 2, s > 2 and I = 3 ( m o d 4 ) . s 1 ifp = 2,s?2. Proof. T h e p r o o f is again b y i n d u c t i o n o n r, the power of I i n n = Vp s Case 1 (r = 1): Let u(x) G Y-p. Y being the direct p r o d u c t rLi|n WC ^)' 3 * n e projection of u(x) into each of these components must be invariant under the a c t i o n of Gi. W h e r e u(x) = f[w (xP )llw (x P ) i i=0 this means that w (x ) p3 as a- G AHn/pi l ai J /3j j=0 = 1. Further, for 0 < i < s, at, must be o f the f o r m ( ^ G ; ) a ^ where w i t h (/ - l ) a j G A # . Now 2 n / p i v(z ) p i E G '^.(^ T p r l _ 1 -+ ±i i n Z [ ( ] - I n fact n ^a;(^ p i ) E G '^(^ p i ) r r l _ 1 -» ± 1 i n Z[C/pj] for 0 < j < s. W e rewrite u(x) = ui(x)n2(x) where s-l «i(ar) = [^^(^^^^(x^Vr -! 1 i=0 83 Chapter 7. Calculation of Group Orders and 5-1 Mx) = Y[wp>(x ), lpj 3=0 with/3^. = 5 - ( r f - l ) a ; . . 1 / j I n fact, u (x) = 1 w h i c h we see as follows. Since b o t h u(x) a n d ui(x) 2 — > ±1 i n each o f the Z[C(pj], U2{x) must as well. B u t u (x) is actually a function of a;', i.e u (x) = u (x ). W i t h the l 2 note t h a t —1 is not liftable from 2 2 7L[Q i\ to YC *, this gives p P u' (x) -> 1 2 i n Z[C i], 0 < i < s. Hence p u (x) = u' {x ) ->• 1, l 2 2 i n these rings as well, since 77 is a n a u t o m o r p h i s m i n each o f these rings. T h i s means that U2{x) = 1, since it is t r i v i a l i n a l l characters. T h u s , any u(x) € Yp has the form s-l u(x) = Y[w (x )Z w {x ) >~ - . p, Gl lpi < T 1 1 < (7.1) i=0 T o take into account that some factors i n this product m a y m a p to — 1 i n fi(Cn), we define Y- = Y^/Yp ( - (-l)DYC \V). 1 x n is at most a group of order 2. (7.1) then gives a direct p r o d u c t d e c o m p o s i t i o n s Yv =Y[W(x ), pi i=0 where w y ) = w{x )w{x ) n Y^. pi Defining lpi W(y ) to be the image of W(x ) i n Y(C\V) we have pi pi W(y ) 'pi T 1 C Wip) C W(y ). pi 84 Chapter 7. Calculation of Group Orders This gives us that =n im[n/(]Yp n-l Y(C \V) n/l W{yP') 'r i=0 and we evaluate the order of the group on the left through this decomposition. Prom PBO we have the following diagram for each of the i's,: ^ { ± 1 } C % ] W(x ) pi W(yP ) WC s-i C { {1} C p W(z )~WCp.-i-i. p, Consider an element Vi(x) where a' G AH s-i lp =w (x jZ w , (x ) t pi G, < with (/ — l)a' lpi T 1 - a i 2 lpS v (y) = w ,(y ) - ' pi W{x ), G pi If the image of a\ in G A H -i. i 1 l A H ,-i 2 p €Y(C \Vy- . T 1 a n/l What, then, are the conditions under which W(y ) is properly contained in pl The condition Vi(x) = 1 G W(2 ->• w .{z ) ~ pi l n al occurs if and only if —»• 0 in then P < ) means that (l - TjJaJ = W(x ). 0 in AH pt This -i-i. pS AH ,-i-i. p We consider the maps AH, AH -i A H, A H s-i AH s-i-l n3 2 p (7.2) A H -i-i' 2 2 p pS The order of the first group is Z — 1 times the order of the second. Where 0 < i < s — 1, the order of the second group is p times the order of the third. We use the fact that (I — l)a' A H s-i 2 i in the kernel of the second map. Iip\(l- 1), then a- G A H -i 2 pS giving W&) = 85 W(y Y'- . pi 1 p is Chapter 7. Calculation of Group Orders for these cases. If p | (/ — 1) ( a n d s > 2 i f p = 2), Hi -i contains a n elementary p-subgroup o f order p 2 pS H -i pS is cyclic. Hence, we c a n find a n a\ € AH ,-i w i t h pa\ E A H -i b u t a' 2 lp lpS while A H -i. 2 pS T h e n we have P- W(yP') <T W h e r e i = s — 1, (7.2) becomes AH,i A H,„ H AH A tf„ n P 2 2 <_1 p ± * 1. P T h i s was considered for the case n — lp [7]. Here we have = W(yP ~ ) 's l T (/-l,p-l)/2. 1 T h e final consideration here is to determine the order of Yp /Yp. F o r t h i s we look at the value of Vi(Cn) for Note that ( T a - 1) E G j = ( r - 1) ( ( £ Gj) - (/ - 1)) + ( r - 1)(Z - 1). F r o m (iv) a n d (vii) of a a L e m m a 2.3, If n is o d d this has the value 1, a n d consequently Yp = Y-p. T h e n , we have = p*- (Z-l,p-l)/2. 1 Y(C | V)«~ l n/l W h e n p = 2, WC2 a n d WC4 are t r i v i a l , so we need o n l y be concerned where s — i > 2. W e go back to the consideration where (I — l)a' E A H s-n 2 2 choose here is a' = r — 1 where a = 2 s - l - 1 a W h e r e I = 1 ( m o d 4), a ~ l = 1 (mod 2 * - l S 2 2 T h e representative we Z + 1. i + 1 Z ) , so that xyi-i) (Cn*) = 1. W h e r e I = 3 ( m o d 4), a ' " = 2 ~H + 1 ( m o d 2 1 b u t a' £ A H =-i. s i + 1 86 Z ) , so that v «_D (0?') = - 1 . ( Chapter 7. Calculation of Group Orders Y v / Yp is n o n - t r i v i a l only w h e n / = 3 (mod 4) a n d p = 2 w i t h s > 2. S u m m i n g u p these cases for p = 2 we have im /*]Yp [n = 1 Y(C /i I V) '~ T l if s < 2. n = 2 if s > 2 s - 2 = 2s - 3 Z EE 1 and s > 2 and if / = 3 ( m o d 4). ( m o d 4). (7.3) Case 2 ( r > 2): L e t u(x) € Y p . u{x) is of the form u(x) = (f[w (x )) (x ). pi ai l Ul i=0 A g a i n we use the fact that u(x) is invariant under the actions of the group Gi — {r a + 1,0 < i < I — 1}. However, i n this case, the order of G; is I, since OJJ'S a\ € A i T ^ s - ; w i t h la\ 6 must t h e n be of the form ( ^ G j ) o 4 where : a = + 1,1) = 1. T h e A H 2 l v p s - i . For 0 < i < s — 2, we have W(x )^W(z ')cYC pt p n/lp a n d is the o n l y component of Y w h i c h maps into W(z ). Therefore, w (z ) pt pt Qi = 1, — Wiai\z ) pX w h i c h implies that a\ = 0 in A H p - i p s - i - i . W(x ) a n d W(z ) are pl T h e a u t o m o r p h i s m groups of pl H ^ p s - i and H [ r - i p s - i - i ) respectively. C o n s i d e r the maps AHlrpS-i A 2 H l r AHlT-lpH-i p s - i T h e order of the second group is A 2 H l r - l A H l T - l p S - i - l p s - i A 2 H t r - l p s - i - l ' p times the order of the second, la' = 0 € therefore i m p l i e s that a' € A ifjr-i -;. N o t e that 2 p3 w ,.