THE BEHAVIOUR OF GALOIS 'GAUSS SUMS WITH RESPECT TO RESTRICTION OF CHARACTERS Michael William Margolick B.A., Cornell University, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA © M i c h a e l William Margolick, 1978 In presenting an advanced the Library I further for degree shall agree scholarly by his of this written thesis in at University the make that it thesis purposes for freely may It for University gain Mathematics of British October 10, 1978 of of Columbia, British Columbia for extensive by the is. understood permission. of fulfilment available be g r a n t e d financial 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date partial permission representatives. Department The this shall Head be requirements reference copying that not the of agree and of my I this or allowed without that study. thesis Department copying for or publication my - i i- THE BEHAVIOUR OF GALOIS GAUSS SUMS WITH RESPECT TO RESTRICTION OF CHARACTERS Abstract: The theory of abelian and non-abelian L-functions i s developed with a view to providing an understanding of the LanglandsDeligne l o c a l root number and l o c a l Galois Gauss sum. The r e l a t i o n - ship between the Galois Gauss sum of a character of a group and the Galois Gauss sum of the r e s t r i c t i o n of that character to a subgroup i s examined. In p a r t i c u l a r a generalization of a theorem of Hasse- Davenport (1934) to the global, non-abelian case i s seen to r e s u l t from the r e l a t i o n between Galois Gauss sums and the adelic resolvents of F r o h l i c h . - iii Table of Contents ACKNOWLEDGEMENTS IV INTRODUCTION V §1 Review of Tate's Thesis A) B) C) D) §2 1 Characters and Quasicharacters of a Local F i e l d of Characteristic 0 Local ^-functions Global ?-functions Comparison with Hecke's L-functions . 1 2 4 5 Non-abelian L-functions A) The Unramified F i n i t e Primes - F i r s t D e f i n i t i o n of L-functions The Ramified F i n i t e Primes - F i n a l D e f i n i t i o n of L-functions The I n f i n i t e Primes and Exponential Factor; The Extended Function A; Analytic Continuation 11 Appendix 16 §3 Local Constants 19 §4 Structure of Local Root Numbers §5 Gauss Sums for Hilbert Symbols 29 §6 Behaviour of Gauss Sums With Respect to R e s t r i c t i o n 43 §7 The Locally Free Class Group, Resolvents and Gauss Sums 52 ' B) C) D) Bibliography . 8 9 24 64 - iv - Acknowledgements I would l i k e to express my warm thanks to Professor Larry Roberts for his guidance throughout my graduate studies, and to Cathy Agnew for her mastery of the mathematical typewriter. Many thanks also to Professor Roy Douglas and Professor Peter H i l t o n for advice and support. - V - Introduction C l a s s i c a l Gaussian sums, as defined i n the nineteenth century, are certain sums of roots of unity whose properties have played an important r o l e i n the theory of cyclotomic f i e l d s . Of considerable importance i s their f a c t o r i z a t i o n as algebraic integers, f i r s t discovered by Stickelberger i n 1890. This f a c t o r i z a t i o n led to r e l a t i o n s i n the ideal class group of cyclotomic f i e l d s , an old and largely open problem i n c l a s s i c a l number theory. More p r e c i s e l y , a c l a s s i c a l Gaussian sum i s a certain f i n i t e Fourier transform of a character the ( f i n i t e ) residue f i e l d x(x) = _ xdF 2F^ and °f the m u l t i p l i c a t i v e group of (say JF^) of a l o c a l number f i e l d respect to the additive group of the form x 3F^ . Hence c l a s s i c a l Gaussian sums are x(x)*(x) where * i s a fixed character of q i s a character of W_ . x with (We set x(0) = 0) . One can generalize this concept to certain character sums taken not over the residue f i e l d of the l o c a l number f i e l d representatives chosen as follows: and to U the units of 1 mod p K but over a system of Suppose K . Let U^ ^ n p i s the prime of K denote those units of K congruent . We take representatives of the cosets of U^ ^ i n U . * (n) Any character x of K i s t r i v i a l on some U , so that these n n character sums can be defined for characters of K . Let us c a l l this generalization just a Gauss sum as d i s t i n c t from the c l a s s i c a l Gaussian sum, which arises as the case n = 1 smallest integer such that i s t r i v i a l on x above. tamely ramified, f o r arithmetic reasons. In case U^ ^ n n = 1 we say When the least n i s the x is such that - vi - X i s t r i v i a l on U n i s not 1, the associated Gauss sums divided by i t s complex norm i s a root of unity. The l o c a l abelian root number may divided by i t s complex norm. as a constant of Tate. be defined as a Gauss sum The l o c a l abelian root number appears i n the functional equation of the l o c a l zeta function This l o c a l zeta function i s i n turn a kind of Fourier on the m u l t i p l i c a t i v e group of, a l o c a l number f i e l d . transform If we f i x a global number f i e l d and take the product of the l o c a l zeta functions at a l l the completions of that number f i e l d , we get a generalization of Hecke's c l a s s i c a l L-function and i n p a r t i c u l a r the functional equation of that L-function together with i t s constant, the global abelian root number. This turns out, not s u r p r i s i n g l y , to be the product of the l o c a l abelian root numbers. It i s more productive to view the Gauss sum i n i t s role i n the functional equation of zeta or L-functions because this theory was generalized to the global non-abelian case by A r t i n i n the 1920's. In §2 we define the A r t i n L-function. This i s defined for characters, not of the idele class group (Tate) or i d e a l group (Hecke) as i s the abelian L-function, but for characters of Galois groups. The connection between A r t i n and Hecke L-functions i n the abelian case i s the fundamental construction of class f i e l d theory, the r e c i p r o c i t y map. Abelian L- functions were used (before the introduction of cohomology to class f i e l d theory) to prove fundamental theorems of arithmetic of abelian extensions. Therefore discussions about non-abelian L-functions by implication discussions about non-abelian class f i e l d theory arithmetic. are and In §2 we prove the functional equation of the A r t i n L- - vii - function. The constant appearing i n this functional equation i s the global non-abelian A r t i n root number. So far i n this discussion we have global abelain and non-abelian L-functions, l o c a l zeta functions, l o c a l abelian root numbers and global abelian and non-abelian root numbers, but we don't have l o c a l non-abelian root numbers. abelian root number was This i s the subject of §3. The l o c a l non- found by Langlands and Deligne and i s character- ized by the facts that i t works well with respect to addition and induction of characters and agrees with the l o c a l abelian root number in case the given character i s 1-dimensional ( i . e . factors through an abelian quotient of the given Galois group and hence can be as a l o c a l f i e l d character v i a the l o c a l r e c i p r o c i t y map). considered The product of the Langlands-Deligne root numbers i s the global non-abelian root number. In §4 we examine the structure of the Langlands-Deligne l o c a l root number and find that i t i s a root of unity times a c l a s s i c a l Gaussian sum divided by i t s complex norm. This i s obvious i n the abelian case, from above remarks. The contents of §5 are c l a s s i c a l i n nature and refer Gaussian sums. to c l a s s i c a l We prove a few scattered statements on these sums and the two known theorems on r e l a t i o n s i n ideal class groups of f i e l d s a r i s i n g from Stickelberger's f a c t o r i z a t i o n . Gauss sum cyclotomic The l o c a l Galois i s defined naturally from the