(x )^ 'w (x )pi a G lpi ai 87 1 -> 1 A 2 H l r - i p S - i - i Chapter 7. Calculation of Group Orders in all the rings Zforp,-] 0 < j < s and also in 0,C p . n and its image in YC p, n w ,{y )^ {y )pi lpi a Setting Consequently p pi pl G lp a l T a Y(C \Vy>- . 1 n/l for the i's where a- e Vi(x) = w i.(x )'£ 'w < (x ')- , a e = w ,(y ) - ' 1 Wai 1 i we further consider A H r-i -i, 2 l pS the simpler unit s-2 = u'(x) JJWJ(X) - w _ (x '~ )w (x ')u' (x ), p = 1 at 1 p 1 l at 1 i=0 where u'^x) = u {x ) UtZo w (x '). l lp x ai We consider now the cases i = s — 1 and i = s. In the map YC„ -> the preimage of W ( z w _ (x " and w ) p aa 1 1 pS 1 We will show first that the elements p 1 pS 1 p are not independent, in fact that a = —T~ CX -I in Hp. (x ) s )w (z ) p l n ) is the product W(x ~ )W(x "). as Since w _ {z " 0ls pS_1 YC /i S = 1 and T is an automorphism of W(z " *), pS p cts P a -i + ra s p = 0 in s H T-i. t Thus v . ( ^ k ( / ) T M i e i y ( / ) 1 and hence is trivial for all the characters in V. Now we use that OJJ = a' ^ G\. Then t ct' _ + r a' = 0 in if/r-i, s since (a' _ + r o;' ) ^ s 1 p s = /(Q;' _ S 1 1 p s +T a' ) = 0 in Hp-i, which is torsion free. For the remaining p s characters Xk, f ° which V \ k, r Xk(w ._ (x ')w Ax ') >) = (1 P a 1 P T a 88 -Cfc) ' Q s 1 + T X = 1 Chapter 7. Calculation of Group Orders modulo torsion. Thus, w _ (x ")w 1 p as ct -i 1 s p s Tp + Ta s and a = —r a -i is a torsion unit and must in fact be 1. Hence (x ") p cts P = 0 in H r, s t as claimed. Further note that the automorphism group H^ of W(x ') p s implies that a ' E A HIT la' _ 2 l s - 1 G is cyclic of order Z | i ^ r - i | . Thus, A H r-i. 2 t The question left to consider is to find when it is possible to have an IOL' _I E A Hir 2 S p but with a' _ s t p p a E Y(C \V) '- . T a with p AH r-i . 1 G A H r-\ . If I \ (p — 1), then the Z-subgroup of Hir is cyclic and a' _ and v -i(y)v (y) G AHir s Then v -\(x)v (x) 2 l 1 p a E Y-p s Then 1 n/l This completes the inductive step. <_ (^" X_ (^) 1 1 1 T p " eY(C \V), n/l and we are left with determining the corresponding index where the order of the group C /i n has dropped by a factor of I. Thus for I \ (p — 1) \m / ^YjCn / i\V) Y(C \V) Tl-l im Y{C \V) [n/l] [n n Y(c \vyt-i n/l n/l2 and is by induction the same as the index calculated in Part 1. Where Z | (p—1), the situation requires a more careful consideration. What we will show is that for Z odd with Z r _ 1 For some /3 G A Hir and a E 7L with a = ± 1 modulo Vp, o^ _ = (r - 1) + (5 where 2 1 p Z(r — 1) G A Hir . 2 a p \ (p—1) the index is increased by a factor of I at this inductive step. s We can write a = a;a for some r , G p a copies of these groups in H\r . Since this means that (r ) 1 p ai 89 x a and r a G G where we take the p p = 1 in Hp and hence that r i = 1 a Chapter 7. Calculation of Group Orders in HIT-i. Using Ta ai ~ 1 = (T ~ l)(T p ai - 1) + (r , - 1) + ( r - 1) ap 0 0p we can rewrite <x's-l = (T -l) + ai where /? = 0 + (r - l)(r a; ap (Ta -l)+P' p - 1). Using the norm relations on images in Z[C ] we have the following n ^(cr )^ 1 r -i(cr ) _1 =±i, EG; ap ^(cr ) _1 = Mci ) EG; pS_1 Then ±«a (d '" )^(C^'" )«'a-(C i '" K(0 = 1, p 1 1 p J> / which means that ± w Q p (£ a— 1 ) is the image of some constructible unit w(x ) € QC . This p l n n implies constructibility for ±v (Qr-i ), a ZC;r-i . p 1 I i.e. that it is the image of a constructible unit from p Where k is the highest power oil dividing {4>{l ~ ,p)/2, this means that a must be r l p a 2/ th power in G . Further, to have a _ i £ H -i fc p s lr p we need that 2 / fe+1 | (p — 1). For this we need 2f- |(p-l). 1 This is a necessary condition. We next prove sufficiency. If 2Z ~ r 1 | (p — 1) we take r ((f>(l ~ ),cf)(p)). r a — E G C G V such that a has order kl modulo p where k — p p With this choice, a = 1 mod l and r 2 but r fc a T 1 ^ A H r-i . 2 l p We know that 90 a 1 in Hp. Also, /(T fc_ - 1) € 0 x A Hir 2 p Chapter 7. Calculation of Group Orders W h a t we want to show is that for some w(x ) € y, l a n d further that u(x) := w -\(x pS 'iw(x )~ 1 G Tak have that u(y) $ l € Yp. B y this construction, we w i l l as well l Y(C \V) '- . T 1 n/l For some integers ki, k , k = ki(f>(l ~ ) + k (fr(p). U s i n g r 2 p p and we o b t a i n t>.(cr~Y»-(cr~> E G ^^ or where 71 = ki^2G/r-i further that T k_i a + k Y^G P —1 = (r - k p a — 1) € AH[r , Y^i=o k (l - r , ) a n d - 1 72 = p k (l - T~ ). 1 73 = t p Note gives us a T ^(cio=^(cr )Sr -i (cr"), _1 a where 74 = J2i=~o d ~ k. C h o o s i n g T (x ~ ) lpS w )74 1 = W( _ {x ° lp Ta lh4 1 )u>( _ ) (x' ' p To 1 7l 1 ) _ 1 ^ ( T a - i ) 7 2 ( x r p 3 1 ) (r -1)73( w a x l p S ) gives us what we want. C e r t a i n l y b u t it also maps to 1 i n Z[Cjipj] for 0 < i < r a n d 0 < j < s since r a Thus « ( x ) G 1>. 1 91 1-4 1 i n a l l the groups f f y . Chapter 7. Calculation of Group Orders T h e results from L e m m a 7.2 a n d P r o p o s i t i o n 7.3 coordinate w i t h the results from L e m m a 3.5 g i v i n g the size o f unit groups i n the finite rings i n the lower right o f P B O a n d T h e o r e m 5.8 g i v i n g t h e index o f constructibility i n fi(C ). T h i s is displayed i n the proofs o f P r o p o s i t i o n 8.2 n a n d T h e o r e m 8.3 of the next chapter. 7.2 Images of constructible kernels in SlC /i n p In analyzing [n/ip]^C im we use the filtration (YC y>- (YC yr-*> C im Y l n/lp n/lp [n/lp] YC C K n/lp set u p b y the following lemma. Lemma 7.4 im Y [n/lp] D K (YC y- (YC yr- . l P r o o f . F r o m L e m m a 7.1 (YC /t) i~ T Similarly, Y C / n ) p" C im l p T ( p n = im[n/i ](YC /iy~ l n n/lp C im[ /;]yic. T h u s l n (YC /l y~ p n/lp P p C im[ n/Ip n ]Y,c. [ B / , ] (im- qY/c) = p n/ im Y . [n/lp] K I O u r calculations start w i t h t h e index o f the first group i n the last group of the filtration. L e m m a 7.5 YG n/lp YC -i YC r-l t pS {YCn/lpY ' { C /i Y T Y n p " T P r o o f . L e t z b e a generator of C / n (YC^y^ (YC -r) l Tl pS a n d w(z ) € d lp W(z ). d Selecting k\ a n d k so that k\l + k p = 1, we have 2 2 k\(l - n) + k {p - T ) = 1 — km - k T 2 P 92 2 p Chapter 7. Calculation of Group Orders and P{1 - + i(p - T ) n) d W^djl-kiTt-klTr e l d p l Tl d p p z T a n ( p J ( d}-Tt(p-T )+T (l-T,) £ W z p W(z ) - W{z ) - rW{z y- W(z ) - r d -Tl{p-T ). P and w(z ) ~ p therefore generate the same subgroup of Tl ( d)l-klTl-k2T W P T {l-n) = The elements w(z ) ~ + IT = -pn p T pd Tl and we are able to rewrite ld p = T (YC /i ) ' (YC /i ) ~ lT n / p p p n ^pd^l-^ W as YC /i n p ^ldy-T _ W p T h u g W{z ) - - rW{z y- W{z ) - r, d Tp a l klTl k2T pd Tl ld p T s p ^ l-klT -k Tp I Yl 2 W(z ) {YCtr-iYCps-i) lipj 1 {YCir-iYCps-i) ~ n p (7.4) Tp 0<t<r-l \0<j<s-l where Y C ^ - i = Ilo<t<r-i { ' ') and YC .-i ators z C -i. w pS 1 for Q r - i and z lr Since ( y C , , - . r c i ) l l T lpS for 1 = IIo<j<s-i W(z lT p ), choosing the gener- lpj pS (YCV- YC -i) N r (YC r-i) ~ ' z 1 p _ T p p 3 is the direct product ( y C . - i ) p H (yC„.-i) " x p Tp ( y C , r - i f " , the lemma follows by showing that T p YC, n/lp {YC fip) 1 n Tl Y CIT—i YCps—i (YC /i ) P-Tp n (YCtr-iYCps-i) ^ 1 p (7.5) (YCir-iYC s-i) ' 1 p Tp p This in turn follows by showing that we can drop successive terms from the top and bottom of the quotient group rio<i<r-l 0<j<s-l II o<i<r-i W(z ) lipi 0<j<s-l ) ^ W(z ' ))YC r-lYC -l ' l pJ l " {YCtr-.YCps-,) 2 1 pS Tl (YCr-iYCr-i) P-T p 1 without changing the order. We start with a multiplicative basis, B , for each W(z ), d \ n/lp. d can be extended to a basis for (YC /i ) n ( Y C / r - i Y C - i ) * ' (YCir-\YCps-i) T p S p ' (YC /i ) T p p n . p Tp l-k\Tl-k2T U <i<r-i B, 0<j<s-l i p 0 using only elements of the group We now set up a matrix whose rows consist of the Tp powers the elements of the successive B s d in the expansion of elements of the second basis. The determinant wil give the order. If we start with the basis of W(z) the first columns will 93 Chapter 7. Calculation of Group Orders have l ' s o n the diagonal, these b e i n g the o n l y non-zero entries. D r o p p i n g these first rows a n d c o l u m n s keeps the value of the determinant, a n d corresponds t o d r o p p i n g W(z) from t h e t o p a n d W(z) ~ '~ p 1 klT from b o t t o m of the quotient. C o n t i n u i n g i n this way, we arrive at (7.5). k2T I W e now analyze the group YC^-ij ( y Q - i f " ' , using z as a generator for Q r - i . (YC[r-i) ~ l Tl T f For 10(2;) e Y C j r - i , wizf = u(z f~ 1 T h u s , YCp-i l 2 = ...= u(z r _ 1 ) - 1 (mod ( y C r i ) ) . H I (yQr-O " ' a n d consequently Y Q r - i / ( V C ^ - i ) " ' ( y c V i ) " " is a n Z-group. 1 1 1 p 7 T L e t f i b e the order o f < T > i n H i, a n d g p its index. T h e n we have the following l e m m a . P p i Lemma 7.6 If f n p> t p = 1 then the group WCii/(WC[i) ~ p °f 9p,l' ~ 1 cyclic groups, each of order p^p.' If fp,l* 7^ 1 then the group WCii/(WCii) ~ p 1 — has rank g \i Tp p and is the product of g n g i p pl groups, each of order p^ ^ — 1, and a cyclic group of order p Proof. {r T h e actions o f < T > d i v i d e H i into g i P '• 1 < i < 9p,i<} g i ai m u l t i p l e s of a r a i v e 1 and is the product 1. has rank Tp — t (j/p-'' — 1 cyclic l ) / ( p — 1). orbits, each h a v i n g f i pl — elements. L e t pl representatives from each orbit. M o d u l o p — r , successive p-power p w i l l give the other elements i n its o r b i t , w i t h p^p.'