Langlands-Deligne l o c a l root number i n such a way that the l o c a l Galois Gauss sum divided by i t s complex norm i s the Langlands-Deligne l o c a l root number. The actual subject of the thesis begins i n §6. Let E ^ L K - viii - be l o c a l number f i e l d s with E/K Galois. We wish to find a r e l a t i o n x between the Galois Gauss sum of a character Galois Gauss sum of the r e s t r i c t i o n , Res x> of of x Gal(E,K) t o and the Gal(E,L) . This r e l a t i o n i s primarily determined by the nature of the ramification of L/K . Let us suppose unramified. E/K i s abelian and x i s tame or This means that we are looking e s s e n t i a l l y at c l a s s i c a l Gaussian sums. A theorm of Hasse-Davenport says that i f L/K i s TL *Kl unramified, then r(Res x) to i(Res x) of L/K . The case L/K a r = ± (x) T ' • Results relating;;: T ( X ) given under two more assumptions on the ramification e L/K a r b i t r a r y may be derived on decomposing into a tower of extensions of the 3 given types. The l a s t theorem of the section indicates how the r e l a t i o n may be generalized to the case E/K to be pleasing. non-abelian. However the result i s too complicated The global Galois Gauss sum i s defined to be the product of the l o c a l Galois Gauss sums. Section §7 contains a summary of some recent work of F r o h l i c h , r e l a t i n g the following three constructions: 1) The l o c a l l y free class group of the group ring of a Galois group of a tame extension of number f i e l d s over the integer ring of the ground f i e l d . every prime prime to p of E/K K, i s a tame extension of number f i e l d s i f for the ramification degree of p in E is p . 2) The global and/or adelic resolvent. 3) The global Galois Gauss sum. The f i r s t of these i s a K-theory construction and y i e l d s information about how much a tame extension of number f i e l d s f a i l s to have a normal - ix - i n t e g r a l basis. of E E/K has a normal i n t e g r a l basis i f f the integer ring i s free as a module over the group r i n g of integers of K . Gal(E,K) over the The second construction i s a generalization of the c l a s s i c a l resolvent, going back to Lagrange. The connection between these three constructions i s forged by a characterization of the l o c a l l y free class group as a quotient of a certain group of Galois homomorphisms defined on the free abelian group on the set of i r r e d u c i b l e characters of Gal(E,K) . The product of a global Galois Gauss sum and a global or adelic resolvent can be seen as an element of this quotient. A key theorem relates their f a c t o r i z a t i o n s . We prove a theorem generalizing the above r e s u l t of Hasse-Davenport to the global non-abelian case: fields. Let a character of E £ L £K Gal(E,K) with and Let L/K E/K be a tame extension normal and unramified. Res x i t s r e s t r i c t i o n to Let of number x be Gal(E,L) . TL *Kl Then x (Res x) integers. a n < i T (x) * have the same f a c t o r i z a t i o n as algebraic - 1 - §1. Review of Tate's Thesis Tate's thesis lays the foundations of adelic Fourier analysis. The depth of Tate's re-interpretation of Hecke's c l a s s i c a l functional equation can be seen i n the success of Jacquet-Godement's reworking [8 ] of the theory of L-functions attached to automorphic representations of adele groups [10] (the so-called 'Hecke theory for GL(2)'). We w i l l summarize the main points of this celebrated work, concentrating on the comparison with Hecke's L-function. A) Characters and Quasicharacters of a Local F i e l d of Characteristic 0. Let of Q k be the completion of a number f i e l d , at the r a t i o n a l prime below that of primes are p n A: IR —>-R/Z by X: — y Q./Z Q P a and p respectively. A(x) = -x mod by setting Z . If If k k R the completion and suppose these k i s archimedean l e t i s non-archimedean l e t A(x) = unique r a t i o n a l (mod Z) with only 0 p_-power i n i t s denominator and such that "E 0 A(x) - x e . Let P A: k —> S A(£) = X(Tr^ (0) . A i s called the canonical character, of (the additive group of) k because Theorem: be defined by 1 k i s canonically isomorphic to i t s dual group under the correspondence If k so that R n «—> [5 — y exp(27ri A(£n))] . i s non-archimedean, l e t n e 6 5 be the absolute d i f f e r e n t of i f f "^k/R-^^k^ — ^p character associated to n ' i s t r i v i a l on I t : 0 i s k e a s Y t o s e e iff n e 6 t L n a t . t n e k, - 2 - For k , one considers continuous m u l t i p l i c a t i v e maps k —*• <E c a l l e d quasicharacters. A quasicharacter i s unramified i f i t i s t r i v i a l on the units of k . Two quasicharacters are equivalent U i f their quotient i s unramified. Remark. Let k sep ab and ft^. l t s be the algebraic closure of k, JL =Gal(k ,k) ° k sep * ab maximal abelian quotient. Let 0: k —>• ft be the x: ^ l o c a l r e c i p r o c i t y (Artin) map and x ([W ] (XII §2)) defines L ( i . e . Ker x .X = Gal(k a s —• S 1 9 Weil unramified i f the c y c l i c extension ,L )) i s unramified. sep x class f i e l d theory, x( (°0) a character. However, by l o c a l i s t r i v i a l for a l l a e U i f f X is unramified and so the two d e f i n i t i o n s are " e s s e n t i a l l y " the same for characters. Theorem: for s e (write (See [W ] XII §3 Prop. 6). Quasicharacters are maps of the form C where c a = aa with i s a character of U, a > 0 |a| = exp(s log |a|p) • s Re s c if k and i s called the exponent of a — • c (a) = c(a)|a|"" a = unit part of a i s archimedean) and s are uniquely determined. c . As a result of this theorem, the problem of quasicharacters reduces to finding characters of U . Define the conductor of such that c = p B) n c . n c i s t r i v i a l on as follows: 1 + p n take the least p o s i t i v e and set fj(c) = conductor of i s f i n i t e by the "no small subgroup" theorem. Local g-functions * Let measure us select a Haar measure on 1 . k which gives the units n - 3 - Definition: If f i s a complex function on k (satisfying c e r t a i n i n t e g r a l convergence and continuity properties) and character of k * of exponent > 0, let ?(f,c) = c f J a quasi- ^f(a)c(a)da, k the l o c a l £-function. Remark: Each equivalence of the form class of quasicharacters consists of those c(a) = C Q ( a ) ( a ) for a fixed representative s c . n each class can be considered as a Riemann surface, parametrized s . So by In the d i s c r e t e case, t h i s gives the complex plane i n which 27Ti points d i f f e r i n g by an i n t e g r a l multiple of since l a Theorem: i2iri/log N(p) =1 . log ,. „ for a l l a . N are i d e n t i f i e d , N(p) N , denotes absolute norm. 1 n A ^-function has an a n a l y t i c continuation from the domain of a l l quasicharacters of exponent s t r i c t l y between 0 and 1 to the domain of a l l quasicharacters given by the functional equation 5(f,c) = p(c)£(f,c) . Here, f(?) = 1 - exponent c ) . f, p(c) function on the surface In the p-adic case p (c and c i ,. . dn = additive 0 k i i -1 A Fourier transform of , \ -2TTiA(ri?) , f(n)e c(a) = |a|c i s independent of (a) . f (Exponent c = and i s a meromorphic • C on which c l i e s . s ) = N k/Q, (<5f)(c)) W(c), 0 £ where W(c) P i s the l o c a l root number of the character c . form l a t e r . The theorem uses a Lemma: 0 < exponent c < 1 For C(f,cK(g,c) and (any nice) We w i l l discuss i t s f and g, = C(f,c)?(g,c) . The theorem i s then proved by f i n d i n g , for each equivalence Q class of - 4 - quasicharacters C a function f^, such that a meromorphic function of c for This quotient 0 < exponent c < 1 . ?(f^,c)/?(f^,,c) ( i . e . has not i d e n t i c a l l y zero denominator) p(c) w i l l then e a s i l y be seen to have an a n a l y t i c continuation to the entire surface i.e. to c of any exponent. or exponent c Since, for any i s positive, ^(f^,,c) C. 0 < exponent c < 1, c, C, either exponent c has an a n a l y t i c continuation. F i n a l l y the lemma allows us to replace for is f^ by any suitable then for a l l c f (first by analytic continuation). Global ^-functions Let k now be a number f i e l d . We consider quasicharacters of * the idele group J t r i v i a l on ideles of norm 1 . If c k (1) . Let be the subgroup of i s a quasicharacter t r i v i a l on it depends only on |a| and so y i e l d s a continuous m u l t i p l i c a t i v e + A * . * homomorphism ~R —>• C . (Imbed E. or <E i n the idele group i n the usual way and c a l l image elements j • ^ () x o w a c(|a| ,) arcn for c t r i v i a l on Any such map i s of the form t —> t character i s t r i v i a l on Since, for any we have c, S for a l l a . by x —> c(x g for c This C a = 1 arcn s e <E . ) i s the map), Therefore a quasic(ct) = |a| . S is*quasicharacter t r i v i a l on for some Now we define the global exponent > 1 ; c i f f i t i s of the form a —> |c(a)| |c(a)| = |a| |c(a)| > 0 K. r s s e (C . Indeed s , i s r e a l , because i s the exponent of c. function for quasicharacters of £(f,c) = f(a)c(a)da, where f i s a complex- k valued function on the adele group s a t i s f y i n g certain integral convergence J - 5 - and continuity conditions. by the fact that II (1 - Np The 'exponent > 1' r e s t r i c t i o n i s forced -s -1 ) converges only for Re s > 1 (see p<oo D) below). Theorem: The global analytic continuation theorem i s : By analytic continuation we may the domain of a l l quasicharacters. except at residues c.(a) = 1 -Kf(0) and £(f,c) = £(f,c) In the above (i.e. and ^-function to The extended function i s regular c(a) = |a|, xf(0) extend the where i t has simple poles with r e s p e c t i v e l y . The equation is satisfied. f = Fourier transform with respect to adele group f(£) = f(n)e _ 2 ^ i A ( n 5 ) dn where •'A A(x) = £ A (x )) P P c(a) = |a|c ^(a) P and r r 1 2 - 2 (2TT) hR . w/| a | the residue of the c l a s s i c a l zeta function of k at 1 . • D) Comparison with Hecke's L-functions Let S primes. be a fixed set of primes containing a l l archimedean Let us consider idele characters unramified outside Such a character i s a product where Let S c I P i s unramified i f S c(a) = II c (a ) p P P p i S and of l o c a l characters II c (a) = 1 P P for a e k * be the part of the i d e a l group generated by primes outside g ord a and l e t <b : J —> I be the usual map a —> II p . Set P^S P b * S . c (a) = II c (a ) . P^S P P P k Then there i s a unique ideal character Y given by ord a x( rg( )) < = 0 => P p e S, function () • c a c (a ) = 1, P P For = a £(f ,c ) P characteristic for p I S . P : i s defined, and f o r p / S function of the integers For exponent c > 1 (c 0 let f of P P s k be the . P and f o r f p ' £(f,c) = 11 ? (f ,c ) P P n P S P any quasicharacter) s u f f i c i e n t l y nice (e.g. as above), f (a) = E be any n o n - t r i v i a l function for which the l o c a l P Theorem (a )x(<rn(oO) II c Write c(a) = P let f P This i s well defined, since P s where . P f (a ) . P P Sketch of proof: 1) For p i S U f (a )c (a )da = P P P P P ( t r i v i a l character) da (char. f n . of 0 ) = 1 P 2) f (a)c(a)da = lim , Jj k T# .. ,- / x ink* peT nu f p ( a P p , x ) c p ( a P . , ) d a p' the set of a l l primes and) containing a > 1 . h e r e pj!T the l i m i t i s taken over f i n i t e sets of primes important fact that w f II p<oo S . (This uses the f (a ) I I a I da P P P P 11 1 (approaching < 00 for See remark i n section C .) 1) and 2) imply 3) ?(f,c) f (a)c(a)da = lim, II ? ( f »c ) T£S peT p = n p (Fubini-type lemma) C ( f , c) P P P By analytic continuation the theorem holds for c of any exponent, - 7- II c (a )x(<J> (a)) then peS one shows by a d i r e c t computation of the integral that f o r p i S, in p a r t i c u l a r f o r characters. If c(a) = C P C (f ,c • ^ ) = — s P for Re s > 1 . P P " N6" P P 1/2 1 - X ( P ) N ( P ) - P S S Therefore n ? (f , ) n N 6 5(f,c|| )= s _ 1 / 2 C peS P p^S P P n c (f ,c ) n P P peS P p^S X(P)NP" S ) _ 1 p^S /2 -1 = n (i- N6 X / Z L(S, ) X P S L(s,x) where i s the Hecke L-function II (1 - x(p)Np ) . This P^S explains why global t, functions are i n i t i a l l y defined only for quasicharacters of exponent > 1 . Writing down t h i s equation f o r f fact that f o r p e S and "cf"p\ and using the the choice of f 's i s immaterial P (local functional equation), the global functional equation of Tate expresses exactly the c l a s s i c a l functional equation of Hecke (see §2D)). - 8 - §2. Non-abelian A) The Unramified F i n i t e Primes - F i r s t D e f i n i t i o n of Let E/K group p L-functions of be a f i n i t e normal extension of number f i e l d s with Galois G . Let V a be a finite-dimensional complex vector space and V-representation with character K, P/p E/K Op = [ ] in E and assume p x • L P be a prime ideal i s unramified i n E . Let P over p . Then the complex number -s -1 d e t ^ ( l - p(o~p)Np ) doesn't depend on the choice n p unramified series converges f o r Re s > 1. P e t be a Frobenius of (Euler factor) of L-functions and we set L ( s , x ) = d e t v ( x ~ p(o )Np -s -1 p Right away one sees the change from one-dimensional ) . The (continuous) representations of the idele class group or ideal group to higherdimensional representations of the (not necessarily abelian) Galois group. The l i n k i n the abelian case i s class f i e l d theory: If G i s abelian and p i r r e d u c i b l e then p i s a character into S . g Let 6 : I —>• G be the global A r t i n map (S = a l l archimedean primes 1 s •and those ramified i n E; Then p • 6 L(s,x +X ) 1 i s the ideal group outside i s a Hecke character"^, (Hecke)L(s,p*6) a) I 2 = (Artin)L(s,p) = L ( »X )L(s,X ) s 1 2 6(p) = (Additivity) I subgroup of p r i n c i p a l ideals generated 1 mod p p e S . and . We have the following properties: A Hecke character i s a character of to S) . R t r i v i a l on some (congruence) by elements of K congruent - 9 - b) Let H of G/H be a normal subgroup of with character character x' • Then x L(s,x') number of Euler factors. c) Let H ation and and Ind x b) p' and p be a representation i t s l i f t i n g to L(s,x) G with d i f f e r by a f i n i t e (Lifting). be a subgroup of L(s,x) Remark: and G . Let G, x the character of a represent- the induced character on G . Then L(s,Ind x) d i f f e r by a f i n i t e number of Euler factors. i s clear from properties of the Frobenius. (Induction). For c) see, for example, Heilbronn i n [CF] . For both b) and c ) , l o c a l factors agree wherever both are defined but there may be primes i n K ramified (respectively unramified) whose d i v i s o r s i n F i n the intermediate f i e l d are unramified , Before going on to other facts (respectively ramified) F in E . about L-functions derived b a s i c a l l y from these properties and the Braiier induction formula, we give the f i n a l d e f i n i t i o n of L-functions. B) The Ramified F i n i t e Primes - F i n a l D e f i n i t i o n of L-functions The problem i s to define l o c a l factors at ramified prime ideals such that a), b) and c) above hold with equality. prime p and l e t D^ (respectively I ) p (respectively i n e r t i a l ) group of a d i v i s o r 0 be the Frobenius i n B /I space by V of Ip, V exists a p . G and i f we write V Consider any be the decomposition P of p acts on the representation f o r the set of vectors fixed i s a representation space f o r ~D^/I R p., making i n E . Let i . e . there - 10 - D • p(D ) p p h invariant subspace V h i Dp/Ip GL(V) h T commutative, where c p • GL(V r ) i s r e s t r i c t i o n of automorphisms to the p- from the normality of I . That p in V D p p-invariant i s immediate is . Now set -s -1 L(s,x) = n det P<" Theorem: (1 - p (a )Np T V ) , independent of choice of P p a), b) and c) above hold with equality. • Remark: p to This i s made plausible anyway by the fact that p^ "replaces" each l o c a l extension one, with Galois group D E p/ p K by an reducing unramified p/ip • It i s clear from the d e f i n i t i o n that zeta function of the number f i e l d regular representation of G of the t r i v i a l subgroup of I X(l)x X irreducible Corollary: and G then LCs,!