* r = r . T h u s each element ai a ; of the group Z i 7 ; i / ( p — T ) has order pf < — 1. T h e group is the direct p r o d u c t o f g n cyclic p li P p groups of order p^p.'" — 1, w i t h generators given b y {r }. ai T o arrive at t h e order o f~S-H i/(p - r , E Hi ) — WCii/(WCti) ~ , { X n o r m r e l a t i o n E Hi* a s w e p we must d i v i d e o u t the Tp p U - Since p E Hi* = TpYj Hl = E Hih i t { n a s order p - 1, so the order we are seeking is (p-^'* — l ) / ( p — 1). Yli=i by T 1 ([ a;]) i s t i l l direct i n Z i f y / (p — T T s has order {p^ li 3p ( i P , ^2 Hit). — 1)/(p—1), a n d has generator 94 M o d u l o this p r o d u c t , the group generated T . E < P >] > T Q 9; t n e s u mo v e r elements Chapter 7. Calculation in the orbit of r a of Group Orders .. The product n ^.]) (K, E p ]) x <r > ii is direct and equals Z-ffy / (p — r , ]T} -#/•)• ^ p If p has even order mod I, then p^p< = — 1 mod /, and (l,p^p- — 1) = 1. Then, for each li i, has order prime to I. Since YC^-i/(YCir-i) ~ p WCii/(WCii) ~ p li Tp p is isomorphic to a direct T product of such groups, it as well has order prime to I and the group 7 C | , - i j{YC\r-\) ~ P p (FC^-i ) ~ T l is trivial. If p has odd order mod /, then p*p< = 1 mod I, and / divides these group orders. One thing to li note is that if I \ f , then p p>' - 1 = {p p.' f { f i/l p>li - l)$t(p ,' ). fp i/l In this case then, I | ^i(p p^ f il1 and must divide (p*p< — l)/p — 1 as well. Consequently, the rank of the Z-Sylow subgroup of li each WC i/(WCii) ~ p Tp t must be the same as the rank of the whole group. Let ri(n) be the sum of the ranks of the Z-Sylow subgroups of the groups WCii/(WCii) p T p , 1 < i < T — 1. Then we define with the corresponding definition for g {n). We then have the following proposition: p Proposition 7.7 YC,n/lp {YC /i ) 1 n p ' T (YC /i ) P-Tp n = 9l{n)g (n). P p Proof. We will simply show a step in the inductive proof that = 9l{n). For the case where p has even order mod Z, it is shown above that this index is trivial. Since the orders of the groups WC^ j (WCii) ~ , p Tp 1 <i <r — l i s prime to Z, the ranks of the Z-Sylow subgroups are all 0, so that gi(n) = 1. 95 n Chapter 7. Calculation of Group Orders Where p has odd order mod p, the group WC^-iI{WC T-\) ~ P v is a direct product of cyclic T 1 groups of order divisible by I. Taking the subset of {r } giving generators of this group, we can ai again set up a matrix to determine the index we are seeking with Vs in the diagonal positions of the first columns with O's being the remaining entries in these columns, where p is the rank of WCir-iI(WCir-i) ~ p, p the determinant will be l times the determinant of a reduced matrix T p giving the order of YQr-2/(YC^)' "' -7 ( 7 ^ - 2 ^ . I The proposition follows from the definition of gi (n). We now concern ourselves with the map ' [n/lp]YfC lm Y K (YC„/; ) ' (YC /ip) p p Tp n Let K\ be the subgroup of Y/c generated by elements of the form u{x) = where dd\ | n, a € AH / n d w {x ) ~ ^ w {x ')-^ d i :Gd, d a with d'a 6 A H j (7.6) dd a , and 7 = 1 if lp | n/dd', 1 — r 2 n d if I \ n/dd' or 1 x 1 — Tp if p \ n/dd'. We separate the discussion into the following two lemmas and, finally, a 1 YC,n/lp proposition which gives the order of Proof. If w (z ) - > d a l T s I im . i/ipytc r Lemma 7.8 Kx -» {YCn/lpy vc Y YY V C {YC )\P- P T n n/lp € {YC ip) ' where a G A H , 1 T n/ 2 n/d where 7 is as above. Thus im[ /; ]Ki D (YC /i ) 1 n p n T i p then . Likewise im[ /;p]ii'i D {YC /i ) p n We establish the reverse inclusion by induction on powers of / and p If di = I in (7.6), then u{x) i-> w (z ) - '. d a l T For d = Z V , i > 1, x 96 n p . Tp Chapter 7. Calculation of Group Orders where 7' = 1 i f p \ n/ddi a n d = 1 - r " into (YC /i ) 1 n T p '. YjG ',nld t n n • 1 where G' is a subset o f H = HGd'/i,nldYjG', d C Hi /dd Gi i /ddi otherwise. T h e second factor o n the right maps 1 mapping to n/d W e now define ct\ — a E G' a n d continue the i n d u c t i o n t h r o u g h the powers of I, a n d t h r o u g h the powers o f p i n the same way, showing that u(x) is a p r o d u c t o f terms m a p p i n g into {YC ) l {YC ) - . Tl p n/lp T h i s gives the reverse inclusion, finishing the proof of Tp n/lp the l e m m a . I L e m m a 7.9 Y — - is cyclic of order m'. K\ K Proof. T h e first step i n the proof is t o show that any element i n Y/c is equivalent ( m o d K\) to a n element i n W(x)W(x ')W(x '). 