^) = £ (s), the G K K . Further, i f r„ i s the G 1 the t r i v i a l representation Ind 1 = r 1 a n d has character G we have: 5 (s) = II £ L(s,x) X ( 1 ) . X irreducible Note that i f dim V = 1 and p = X coincides with that of section A). An outstanding • i s f a i t h f u l , then this d e f i n i t i o n (For i f 1^1, then V = 0,) problem i n L-functions i s the A r t i n Conjecture: It w i l l be shown that L-functions have an analytic continuation to - l i - the whole plane. A r t i n conjectured, i n analogy with the dimensional case, that i f L(s,x) then i s entire. x does not contain the t r i v i a l character, It seems that the only characters for x which a direct approach i s e f f e c t i v e are those x^ with one-dimensional and a^ = I i a ind X. r a t i o n a l and p o s i t i v e . In L(s,x) this case properties a) and c) show that some power of entire. one- is However, a much more powerful approach i s available through the theory of L-functions of automorphic representations (see e.g. [7 ] ) . L-functions l i v e most n a t u r a l l y i n the following setting: Let p: Q K —• GL(V) be a continuous representation with open kernel. b) shows that we may define an L-function for p and a) allows us to extend the d e f i n i t i o n to any v i r t u a l representation C) of Q, . K The I n f i n i t e Primes and Exponential Factor; The Extended Function A; A Analytic Continuation A ( s , x ) = A(x) w i l l be of the form functions A and y and where the functional equation A(s,x) W(x) 1 . of absolute value i s the character of 1 2) ' : G —> GL(V') (V p: x L = Y (s)L(s,x) X for some constant i s the conjugate of V), for some i s as i n B), and w i l l s a t i s f y W(x)A(l-s,x) G —>• GL(V), = dual of s/2 then x: if X x i s the character of -1, defined by p'(f)(x) = f(p ""(x)) s s 2) The group of v i r t u a l representations of ft i s , by d e f i n i t i o n , K the free abelian group on the i r r e d u c i b l e continuous representations of tt v K . - 12 - D e f i n i t i o n s of y (s), A(x): X Set y(s) = TT• "' r^( j)) S / 2 and S II " y ('s-) v infinite y ((s) s) = where vv v 1 y is v as follows: v complex: C v real: y (s) = [y(s)y(s+l)J^ ) . V Write Gal(E ,K ), W V V 1 X V = V V V o = eigenspace of dim V v X and 1 a w generates p(a ) corresponding to w corresponding to -1 . Set r A(x) Fix a prime i d e a l p and l e t oo {G.}. . l 1=0 be the with respect to g. i and set be the number of elements of -, G. v —x i 2. 8n 8- codim V , i=0 G. x 00 (x) = and we need the notion of the A r t i n ramification sequence of G(E,K) = G, n +1 p(o ) w dim V y(s+l) In order to define conductor: where i f wlv = eigenspace of + V Y ( s ) = Y(S) « V , v + v where V G. i p . Let i s the subspace of a l l 1 v in V fixed by (Remark: G. . x This i s a purely l o c a l construction and can be related to conductors of class f i e l d s as follows: for now) i s l o c a l and l e t character of £2 . is. (see §1A)). Let U Let (r) x ^ 1 L be a (just for now) be the units of K (indeed any f i n i t e normal extension of f p , where N L/K^I, ) . where <J> i s the least (just one-dimensional r such that congruent to K) U (r) function and c 1 mod p has a conductor i s contained i n It i s well-known (see §5 lemma 4) that i s the Herbrand K be the c y c l i c extension attached to L/K f Suppose f = <f>(c) + 1, i s the largest integer x r - 13 - such that G ° character of f 1 . But J g.gi=0 <j)(c) + 1 = . x Let be the n 1 Gal(L,K) x • a r i s i n g from We have dim V = 1 and G. X^ 4- 1 so codim V =0 i f G^ = 1 and 1 if G ^ 1, cannot be t r i v i a l on any n o n - t r i v i a l subgroup of x without L/K also being t r i v i a l ) . Now let x where d • Of course this i s 1 •) = |$(x) unramified i n Gal(L,K) The f i e l d - t h e o r e t i c conductor of i s the same as the A r t i n conductor of also clear i f x (since = E . n p ^ ' ^ ; we have n(x,p) = 0 p<°o F i n a l l y , we define A(x) = I d I K n x iff p p i s the absolute discriminant of $(x) an integer follows from the formula K . dimensional character and for some integer c . (j{ (x) ) » K/Q That = <t>(c) + 1 n(x,p) i s ($(x) such that j[(Ind x) to x for the theorem of Hasse-Arf (which says that i n this case i n t e g r a l ) , a formula r e l a t i n g is G one- a 4- G ,., , c c+1 $(c) is (see immediately below), and the Brauer induction formula. The Functional Equation: The method of proof i s to reduce to the case of a representation induced by a one-dimensional character by property a) and Brauer induction, then to a one-dimensional character by property c ) . F i n a l l y one shows that the one-dimensional case of the desired functional equation i s exactly the functional equation of HeckeTate, using property b). F i r s t one must v e r i f y properties a), b) and c) for A, i . e . for A(x) and y (s) . X The only troublesome one i s induction for the A r t i n conductor: ([16] §4.3 Prop. 4) j$(Ind x) = p / j ^ d N F / K ^ X ^ " L E T U S A S S U M E T H A T A ) ' B ) A N D C ) - 14 - are v e r i f i e d f o r A . Write x = I ^ I n one character of some abelian subgroup F. = fixed f i e l d of x H., Hi 1 1 n f° a R\ of the kernel of r X^ degree a G . Let Y- and 1 F l the 1 fixed f i e l d of H! . F!/F. i s c y c l i c with Galois group H./Hl . i i i i i Let G. = H./H! - Image x . • F i n a l l y , l e t x! be the character i l l l l of x^ • Y property b), G_^ a r i s i n g from x^ Let us work with a notation. Let S B at a time and suppress the " i " be the set of primes of plus a l l archimedean primes. c v i a the A r t i n map A(s,x^) = A(s,x|) • F i n the that ramify i n F', x' defines an idele character Now 8: J_ —> G . Let *' be the i d e a l character s a r i s i n g from *' c as follows: If § : J —I-r- i s the unique ideal character such that i- the usual map, s ip' (<j>„ (a) ) = II c (a ) P^S s for a e J P P . r We know from §2A) and the remark i n §2B) that (Hecke)L(s,*') = U. (1 - * ' ( p ) N p " ) ps^S S = (Artin)L(s , ') _1 . Now we X w i l l write down the Hecke-Tate functional equation for compare the exponential factors "A" and factors *' and "y" corresponding to i n f i n i t e primes i n i t , to those factors as defined above. superscript The '"" r e f e r s to a factor i n the Hecke-Tate equation. s/2 Set y ^ i ( s ) L ( s , * ) . Here, we l e t A'(s,*') = A ' ) A* (*') = [ D | K S N K /Q(<5 1 Cc) ) ] where ^(c) i s the product of the l o c a l conductors of the p' > defined i n §1A) and y',(s) = n Y(S)Y(S+1) II y(s) n Y(S+1), with v complex v real v real c = 1 c 4 1 V V C s a s y as above, From the l a s t l i n e of Tate's thesis (see D) below) we have A'(1-S,TJJ') = W(ip')A' (s,<p') f o r some constant W(ip') of absolute - 15 - value 1 . We must show that the factors factors y j and 1 coincide. To show equality of the A r t i n conductor $(x') The p-part of the l a t t e r i s the least A and A = A' A' and the we must prove the and the conductor r such that (5(c) . (r) c (U ) = 1 . P P As was shown above, the p-part of the former i s the conductor of Lp/Fp, where L i s defined by: p Gal(F^,Lp) i s the kernel of the r e s t r i c t i o n (to Gal(F',F )) of x ' • This holds, since F'/F i s P P c y c l i c . More precisely, i f we consider the map f which makes res x Gal(F',F ) P P commute, then res *V K L i s the c y c l i c extension attached to the composition R f * Gal(L_,F ) 1 • S . Recalling that the conductor of p r p L„/F P p i s the least where 0^ r fr) * I P ' c Kernel(0 : F P P P such that Gal(L ,F ) ) , p i s the l o c a l A r t i n map, the equality of conductors then follows from the commutativity of the following diagram and the i n j e c t i v i t y of f . 1. ft F >P F To show c = sgn ( i . e . X*(a ) = -1 v A Y p r 0 Y' 1 — ^ — G a l ( F ' ,F) -X- >- S — P — , Gal(L ,F ) p Y'» = c where w e ^1) a v -> S p need only show that f o r v i f f dim V generates = 1 . Now Gal(F',F ) w v real, dim V =1 i f f for w v . The l o c a l - 16 - A r t i n map 8 v r satisfies 6 (-1) = a v v Y'(O ) = -1 v iff a v c (-1) = -1 v i f f c = sgn. v A v i s order x F i n a l l y then, i f 2 for a l l r e a l v i f f c (-1) i s order v and so 2 in the character of the orginal representn. n. ation p we have A ( s , x ) = n A(s,ip!) = II W(ip!) 1 i i _ n. _ n. A(l-s,ip'.) = W(x)A(l-s,x) with W(x) = II W-(ip!) . Since each i 1 S 1 1 W(ip^) D) 1 has absolute value 1, W(x) has absolute value 1. Appendix Let us v e r i f y i n f u l l the Hecke-Tate functional equation and compute the l o c a l constants i n i t i n the one-dimensional ip (the ip' ip(<f> (a)) = b case. Let us write i n section C)) for the unique ideal character s a t i s f y i n g II c (a ) P^S P for a l l a £ J P The l a s t l i n e of Tate's thesis i s : ?(i-s,ip 1 ) = where the functions n peS p P (c || ) n p^S N(6) s P P P s <p ( 6 K ( s , i p ) , % x P P are defined as follows: I | (2^)1_sr(s + M ) n ^ ) = (-i) 1—r where * * (2,) r(i-s +-IfL) , iO. in 6 „ Cp(e )= e , n e Z. M s 1 1 s p pH ( , P p vvoSy S) iisv ( S S n | 1 } = (-i)y(s+l) Y(2-s) - 17 - p p (c II ) P P finite: = N(S &(c ) ) \ ( j $ ( c )) ^x (c ) P P ° P P P s where S U i s the l o c a l Gauss sum £ c (XTT ) e x p ( 2 T r i A(XTT ) ) . x m x (c ) P P Here m P m = ord (6 |(c ) ) , TT P P P exp (2Tri. . .) !) (not to be confused with the i s any element of order over a set of representatives of 1, IT i n and the sum i s taken 1 + $(c ) in p U . p The i s that of §1A) of course, not an extended L-function. v e r i f i e d that the sum x A here It i s r e a d i l y i s independent of choice of uniformizer p ir . Now we are ready to compute: cd-s,^" ) _ p 1 ? ( S n n '^ v complex N(6 f(c )) p J S _ % p II ) s ( c P y ( s v real " c = 1 V P y ( 1 N(^(c ))" x (c ) n % p n ) n p S ) V real c 4 1 V N(6 ) " S p T(s+D-(-i) % p ^ _ 1 y ( 2 " s ) (6 ) . p peS If v i s complex, ' r l o c a l A r t i n map. c =1 v because c v arises from the ( t r i v i a l ) For the complex quasicharacter || , S l+s, l - 2 s Y(s)y(l+s) l-s 2(l-s)Y(2-s) r I, s. V" T(s) T(l-s) (2TT)- 1 = J / S U = 2 m 2 r ( ; i T = ) r ( by the Legendre duplication formula. S ) Y Furthermore IT N(6 ) p p«x> r Now define constants v complex: W(c^) S 2 = |d„| r as follows: W(c^) = 1 . rl if c = 1 W(c ) = i - iif c 4 1 . v v v real: V v f i n i t e ; v = p: c P l W(c ) = N()J(c )) i s unramified and 2 T p ( -ord W(c ) = c (ir P P P c p ) • (Note that i f p I 5 _^ ) = ib (6 ) . P P S, S - 18 - We get: S(1-S,T[) ) _ C(S,I|I) 1 V^tf v real 4 c y(s) Y(1+S) . Y(l-s) Y((1-S)+D v complex l | d F | N ( < ( c ) ) , S 1 n w c ) all v ( W -) 1 v A'C*)'1-" W(ip) = n W(c ) . F i n a l l y then, v A'(l-s,ip) v real c = 1 V l w where ~* Y(S) Y(l-s) = W(ijj)A' (s,!p) 2 A'(1-S,IJJ) = W(i|>)A' as was to be shown. (s,\b) , or - 19 - §3. Local Constants The following theorem i s due to Langlands and Theorem: Deligne. Let K be a l o c a l f i e l d of c h a r a c t e r i s t i c zero with valuation v . Let J2, = Gal(K ,K) and l e t p be a v i r t u a l K sep representation of Q . Then there exists a unique function W K• from the group of v i r t u a l representations of Q, to such that K T W( i) P l +p ) 2 = W(p )W(p ) 1 2 P-^p^. for a l l Let p be a v i r t u a l representation of degree ii) an induced representation. If p iii) ab fi K and i f Tr W(Ind p ) = W(p) Then i s i r r e d u c i b l e and of degree c v i s the character of l o c a l A r t i n map, then * K 0 and Ind p . 1, hence a character of defined by p through the ° W(p) i s the l o c a l constant for functional equation of the zeta-function ( i . e . the c i n the W(c ) of §2D)). v • Note (after Deligne = W(p) c(f ,<l?)5— , [ l ] ) that we may also write, i n case i i i ) above, v for any f such that £(f ,c ) i s defined. S(f ,c || ) % V V A good framework in which to prove this theorem i s due to Tate ([18]): R(K) p If K i s global or l o c a l c h a r a c t e r i s t i c 0, denote the set of pairs (L,p) i s a v i r t u a l representation of R(E/K) denote those pairs (L,p) such that L/K i s f i n i t e and 9. . I f E/K i s f i n i t e , l e t such that E£ L£ K i s a v i r t u a l representation of Gal(E,L) . R(E/K) a subset of R(K) = R(K) and we have let and p may be considered u R(E/K) . How do E=>K one- - 20 - dimensional representations f i t in? pairs (L,x) if i s l o c a l (respectively of K where L/K Let R^(K) i s f i n i t e and C , x denote the set of i s a character of L the idele class group, i f J-4 K i s global). R^(K) imbeds i n R(K) as follows: A Any character group. x L (resp. C^) factors through an open sub- By the existence theorem of class f i e l d theory, this can it N(L' ) (resp. N(C be taken to be L'/L of . Considering x L* ,)) for some f i n i t e to have domain extension L /N(L' ) (resp. C /N(C J_i and r e c a l l i n g that the l o c a l (resp. global) A r t i n map isomorphism onto taking (L,x) Definition: Gal(L',L), to we can imbed (L,xe ) . F extended to R(K) a) F(L, +p ) b) If then P ] 2 = F(L,p )F(L,p ) 1 Li 2 L' Li p), induced from i s extendible i n n R(E/K) taking i s extendible i f i t can be (L,p ) e R ( K ) . for ± p = 0 with dimension F ( L , p ) = F(L',Ind Q , F R-^(K) , by satisfying: ( L , p ) e R(K) ation of Say R(K) ± i s a function defined on values i n some abelian group. into i s an R^(E/K) = R (K) F i n a l l y , set -1 Suppose R-^(K) 6 ,)) Li E/K where p . Ind If L' L £ L' £ K, i s the v i r t u a l represent- -LI E/K and i s f i n i t e , Galois, say i f i t can be extended from R^(E/K) s a t i s f y i n g (the appropriate modifications of) a) and b). strongly extendible i f b) holds for p to Say F R(E/K) F is of any dimension. • If F i s extendible, then the extension i s unique, e s s e n t i a l l y by Brauer induction. As a consequence i f f i t i s extendible i n that i f K i s global, E/K for every (L,x) >-»- A(s,x) F i s extendible i n R(K) E . We have shown i n §2 and (L,x) I— W.(x) y are - 21 - extendible, even strongly. Other familiar extendible functions are: i) (L, ) ^> 8 ii) (L,x) X N L / K ^ ( x(°) v - > ( X ) ) f° ( r K c a l o b a l o r P-adic). fixed element of C ( i f K i s global) K ft or K (if K iii) is local). (L,x) I * a(L) where - Now a i s a function of L only. the above theorem looks l i k e : Theorem: If K i s local characteristic 0 (L,x) —• W(x), then the l o c a l root number, i s extendible. • Remark: It i s not hard to see that The condition "dim p = 0" W(x) cannot be strongly extendible. i s therefore s i g n i f i c a n t ; i t shows the necessity of working with the entire representation r i n g . X, W(Ind x ) / ( x ) one has i w sa For general fourth root of unity (see §4). The key lemma for this theorem i s roughly as follows: Let a * be a character of K (K non-archimedean) and l e t 8 be a character ft of L with L/K f i n i t e . Let = a • . If a i s " s u f f i c i e n t l y highly ramified" and 8 i s " s u f f i c i e n t l y lowly ramified r e l a t i v e to a", ft then there exists a c e K such that W(8a ) = 8(c)W(a ) . The theorem LJ JLi can be proved i n several steps: 1) Fix in E/K 2) Find a global extension v of n E/K . k finite. One may It s u f f i c e s to show also assume e/k with only one d i v i s o r Gal(e,k) - Gal(e ,k u 0 ) v 0 K (L,x) l ~ > W(x) i s extendible non-archimedean. with u n k in t o t a l l y complex, and a place e such that i s canonically isomorphic to Gal(E,K) . Hence - 22 - there exists an isomorphism R(E/K) - R(e,k) ( £ , p ) < — y (H , p ) where w w Q and p w 0 3) y W 1 t ' i ex o c a ± for n r o o in % to Gal(E,£ ) . One must show w 0 number i s extendible i n e/k . t One may (by the Griinwald-Wang Theorem ([AT] Ch. 10)) choose a single idele class character a) i s the unique d i v i s o r of v 0 n i s the r e s t r i c t i o n of p (&,x) — ^^w ^' that w 0 Q given by For every a v v 4 v , a of C^ such that: the conditions of the key lemma are s a t i s f i e d n and every ( a l l ) (F,B) i n R^(e ,k ) . This i s possible u because the conductors of a l l possible B's f o r a l l possible v's are bounded above. b) a V = 1. 