1 pS W e now consider a n element of the form u(x ) = w (x ) (x ) d i Ild|di,di|n W{x i) ~ YC n d a d Ul w h i c h are i n Y , a n d u (x ) € d d n/d K U d ^ ^ n ^ d W ^ 2 1 I Fl P ) I M N t h e n a is o f the form ,n/d + a P G 2 = 71 ^2 l,n/d ,n/d + 72 Y2 P,n/d G + l G k ^2 l ,n/d P ,n/d, + &2 ^2 i G i G k ,k € 2, l ,p> each | n/d. T h i s follows since elements of YJc where 71 a n d 72 are i n AH /d, l x n 2 have a n expression i n terms of n o r m relations i n Z [ £ ] . Since k\ € b u t where l \ n/d, p> n i+l \ n/d, / c i ( r f - 1) e AH i+i . 1 n/l Therefore, l i+l p AH / i+i n t p only i f ki — 0, \ n/p, a n d u{x ) is +l d of the form u{x ) = w {x )w^{x )-^ d d w, (x )-^ ld where u {x ) G Ud^^d^d ^ pd a 2 l ( T r l _ (^ )^ W{x ) and v is either 1 or 1 - dl dl 2 - _ d 1 ) x 2 ( T T, 1 - 1 , (^' )n (x^), d 1 ) 2 V is 1 or 1 - T~ . 1 2 N o t e that w {x )w {x )-^ {x )-^w ^ d a ld lx pd Wl2 k E nfc EG, ,„ +faEG,,„ )( VM r -i)( ') s 1 i / d / J 1 T / l t 'MT - -i)( 1 p / r f ) (modtfi). (7.7) 97 Chapter 7. Calculation of Group Orders If I \ n/d then a = a' E G / Pjn for some a' and d u{x ) = with w (x )w >(x ) d pd a d d a l d a € K\. Let w(x) = n^dl ^) 1 3 a (7.8) w {x )w ,{xv )- uz{xv ), d Starting with d = 1, we inductively e prove that II "(*.<£<^ B / , ^E^^^ (mod i f O , + d\(n/lp) for suitable fei^'s, fc d's, id's, j s. 2) d Let a € AH with cv H->=fci£ G j i d n d w and each a Y, G / d 5 of [7]. YJC/KI djU d i f l / d (x)£ a d G d + A; £ Gy / e AH / . 2 > n d - = ^(x *) n d Then (mod tfi), 4 € (Z.H ) Y, G\ + (Z-H ) ^ G y . This is the same situation as in Section n r n is then cyclic of order m', using these results along with the modification of of Chapter 5. I YC n/lp Proposition 7.10 l™[n/lp] K 9l(n)g (n) Y p Yy,c YvYc m' Proof. Since Y-p^c is the kernel of the map Y& —>• YC /i , n p we have \m i Y Y [n lp] K K We prove the propositon by showing that Yp c fl K\ = YpYc- This gives the isomorphism u KiYpuc K x Ypuc Y-pYc Then we have YG n/lp (YG /ip) 1 n n (YC Hp) P-T P n 98 Chapter 7. Calculation of Group Orders Lemmas 7.8 and 7.9 then give the result. Let w(x) = rjd| Wa {x ) G Y-puc fl K\. ot\ has the form d ra d « i = ai,i ]T Gj + a XiP ^ G p H-> / ai,, + P«i,p = 0 G Thus, both aij and a i A H /i. 2 n = 0 in A H / n ip (7.9) 2 n/lp so that a^, X) Gj 2 ) P AH . 0 G AH/ and a i 2 n p ) P 2~Z G i-> 0 G p In this way, tu(x) consists of factors which map to 1 in either YC /i or in YC / , n n p so that u;(x) G Yp YcWith our definition of K\, elements of Yp and Yc are in K\ as well (see proof of Proposition 7.3), giving the reverse inclusion. I This shows that Yp^c fl K\ = YpYc and the proof is complete. 7.3 R e s t r i c t e d circular indices The groups fiC^, QC^i and nC^ p arise in considering the pull-back PB1 p nC ji X-n/lp nC j n > n,C / n p n p nc,n/l QC^ , t QC^ p n/pl- and nC^^ are the images of nC p x /i p n n nC / p in these corresponding groups. n p This leads to the adaptation nC /i x n n / i p SlC / »• QC^ip n p net. 'n/l n "^n/pl* In the inductive process we calculate the circular indices c{C /i ), c(C p) and c{C / ) where, n for example n/lp (C /i ) c n YC,n/lp P 99 p n n p Chapter 7. Calculation of Group Orders The new indices we are seeking i n this section are divisors o f these respective c i r c u l a r indices. If c{C ji ) n = 1, t h e n these new indices are precisely the c i r c u l a r indices. T h i s is c e r t a i n l y t h e p case for n = Ip, 4p, 8p, 9p or 36 a n d also for n = lp w h e n p is regular. s We i n t r o d u c e the following n o t a t i o n to denote these (possibly) new indices: n(C /l) ' = c L n If c ( C / / ) n n L p = Cn(C /i ), n p Q.G nc< n/l YC n/l n/p n{C /p) — c n(C /i ) c n YC, n/p n p — L QC,n/lp YC, n/lp then the other two indices are the same as the c i r c u l a r indices. A t t h e present, we are not able to give m a n y other general results o n this question. W e are able to give a n extension for the case n = l p 2 where c ( C / ) may not b e t r i v i a l . 2 p Lemma 7.11 All units in fiC; are liftable to fiCj2 p . p Proof. F o r n = l p we have the pull-back P B 0 ; 2 2 j ) nc^ ilCip YCip). u i~ T l 2p x, QCp Let u € QCip w i t h u G YCi . k > n(c, ) > F,[i]/($p(t')) x Fp[C ]. P Suppose first that (k,l) = 1. u G YCi , Tl p !-»• 1 i n ¥i[t]/(%{t )). Since Y{C\K) 1 -» YC{ ~\ p so that u '~ = u ( m o d T l we c a n find a constructible u n i t 1 v i n VtCp m a p p i n g to the image o f u '~ i n YC\ a n d also m a p p i n g to 1 i n F [ £ j 2 ] . T l p therefore lifts t o a u n i t i n ClCpp. Hence u must lift to a u n i t i n QCp p (u '~ ,v) T l as well. Suppose now t h a t A: is a n /-power. W e need i n this case to look at P B 0 z : p ncip—-—>m ) P YC P Since \L(Q /Y((i )\ p Xip- x Yd • Fj[C ] x F [CJ ]. P P 2 is p r i m e to / , u is equivalent ( m o d YCi ) p v u\ maps t o a constructible unit w\ i n YC P a u n i t wtp G Q(Ci \p). p a n d the u n i t (wi,l) A l s o , the order o f YC /im^Y(Ci \p) p 100 to a u n i t u\ i n t h e kernel o f p i n YC P x YCi lifts to is p r i m e to / as well, so t h a t Chapter 7. Calculation of Group Orders w\ E Y(Ci \p) v for some i with p = 1. Then, u = u = ufa^ 1 2 (mod VCjj,), and with these adaptations, u >-> 1 in both 0(C/p) and fi(Cp)- Going back to P B 0 p 2 p «2 -> 1 € F,[*]/($ (i')). p Since E YCi , v p = u (mod YCi ). As above, u^" p P can be matched with some v E OCp mapping to 1 in F [Cp], and (ii2 ~ ,v) in liftable to a unit in O C j . Hence u is liftable as well. P p p 2p I Corollary 7.12 For n = l p , 2 2 Cn(C ; ) = L n/ Cn(C p). p n/l Proof. This follows directly since both fiC/2 and OCy 2 map onto ilCi . p p p I For n = l p and Z p , at least, the restricted circular indices are the same as the circular 2 2 2 indices. 