0 Choose, f o r v 4 v», c as i n the key lemma. Let a. be the ' 0 v I idele class character of C„ given by (a„) = a • N„ „ I Iw v &„/k_ w v 4) J We have, f o r each ( & , x ) e R-^(e/k), X ( „ ) ((°O ) C w TT tt w W(x v w x, ) w non arch., w 4 w w n u w =w q 0 w arch. The f i r s t case i s the lemma. The second i s from a v i s from 5) k totally = 1 . The t h i r d 0 complex. ) = W(Y )x(c)a(£) with L *w ), independent of x • Y ii) In view of 4) write W(xaT 0 a(£) = II , w f W((a.) w X, q w non-arch. B W a n d iii) above - 23 - X(c) and a ( I ) are extendible. By e x t e n d i b i l i t y of the global root number and Frobenius r e c i p r o c i t y , Hence (&,x) *" W(x ) l— w (&>x) w (x ^) a i s extendible. i s extendible and we are done. 0 • One i n t e r e s t i n g feature of this proof i s the g l o b a l - t o - l o c a l (!) approach, using the known e x t e n d i b i l i t y of the A r t i n A function. This may indicate the d i f f i c u l t y of approaching A r t i n L-functions i n a purely l o c a l way. Another feature i s the examination of highly ramified l o c a l characters. sum over U /U^ ^ P P r This allows us to replace a character by a s i m i l a r sum over U^ ^/U^ ^ . As a consequence, P P X r one can show that f o r a one-dimensional character root of unity i f a i s unramified In the remaining case, a or a, W(a) is a p |$(a) (a i s w i l d l y ramified). tamely ramified, W(a) i s essentially a character sum ( f i n i t e Fourier transform) of a m u l t i p l i c a t i v e character over the additive group of the residue f i e l d norm). (divided by i t s complex These character sums or Gaussian sums are u l t r a c l a s s i c a l objects and have played an important role i n the theory of cyclotomic f i e l d s (see, f o r example, [19]). In the next section we w i l l examine this r e l a t i o n more c l o s e l y . References f o r §1,2,3: Tate's thesis can be found i n i t s entirety in Cassels-Frohlich ( [CF ] ) . Our discussion of non-abelian L-functions follows that of Martinet's a r t i c l e i n the Durham notes ([14]). The discussion of l o c a l root numbers i s due to Tate i n the same s e r i e s . For Deligne's proof of the e x t e n d i b i l i t y of l o c a l root numbers, one i s referred to [ 1 ]. Any well-known facts about arithmetic of l o c a l f i e l d s (Herbrand function, induction formula for A r t i n conductor etc.) can be found i n Corps Locaux ([SI]). - 24 §4. - S t r u c t u r e of L o c a l Root Numbers Throughout l e t F be a l o c a l p - a d i c f i e l d , over 1 Q P F be i t s residue field, l+ord 6 d_ = TL , F order of 1, F . p r We the prime of F, TT 0 , say. Let an element of r F r where 6^ i s the a b s o l u t e r denote the v a l u a t i o n r i n g of F by 0^ different and the units r T7 by We . r have ip,, Let be the c a n o n i c a l c h a r a c t e r r ^(x) = e x p ( 2 i T i X(Tr r x)) where of Tr_ r F as i n 11A) . i s the absolute r t r a c e of F and X(x) i s the unique r a t i o n a l (mod Z) with only a p_-power i n i t s denominator and such t h a t X(x) - x e & . The U p character group of i s the c h a r a c t e r F i s c a n o n i c a l l y isomorphic to — of F g i v e n by g(x) = exp( 2Tri P y v-*- g^ isomorphism i s g i v e n by let F be rn F . where the maximal u n r a m i f i e d gy(x) extension F . Tr=(x)) 0 If then g- = g the F 0 = g(xy) of Q Po- . Finally, contained L e t us make the c o n v e n t i o n t h a t "tame" or "tamely r a m i f i e d " r e f e r s to a c h a r a c t e r or f i e l d extension t h a t i s r a m i f i e d and so, r e s e r v i n g " a t most tame" f o r " u n r a m i f i e d The aim of t h i s s e c t i o n i s to d e s c r i b e numbers. a If complex norm. unramified, find W(x) Theorem 1: m(x) 4 1 If a W(a) for or c h a r a c t e r the s t r u c t u r e of l o c a l sum i s a r o o t of u n i t y . x a n W(x) over i s wildly ramified y character (Dwork [ 2 ] ) . then or tamely r a m i f i e d " . i s a tamely r a m i f i e d c h a r a c t e r a c l a s s i c a l Gauss sum, Let of m(x) tamely F, (i.e. of F , W(a) root is d i v i d e d by i t s o r d ^ ^ ( a ) • >^ 2 ) or From t h i s i t i s p o s s i b l e Q^, up to r o o t s of = ordp(^(x))/x(l) i s a r o o t of u n i t y . • to unity. If ^ - 25 a = x If is i s one-dimensional, - W(a) i s a r o o t of u n i t y u n l e s s a tame. Lemma 1: a) Suppose ippCpQ^x) = g(x) F i s absolutely unramified. x e 0F for a l l . b) For a r b i t r a r y f i n i t e F/Q P I|J (d x ) Then 1 = g (xy) for a l l p Then 0 , let y = Tr ( a , ~ p / F 1 p 0 ) nr 0p x e nr Proof: a(x) a) x e 0p, Choose, f o r = T r - ( x ) mod p_ 2, and i t s denominator such t h a t e Z a(x) b(x) b(x) e Q such t h a t with only a p„-power i n = A(Tr^Cp^^x))mod 2 . r We want to U 2 Tri show t h a t exp(— P a(x) Tr a(x)) = exp(2iTi b(x)) - Pgb(x) e v^TZ (x) = a(x) 0 . Now, I <j(x) = aeGal(F,q) E Tr„(x)mod vJZ v 0 a(x) PQ . since F i s absolutely Q so = 0 mod p^CTrpCp^x)) - p - p^r^p^x) - p b ( x ) e PQ%^ n Q = e(x)/p (by d e f i n i t i o n p„Z 0 b(x) = Tr-(x) , Hence - p b(x) = Tr (x) an i n t e g e r , w r i t e unramified Q p a(x) T h i s i s the same as I a(x) aeGal(F,Z/p ) - Tr (x) Therefore . n P Q • T o with s e e e(x) t n a t e 2 t n i s i s and, actually without c(x) l o s s of g e n e r a l i t y , c(x), PQ - 1 d(x) a(x) r >_ 1 i n t e g e r s and - e(x) = p^ . p n => Write a(x) prime to d(x)|c(x) - Ppb(x) = p A ~[7^y d(x). We see that and a) i s proved. r o r of - For 26 - x e 0_ , Fnr V^x) = *F nr ^ d F / ' P , -1 <P % ( / r x x ) ) d F r T r 0 F nr /F^ ( ^1 n p ( b y ) ; ) l i n e arity of trace) m nr = g-(xy) by a) ( f o r we have F = F ) . • n r ft L e t us c o n s i d e r a tame c h a r a c t e r -ft F (1) - U_/UJ, , r r sum and set r X r T F The as d e f i n e d definition where the sum r system of r e p r e s e n t a t i v e s of and of a(0) = 0 . V -1 -1 (= ). a(d„ x)<p ,(d„ x ) , x (a) p |J(a)) a U^ ^ in m TT on of the l o c a l Gauss i s taken over a and m i s the order of lemma 1 y i e l d ft Prop. 1: Let x (a) = a(d„ p v a be a tame c h a r a c t e r of F . Then _1 _ft T r , _ . (d„ p r i ) ) G 1 ( a ) , where f o r y e F , k / \b u l over F g i v e n by G (a) = 7 be £_ a(x)g(xy) xeF Note t h a t a s e t of r e p r e s e n t a t i v e s f o r taken from and TT that a(y)G £ n (a) = of u n i t y . E/K we E . The •. in U may p (a) . y n • • J L e t us w r i t e a ^ b if ab X be f i n i t e normal w i t h G a l o i s group L U^"^ • G . Let L be and let the f i x e d i s the maximal at most tame e x t e n s i o n of s t r u c t u r e of Gal(L,K) to r o o t s i s a r o o t of u n i t y . G i s the . can g i v e the s t r u c t u r e of l o c a l r o o t numbers, up r a m i f i c a t i o n group of that G 1 nr Now Y n r c h a r a c t e r sum Proof: G' ( a ) w G^ field K be of Let the w i l d G^, contained has been determined by Hasse: so in - 27 - Prop. 2: ( [ H ] § 1 6 ) . L e t L/K q = order o f K, be a l o c a l tame G a l o i s f = r e l a t i v e degree of e = e(L/K) = r a m i f i c a t i o n index of L/K = [ L : K ] , L/K . Then some r such t h a t a and e[r(q-l), x such t h a t and axa X x and Gal(L,K) e c y c l i c , generated by extension f =1, a i s metar = x for = x^ . • If p i s an i r r e d u c i b l e r e p r e s e n t a t i o n of G V, p gives r i s e p Since to a i s irreducible, determining a t i o n of on the v e c t o r representation p ^ = 1, case W(p) or ^ 1 p^ p^ = p . Now Gal(L,K) of G/G^ = Gal(L,K) . Dwork [ 2 ] has so t h a t we may assume ( i n s t r u c t u r e up to r o o t s of u n i t y ) t h a t Gal(L,K) . p i s a represent- i s m e t a c y c l i c hence super- s o l v a b l e and so by ([S2] Th. 20), i t i s a monomial group. t h i s means t h a t every space l V shown t h a t i n the f i r s t G By d e f i n i t i o n i r r e d u c i b l e r e p r e s e n t a t i o n i s one-dimensional or by a one-dimensional r e p r e s e n t a t i o n . I f p i s not oneQ dimensional, l e t H c Gal(L,K) w i t h Ind a = p f o r a c h a r a c t e r induced ri i< a of K , where K i s t h e f i x e d f i e l d of H . Lemma 2: ([18] §2 lemma 1 ) . I f ( L , p ) e R(K) and L £ L' £ K L' then W(p) ^ W(Ind p ) . Indeed the q u o t i e n t i s a 4 t h r o o t i f u n i t y . n n • Remark: A s i m i l a r statement h o l d s f o r l o c a l G a l o i s Gauss sums which are d e f i n e d i n §5. Lemma 2 may be proved j u s t as lemma 9 of §6. By lemma 2, dimensional considered W(p) 'v W(a) . We w i l l see i n §5 t h a t any one- c h a r a c t e r o f a tame G a l o i s group i s a t most tame when as a l o c a l f i e l d character. P u t t i n g t h i s together with - 28 - Prop. 1 we Prop. 3. Then can d e s c r i b e Let W(p) KQ, field p W(p) . be an i r r e d u c i b l e r e p r e s e n t a t i o n of ^ W(a) where a G = Gal(E,K) . i s an at most tame c h a r a c t e r of some a t most tame over K and c o n t a i n e d i n e i t h e r a r o o t of u n i t y or a Gauss sum E . W(a) is d i v i d e d by i t s complex norm. • No uniqueness if i s p o s s i b l e i n t h i s decomposition i n g e n e r a l . f = f(E/K) is 1 or a prime, Gal(L,K) However has the p r o p e r t y t h a t every i r r e d u c i b l e r e p r e s e n t a t i o n i s o n e - d i m e n s i o n a l or induced by a c h a r a c t e r of the subgroup T h i s subgroup of o r d e r e generated by i s e x a c t l y the G a l o i s group u n r a m i f i e d e x t e n s i o n of K contained i n of L E, say x ([CR] §47.14). over the maximal E nr Prop. 4: I f 1) the r e l a t i v e degree G i s i r r e d u c i b l e and then t h e r e i s a tame c h a r a c t e r W(p) Proof: p = Ind = l^ia) . a E a = Ind nr O-dimensional. K E So nr a', a a of E nr not 1, and 1-dimensional, with i s u n i q u e l y determined A l l t h a t remains K i s prime or l 2) p = p and f(E/K) I n a = p d E nr by p (mod 4). to be shown i s uniqueness of a . If then W(a) = W(a'), i s u n i q u e l y determined since (mod a - a 1 is 4). • - 29 - §5. Gauss Sums f o r H i l b e r t Gauss sums f o r H i l b e r t Symbols symbols a r e some of the most important a l g e b r a i c numbers of n i n e t e e n t h c e n t u r y number t h e o r y . They l i e a t the core o f c y c l o t o m i c f i e l d s and hence u l t i m a t e l y c l a s s f i e l d t h e o r y . I n modern number theory they a r e t o be found n o t o n l y i n the f u n c t i o n a l of equation t h e A r t i n L - f u n c t i o n , b u t a l s o i n t h e theory o f p - a d i c L - f u n c t i o n s where g e n e r a l i z a t i o n s o f S t i c k e l b e r g e r ' s Theorem (see below) a r e sought. Let us assume t h a t (r) K f i e l d s and w r i t e Lemma 3: a) b) I f L/K I f Proof: is L e t L/K L/K f o r the u n i t s o f r K, =l.mod p . 7 be an a t most tame e x t e n s i o n . i s unramified L/K i s r a m i f i e d , The f i r s t i s a f i n i t e normal e x t e n s i o n o f p - a d i c N^^(U^ ^) = u £ r \/ 0 h'^ statement J = V K for a l l r ) r . ^ • i s ( [ S I ] V §2 Prop. 3 a ) ) . The second ([CF]I§8 Prop. 2 ) . Lemma 4: L e t L/K be any t o t a l l y r a m i f i e d e x t e n s i o n . be the sequence o f r a m i f i c a t i o n subgroups of be t h e Herbrand f u n c t i o n ( [ S I ] IV § 3 ) . i n t e g e r such t h a t G 4 1 . C contained i n N ^ ^ ( L ) Proof: ^ i ^ - o G = Gal(L,K) . Let c Then the l e a s t Let r Let < j > be t h e l a r g e s t such t h a t (r) U K. is i s <)>(c) + 1 • ([SI ] XV §2) . • - 30 - Prop. 5. L e t L/K one-dimensional be an a t most tame l o c a l e x t e n s i o n and character of Gal(L,K), a a c o n s i d e r e d as a f i e l d c h a r a c t e r v i a the l o c a l A r t i n map. a) I f L/K i s unramified so i s a . b) I f L/K i s ramified, a Proof: I f L/K i s a t most tame. i s unramified, c_ N ^ ^ ( L ) (1) L/K i s t o t a l l y r a m i f i e d , then have to c|>(0) = 0 . The case by lemma 3 a ) . * • _c N ^ ^ ( L ) L/K Since e x a c t l y t h e k e r n e l o f the A r t i n map, the r e s u l t characteristic of — A K . K contains K For a e K m Let p 0 L e t q = order of character 0(x) v^, (a,x) v^= • follows. (HI,PQ) = 1 . there i s the H i l b e r t reduces N^^(L) i s mth r o o t s o f u n i t y . and assume d e f i n e d by t h e e q u a t i o n by lemma 4, as we r a m i f i e d , but not t o t a l l y , A t h e t o t a l l y r a m i f i e d case by lemma 3b) . Now suppose If x = (a,-) where m 9: K —> G a l ( K ( /a),K) i s t h e l o c a l A r t i n map. An important p r o p e r t y of x i s that i t s kernel i s exactly N T- . (K( v^a) ) . Since ,a K( vaj/K m m K( /a) m i s always a t most tame, so i s x • W x a l s o have e SL unramified i f and o n l y i f i n ?— *k X (U ) = 1, a K K( > 'a)/K m / K( /a)/K U c N(K( /a) ) K 3. i s unramified, m for i f and so by l o c a l c l a s s f i e l d theory, i s unramified. w(x )w(x ) a Prop. 6. —rW U b r — ( a b X =/^(a.iJCxP) if X unramified. a J Xb(-D (ab,Ti) x and 7 q" I xeK x^0,l % i f a b x a, X b i X » X f a b ramified unramified. -ord,a -ord^b , [x (1-x) ] ( » X a a b l l ramified. .,. , q 4 ) / m - 31 - If x Proof: Cor. p 5). i a s unramified (In general i f x i s any r e p r e s e n t a t i o n , W ( x the second case ab , ) W ( x . x case. the r e s u l t f o l l o w s from i W(p)) by the f i r s t K a ([14] Prop. 2.2) WXx^WXx^) = x ( l ) However . In dimP -1 -1 = X a b ( t f ( x b ) ) = (ab,7r ) 1 } _ a n b second case f o l l o w s . §2 and one-dimensional and ® p) = x(&(p))W(x) W(x b unramified s ([18] d s o the I n the t h i r d case we a r e d e a l i n g w i t h a p r o p e r t y o f c h a r a c t e r sums; ( i f they a r e c o n s i d e r e d as F o u r i e r transforms, then t h i s i s a statement about c o n v o l u t i o n ) : G ( a ) G (B) = 7 7 l_ xeK a(x)BU-x) G (aB) . (See [L ] IV §3, GS4). 7 V a > V X b > X In our s i t u a t i o n t h i s l o o k s l i k e — T x Since i s e v a l u a t e d a t elements a t h i r d c l a s s f o l l o w s immediately . . p ab' t X ) -x i n 0 K r = £ (a,x)(b,l-x) . xeK x^0,l n o t i n p, the from the e v a l u a t i o n o f the H i l b e r t symbol i n the tame case which we s t a t e as lemma 5. Lemma 5: * c e K ( [ S I ] XIV §3 Prop. 8, C o r . ) . . . , i s g i v e n by (a,b) = c ^ • where q z -. x (ord a) (ord b) o r d b -ord a c = (-1) a b • In case we a r e d e a l i n g w i t h q u a d r a t i c symbols we have ( f o r more s u b t l e reasons i n v o l v i n g r o o t numbers of o r t h o g o n a l representations) w( jw(x ) x — - h r ^ - = (a,b) 7 W ( x ab (see [18] §3 Cor. 2 ) . } R e c a l l t h a t the map e x t e n d i b l e i n R(K) Definition: Let x x f)(x)> the A r t i n conductor, i s . be the c h a r a c t e r of a r e p r e s e n t a t i o n o f ft^ - 32 - x(x) D e f i n e the l o c a l G a l o i s Gauss sum by = T (X) = N K/Q ^ ( X ) ) *W(x) p 0 dimx2 tCx^ D e f i n e the J a c o b i sum J ( x L 5 X ) b 2 J V (x »X ) 1 2 = — T dimx-j^ (x2) T(X X ) 1 2 • L o c a l G a l o i s Gauss sums s a t i s f y a l l the main p r o p e r t i e s o f l o c a l r o o t numbers: a d d i t i v i t y and i n v a r i a n c e under l i f t i n g and under i n d u c