101 Chapter 8 Comparison of Group Orders Our procedure for the determination of c(C ) focuses on the sequence n (nc„|/c) ^ nc L(Cn) n YC Y(C ) («C„|/C) HCn) Y(Cn) Y K n n From this we have c(C ) = n (8.1) splitting this index into 2 factors, which we examine under (1) and (2) below. (1) Proposition 6.2 gives an expression for the index IK. = in terms of indices of groups in SlC /i, ™[U.]YK QC / n and Q,C /i . n p n From PBO, (ClC \JC) p n to ker (3, the kernel in the map or * or /? Hh] F[t] x y p 2 (^(tr )) ( M O ) ' " W / / n/lp ^ n / p 1— > ~ S As well, we have that x (fiC„|/C) imp,,] (OCnl/C) ( y c |/c) im «.,. y n [ ] x: ' since for elements in the kernel of Xn the map SL is surjective. Using the following sequence derived from (3 ker/3 ttC /i n x /i SlC / n p '^[U.]YK 102 p n p . imp is isomorphic Chapter 8. Comparison of Group Orders we have that (new where im/3 is the image of C / z x / ; £lC / n n p n p (8.2) im/3 under /?. Example. F o r n = Ip, we have from (6.4) that YG 'p l-T, WC 1 P im/3 = Pi(YC ) p x P (YCi). YCi wc? T h e powers of prime factors i n the orders of f3i(YC ) and (5p(YC{) p p are c a l c u l a t e d i n d i v i d u a l l y . F r o m the definitions of the defects, we derive di{p)d (l) (flC |/C) (YC \K) p ip m' lp W e k n o w that the invariant m' w i l l d i v i d e each of the defects. O t h e r factors are difficult to predict. D a t a available at present seems to indicate that other possible factors of di(p) m a y be p (if / has o d d order m o d p), factors of m', a p r i m e power w h i c h is 1 m o d u l o a p r i m e divisor of p — 1. A s a m p l i n g of some defects showing these trends is given i n the following table: p 11 11 11 11 11 11 23 23 23 23 47 I 23 353 743 661 1423 1451 71 1013 3083 18493 617 di{p) 11 l l 101 5 151 23 11 23 - 67 3 47 2 2 4 5 47 47 11503 11939 1013 47 . 2 N o t e that m' = 5 for n = 11 • 661 a n d m' = 11 for n = 23 • 1013. For the more general cases, we s t i l l need separate calculations for different values o f n, w i t h defects i n the finite rings appearing. T h i s we analyze more carefully below. 103 Chapter 8. Comparison of Group Orders (2) In Theorems 5.11 and 5.14 of Chapter 5 we have shown that = (8.3) mm' \e,(n/l)-> (-rhn,-rri) e (n/p)J p when (m, Ip) = 1 or when n = 8p and 2 \ c{C ). For all cases, Theorem 5.8 establishes that 4p = m m'<p(n) (8.4) 2 8.1 Defects We shall define the defects for n = l p in terms of the order of images of constructibles units r s in the finite rings, extending the definitions given earlier for n = Ip and n — 4p. With that in mind, we set (YC \V) n/l di(n/l) = \im(3i/(YC \V)\ im[ j]Yp n/l n/ and (YC \C) n/p d {n/p) = p where j3i : fiC^// —>• Fi,n/l[0p] a n ( \im(3 /(YC \£)\, ' [n/p]Yc lm p n/p f Pp '• ^^n/p ^ •^>,n/p[*' * /] — ie a 3 X 6 restrictions of the map /?. We next interpret (8.2) to include these definitions. We start with QC /i x /i Q.C / n im/3 n p QC /i n p x / Q,C / n ker/} n lp n p ker/?/ x , k e r / ? n/ p p Using the commutative diagram (from PB1) 1 >• ker ft x j ker @ 1 n/Zp ^C"n/p n / p p • ker A x ker /3 > im[ /z ]ker j3 • P n p + 1, n/lp we derive QCn/l n/lp ^C"n/p x |im/?| = ker A x n//p ker/? p n/p ker A ker /3 P n/lp im[ ] ker/9 |im/?;| |im/3p| n//p im 104 [n/ip] ker/3 (8.5) Chapter 8. Comparison of Group Orders as i n C h a p t e r 6. A p p l y i n g the same method for /?/(YC /i x /ipYC / ), n n n the restriction of P to YC /i p x /ipYC / , n n n p we o b t a i n \imp/(YC x /lpYC )\ n/l n |im/3j/yc ,| \imP /YC \ n / = n/p p n/p (8.6) YC, im ker/3/{YC x YC ) [ n / l p ] n/l n/lp n/p U s i n g the definitions c ( C f c ) i n Section 3 of C h a p t e r 7 a n d P r o p o s i t i o n 7.4, we modify P r o p o L n s i t i o n 6.2 t o give itc = I -TTF, ^ CniCn/lp) (YC \V) (YC \C) ™[n/l]*/C ' -[n/p]YK n/p n/l {9i{n)9 {n)) — — P (8.7) lm W e now write |im/?| = (i ) (i /Y ) n/Y Y I ViC i m A ((YC \P)) x i m / ? ((YC n/l p n/p \C)) | . (8.8) where imp in/Y imP/(YC x YC ) n/l n/lp n/p and imP/{YC Y/Y x /i YC ) n/l n p n/p 1 impi/(YC \V) VX x im n/l p /(YC \C) p n/p W e now have the following theorem: C n ( C ' / ( ) c ( C / p ) ' - \ ( gi{n)g (n)\ L Theorem 8.1 c ( C „ ) = n n n p fe/y) CniCn/lp) - J \ fdi(n/l)d (n/p) L(Cn) m' Y((n) p Y/Y 1 i Vi Proof. F r o m (8.1) a n d (8.2) we have c(C ) n HCn) IK (8.9) \imp\ T h e theorem now follows using (8.7) a n d (8.8) along w i t h the definitions for di(n/l) a n d d (n/p). p m m 8p when d±(p) = 1, c ( C ) = md^iCp) d {Ap) d (8) . , ,, \e (4p) e (8)/ using T h e o r e m 5.11. T h u s , c(C% ) is n o n - t r i v i a l when p = 1 (mod 4). F o r n = 24 the defects Example. For n - 2 8 p 2 p 2 p 105 p Chapter 8. Comparison of Group Orders are trivial, giving 0(624) = 1- Calculations for other cases where rs > 1 are still incomplete. There is, however, a remarkable coordination between the orders calculated in Lemma 3.5 for the inversion invariant units in the finite rings and the orders calculated in Lemma 7.2 and Proposition 7.3 for quotient groups in YC /i n and YC j . Using this information, we have the following proposition which shows n v many cases where the circular index must be non-trivial. Proposition 8.2 Where (m,lp) — 1, mm!gi{n)g {n) ( v ™ ' ™ ) | c ( C »)• r Proof. From Lemma 7.2 and Proposition 7.3 Y(C \V) n/l Y(C \C) n/p (8.10) m (j)(n) 2 ' [n/p]Yc lm From Lemma 3.5, we get the order of the finite group in the lower right of PBO: From Theorem 5.14 we have = <f>(n) lc.m.{ei(^),e {Cir)}. p Thus, the order of the possible image space for fiC /j x £lC / n n p is l\{l - -l)<P{p )(lh _ i J f t p i C p - - ! ) ^ ' ) ^ _ y r l s 1 x p (8.11) 0(n) l.c.m.{ej(Cp«)iep(C/')} Dividing (8.11) by (8.10), we get m (fc e; the values for ioyy and iy/Y ipt^Tj)' w n i c n i s prime to Ip. This sets limits on - ^ particular, any power of I or p which divides their product n v c must also divide di(n/l) d (n/p). p Using the fact that gi(n) is an Z-power and g {ri) is a p-power, we have that gi{n) g {n) \ c(C ), p using Theorem 8.1. The factor |fi(Cn)/Y(Cn)| mm' gi(n)g (n) p i s separate. Using Theorem 5.14 again, we have m m p c(C ), n e*(Cp«)' 106 e {Qr) p n Chapter 8. Comparison of Group Orders I as stated. Examples: B e y o n d the factors identified as divisors of the c i r c u l a r i n d e x i n the case n = lp, we as well have the new factor gi(ri) a n d g (n) as defined o n page 95, w h i c h c a n be n o n - t r i v i a l p only w h e n the order of one p r i m e is o d d m o d u l o the other. (1) W h e n b o t h I a n d p are equivalent to 3 (mod 4), this must occur i n one case or the other. W h e r e n = 49 • 11 = 539, as a n instance, we have that 11 has order 3 ( m o d 7) while 7 has order 10 ( m o d 11). W e o b t a i n #7(539) = 7 a n d ^ 1 ( 5 3 9 ) = 1. T h e c i r c u l a r i n d e x c(C53g) has 7 as a divisor. (2) F o r n = 7 • 3 7 = 67081, 7 has order 9 ( m o d 37) while 37 has order 3 ( m o d 7). Here 2 2 g (n) = 7, # ( n ) = 3 7 a n d we have 7 • 3 7 | c ( C ) . 2 7 2 37 n P r o p o s i t i o n 8.2 gives some useful i n f o r m a t i o n b u t is l i m i t e d i n its scope. T h e following theorem gives a better s u m m a t i o n . Theorem 8.3 n c c(C ) = n (C p (C / ) L ) C L n n n p gi{n)g {n) p (8.12) g n(Cn/lp) - c L i m / ? • imp. .]fi(Cn) n r where g — 1 if (p, I — 1) = 1 and (l,p — 1), but is equal to the power of p given by dividing the n factor determined by Proposition 7.3 by m ifp \ I — 1 or the corresponding l-power ifl\p—l. Proof: L e m m a 7.2 a n d P r o p o s i t i o n 7.3 give, i n this case, Y(C \P) Y(C \C) n/p n/t g m 4>{n) 2 i [n/ ]Yc im[n/i]Yp m n P \Fl,n/l[0p] xFp,n/p[0l} \ g m 4>{n) X X 2 n using L e m m a 3.5 again. U s i n g this w i t h (8.7) a n d s u b s t i t u t i n g into (8.9), we o b t a i n n{C /iYc {C / Y\ c c(C„) = n n n p CniCn/lpY f gi(n)g (n)\ p ( \Fi, /i[Qp] Fp,n/p[0i]* m m'(j)(n) | i m /3| x n g 2 n 107 x HCn) Y(Cn) (8.13) Chapter 8. Comparison of Group Orders W e now o b t a i n that MCn)/y(Cn)| |fi(Cn)/£(Cn)| Y(Cn) m m'(f){n) 2 (8.14) |im[;.,]0(C )/ (im[j. .]fi(Cn) n i m / 3 ) | n using T h e o r e m 5.8 a n d the fact that L(( ) n r is the inverse image of (im[(. .]fi(C ) f l i m 0) i n VL(C, ). r n n Note that |im/3| | i m , . , f i ( C n ) / i m . , Q ( C „ ) n i m / ? | [ ] [/ ] = C o m b i n i n g (8.15) and (8.14) into (8.13) gives the result. Corollary 8.4 For n = lp, c(Ci ) p |im/? • im ,]ft(C )| . [L n (8.15) I imu r.]^(Gp) Proof: T h e factors i n the first two brackets on the right side of (8.12) are a l l 1. N o t e t h a t i m j3 is the image of YCP x YC\ a n d hence as well the image of YCip following back i n P B O . T h u s i m / 3 = i m p ] Y ( C ) C im^r]Q(Qp) r p Examples: W h e r e (lp, (I — a n d this result follows. I l)(p — 1)) = 1, P r o p o s i t i o n 8.2 shows already that gi(n)g (n) p is a divisor of the circular index. W h e r e (lp, (I — l)(p — 1)) ^ 1, g n sidering where n — 2 p r s w i t h r > 2, g n = g (n) 2 m a y not be t r i v i a l . C o n - i f p = 3 or 5 ( m o d 8). I f p = 9 ( m o d 16), g (n) = gl = 2 ( - ) , showing that 2 ~ divides c(C ). 2 r 2 r 2 2 n M o r e precise i n f o r m a t i o n requires calculation. M o r e general theorems extending T h e o r e m 5.14 for cases w h e n (m, lp) ^ 1 may be posssible. There should as well be more results of the type found i n Section 3 of C h a p t e r 7 regarding liftability of units from one group r i n g to another. T h i s p r o b l e m of liftability is directly related to the question of whether or not i m j3 is d i r e c t l y contained i n im[; ]fJ(C„) as i n C o r o l l a r y 8.4. -r- 108 Bibliography [1] H . Bass, 407. Generators and relations for cyclotomic units, Nagoya M a t h . J . , 27, 1966, 401- [2] H . Bass, The Dirichlet unit theorem, induced characters, and Whitehead groups of finite groups, T o p o l o g y 4 (1966), 391-410. [3] G . H . Cliff, S. K . Sehgal, A . R . Weiss, A l g e b r a 73(1981), 167-185. [4] V . E n n o l a , Units of integral group rings of metabelian groups,^. On relations between cyclotomic units. J . N u m b e r Theory, 4, 1972, 236-247. [5] K . Hoechsmann, 50-54. Exotic units in group rings of rank p , A r c h . M a t h . (Basel) 51 (1988), 2 [6] K . Hoechsmann, Constructing units in integral group rings, M a n u s c r i p t a M a t h . 75, 1992, 5-23. [7] K . Hoechsmann, Units in integral group rings for order pq, C a n . J . M a t h . , 47(1), 1995, 113-131. [8] K . Hoechsmann, (1995), 5-20. Cyclotomic units over finite fields, R e n d . C i r c . M a t . P a l e r m o (2) 44 [9] K . H o e c h s m a n n , On the arithmetic of commutative group rings, G r o u p Theory, A l g e b r a , N u m b e r Theory, W a l t e r de G r u y t e r & C o . , (to appear), 145-201. [10] P a o l o R i b e n b o i m , Algebraic W i l e y & Sons, Inc., 1972. Numbers, P u r e a n d A p p l i e d M a t h e m a t i c s X X V I I , J o h n [11] C . - G . S c h m i d t , Die Relationen (Basel) 31 (1978/79), 457-463. von Gau(3schen Summen und Kreiseinheiten, A r c h . M a t h . [12] C . - G . S c h m i d t , Die Relationenfaktorgruppen von Kreiszahlen, J . reine angew. M a t h . 315 (1980), 60-72. Stickelberger-Elementen und [13] S. K . Sehgal, Units in Integral Group Rings, P i t m a n Monographs a n d Surveys i n P u r e a n d ' A p p l i e d M a t h e m a t i c s 69, L o n g m a n Scientific & Technical, 1993. [14] L . W a s h i n g t o n , Introduction to Cyclotomic Fields, Springer, N e w Y o r k , 1982. 109
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- Units in integral cyclic group rings for order LRPS
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
Units in integral cyclic group rings for order LRPS Ferguson, Ronald Aubrey 1997
pdf
Page Metadata
Item Metadata
Title | Units in integral cyclic group rings for order LRPS |
Creator |
Ferguson, Ronald Aubrey |
Date Issued | 1997 |
Description | For a finite abelian group A, the group of units in the integral group ring ZA may be written as the direct product of its torsion units ±A with a free group U2A . Of finite index in U2A is the group ΩA, the elements of U2 which are mapped to cyclotomic units by each character of A. The order of U2A/ΩA depends on class numbers [formula] in real cyclotomic rings [formula]. Of finite index in ΩA is the group of constructible units YA, for which a multiplicative basis may be explicitly written. The order of ΩA/YA is the circular index c(A). In many cases, for example where A is a p-group with p a regular prime, this index is trivial. This thesis develops an inductive theory for determining c(Cn) where Cn is a cyclic group of order n = lrps, with I and p distinct primes, and also for giving some description of the group ΩCnjYCn. This is a continuation of the work of Hoechsmann for the case n = Ip. It turns out that the methods required for n = lrps are, in general, very different from the ones used for r = s = 1. |
Extent | 4012599 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
File Format | application/pdf |
Language | eng |
Date Available | 2009-04-03 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0079979 |
URI | http://hdl.handle.net/2429/6770 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Graduation Date | 1997-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-ubc_1997-250458.pdf [ 3.83MB ]
- Metadata
- JSON: 831-1.0079979.json
- JSON-LD: 831-1.0079979-ld.json
- RDF/XML (Pretty): 831-1.0079979-rdf.xml
- RDF/JSON: 831-1.0079979-rdf.json
- Turtle: 831-1.0079979-turtle.txt
- N-Triples: 831-1.0079979-rdf-ntriples.txt
- Original Record: 831-1.0079979-source.json
- Full Text
- 831-1.0079979-fulltext.txt
- Citation
- 831-1.0079979.ris
Full Text
Cite
Citation Scheme:
Usage Statistics
Share
Embed
Customize your widget with the following options, then copy and paste the code below into the HTML
of your page to embed this item in your website.
<div id="ubcOpenCollectionsWidgetDisplay">
<script id="ubcOpenCollectionsWidget"
src="{[{embed.src}]}"
data-item="{[{embed.item}]}"
data-collection="{[{embed.collection}]}"
data-metadata="{[{embed.showMetadata}]}"
data-width="{[{embed.width}]}"
async >
</script>
</div>
Our image viewer uses the IIIF 2.0 standard.
To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0079979/manifest