t i o n o f c h a r a c t e r s o f dimension 0 . B e f o r e g o i n g on to d i s c u s s Gauss sums, i n the one-dimensional case i n t h i s s e c t i o n and h i g h e r d i m e n s i o n a l case i n § 6 , sums i n the one-dimensional r o o t numbers. Let Let K case, which we w i l l be the c y c l o t o m i c f i e l d i ' " be the p a r t o f the i d e a l group K dividing Let K Grossencharakter satisfying be a number f i e l d J(fi)J(v) = J(fiv) 1 and m 1 and such t h a t i f are the c o n j u g a t e s o f x, d e c II x I x | . , n n=l numbers Two express i n terms o f o f mth r o o t s o f u n i t y . generated by primes not an i d e a l of ( i n the sense of Hecke) i s a f u n c t i o n x = x^,...,x^ J(xO „) = K of K one p r o p e r t y of J a c o b i m . Definition: and l e t us mention i n the f o r some r a t i o n a l (independent of x e 0 0 is. . A Yft * J : I —> (E K and x E l mod m then integers e n and complex x). ^ G r o s s e n c h a r a k t e r s are s a i d to be e q u i v a l e n t i f they agree on each i d e a l m. o u t s i d e of which they are d e f i n e d . c l a s s e s are i n These e q u i v a l e n c e 1-1 correspondence w i t h q u a s i - c h a r a c t e r s o f C , t h e K - 33 - the i d e l e c l a s s group. order One can show t h a t Grb'ssencharakters of f i n i t e a r e e x a c t l y those f o r which Hecke L - f u n c t i o n s a r e d e f i n e d , i . e . they a r e t r i v i a l on some subgroup o f I of p r i n c i p a l i d e a l s generated K. by elements of Assume integers K K, = 1 mod m . i s the f i e l d of a,c t 0(m) . Define a map w(x J w ( Ja,c(p) = Ja,c: l x X ) a " c P IT — • (C v X TT Theorem 2: r o o t s of u n i t y a g a i n , and f i x by ) % W(X W e i l has proved mth a + c ( _ 1 ) • P P the f o l l o w i n g seemingly i s o l a t e d but i n t e r e s t i n g ([20]) J i s a Grossencharakter a, c fact. i n the sense of Hecke. ^ Note t h a t J interesting a,c w i l l not be of f i n i t e order i n general. I t would be t o extend t h i s theorem (somehow) t o n o n - a b e l i a n Jacobi sums. R e l a t i o n s i n I d e a l C l a s s Groups For this section, l e t E be a p r i m i t i v e mth r o o t of u n i t y , m and l e t K = 0.(5 ) . m p l u s those d i v i d i n g primes d i v i d i n g d e f i n e d by 9: I ^ ( a ) a . Let m . Let J S For (—) be the s e t of archimedean primes of a e K , m . l e t S(a) be S (f) (0 i s the g l o b a l Ar t i n i s defined only f o r K plus a l l be the power r e s i d u e symbol f o r e^K'Va) = (j) ("Va) , -*• G a l ( K ( v / a ) ,K)) m > 3 — K, map P I S(a) . - 34 - ord b Prop. 7: where (a,b) i s the H i l b e r t p P i S Now f i x ordir all = 1 = ^P t e Z . and choose an element F o r such t -1 \"p»^/p ( ^ ' X ) ^ = x, 1 of p of Kp U with /U^ \ that X P I S(x) and T h i s h o l d s f o r any . i s t o t a l l y r a m i f i e d , by l o c a l Kummer m t $ 0(m) i s (totally) ramified i f Kp^/iTp ") p • TT {x} t t x t x ( x , ^ ) = (-) (—) \">"p/ \p/ = \p Kp(/m/rTp) Since = (a,b) , Y symbol f o r Kp . and a s e t of r e p r e s e n t a t i v e s lie in K . X ^OO I f P i S ( a ) , (|) ([CF] E x e r c i s e 2.8). x _ and so t theory, is ^P ramified T (and tame) i f f r>(x <_) r —t TT G ('^•J'> d = t r )G (x Jv_, y "P p f o r G (x _ ) t • t t 0(m) . — .) t TT F i n a l l y , note t h a t for a suitable y . L e t us w r i t e p Let p be the r a t i o n a l prime under A ^P can g i v e the e x p l i c i t f a c t o r i z a t i o n f o r P . We 7 Lemma 6: . (Generalized Euler C r i t e r i o n : P i S(a), (—) i s the unique mth First, [CF] E x e r c i s e 1.4). F o r r o o t o f u n i t y such t h a t N(P)-1 (~) = a m mod P . • I t i s n o t hard t o see t h a t o n l y prime d i v i s o r s o f P i n K(£ P appear i n the f a c t o r i z a t i o n ( i n K(£ P )) of G , ) 0 f o r i t s absolute 0 norm xs a p^-power. Prop. 8. If p n = l(m) and t £ 0(m) then f a c t o r s t h e same as y fc 1 i > j=l E and E. n d p j E p p = \ (—) j=l p th p r i m i t i v e r o o t of A = l(m), g(jy) t so p P Q = z • But Pn For = 5 g(y) and in K^, mod m j P PO" = j n . Let m the order . so that Hence Kp PQ of Obviously if K, ramify be _ (K^) P . we By Prop. 7, • (^) get the * i s the same i s a g e n e r a t o r of (K^) j p ^ mod P r = j 0" and so 1 mod m P . Putting result. • G i s due fc to Stickelberger theorems on r e l a t i o n s i n i d e a l c l a s s f i e l d s - the S t i c k e l b e r g e r Theorem ([12] We and let a K(E. (P ) = P ]L ); P . 0 _ Pn 1 . be VJ p 0 ) a.(£, ) = E . j m m J the prime d i v i s o r s Let . = P. W §4.2) now. such t h a t {P.},. v , -, ^. ^ 3 (3,m;=l,l<j<m ordered so t h a t completely i n w i l l examine these G = Gal(K,Q) u a)(n) z j = z the element of = l(m) in p r i m i t i v e r o o t of u n i t y . in the f a c t o r i z a t i o n of be 3 p„ 0 h P mod m a. F - t n of Therefore t h a t of Mackenzie ( [ 1 3 ] ) . v v P.'s 3 (Z/p.Z) character U i s the source of the o n l y two Suppose P 1 = l(m) Let of K for trivial ind groups of c y c l o t o m i c and i s a non some §3 lemma 1) of t h i s together p g 1 m all in unity. s p l i t s completely i n n PQ as i t s order E. i 0 i s a system of r e p r e s e n t a t i v e s ([SI] XIV ind index of P U r m i s the P n By i 1 K_, - 2/p 2? E 0 n {1,...,PQ-1} t ind 0 0 If V where is a Pn Proof: G , 3 0 P = P 1 . The 1 f o r some the l e a s t n o n - n e g a t i v e i n t e g e r congruent to n mod P. 3 m - 36 - Theorem 3: (Stickelberger (p -l)(l-a)(jt)/m) n Pj [15] p. 355). G f a c t o r s as n in U K(E ) . 0 J The group r i n g Z[G] a c t s on K and on the i d e a l c l a s s group of K i n the o b v i o u s way and we w r i t e 0 e Q[G] be - • P £ u>(jt)o\ (j,m)=l l<j<m t h i s action exponentially. Let . 2 Lemma 7: Let Theorem 4: 6 e Z[G] Choose a prime P = P^ Claim: G^ e K . unity. By i n d u c t i o n ^ t t P P +5 0 P The c l a i m =1 P i s proved. „ m(l-cj(jt)/m) II P 1 1 J 60 £ Z[G] . t 0 _ e mod m . ± S a S U m ^ K o r a n y °^ m t ^ r o o t s °^ 1 = - l> since x 0 i s o f exponent m ir" By theorem 3, G™ f a c t o r s as m Tr by lemma 7 s K , k . But ~~m0 „ _m -td(tj) m t =IIP.nP. = p„ 0 P.. 3 . j 0 K 1 3 3 Since e K I t s u f f i c e s t o show t t +...+5 0 T1 / ^ G E v e r y element of ^ ^2t By Prop. 6, 2 m Then i n each i d e a l c l a s s so t h a t t h e n = E, e Z[G] . a n n i h i l a t e s the i d e a l c l a s s group of r a t i o n a l prime p under i t i s 66 that P i s principal. G 60 ( S t i c k e l b e r g e r s theorem). 0 Z[G] n Z[G] Proof: such t h a t G 4 fc e K K . Now suppose we have • G t P 0 °K = P l G t P 0°K = P l 1 3 P r i n c i P a l - - 37 Proof of lemma 7: since L = K ( m L = K(G ) . Let -m<50 t f a c t o r s as P power of an i d e a l of K and L Q of x p 0 U K and . dividing 69 q, K(£ P of K p|m|p over -1, ) P p K . Ch. 6 §2 Th. . Now d|m Since G ^ g f a c t o r s as the t prime where 4), i f q|p -l, u q i s t o t a l l y ramified e U of m|p -l n over K K . q By i s ramified since n q q mth . in L However a t each prime 0 and n m m i n which case and G L„ = K ( Ju~) Q q q i t must d i v i d e P over = d Hence, l o c a l l y , f o r any ([AT] m, m [L:K] e Z[G], t L o c a l Kummer theory L c K(£ ) ~ 0 Let i s of exponent / G ™ ^ ) 1 - unramified elsewhere. a c o n t r a d i c t i o n unless So f o r any P|PQ> d = 1 . 0 N Theorem 5: < r, — and Then II (Mackenzie [ 1 3 ] ) . s,t,j e 1 f o r any a C O f ^ ' '^ S Let [r] let denote the g r e a t e s t f(j,s,t) = i s principal for a l l [ ] - [^] m j ( s + t ) m s,t e 2 integer - . 3 Proof: We may assume n e i t h e r s nor t = 0 mod G G case the theorem i s t r i v i a l . We have p s t e K m, f o r i n that by Prop. 6 . s+t The f a c t o r i z a t i o n of t h i s q u o t i e n t in K(£ ) P (p -1) (l+w(j (s+t))/m-w(js)/m-o)(jt)/m) I P. . j is 0 Hence i n K i t is 3 l+o)(j (s+t))/m-(i)(js)/m-a)(jt)/n II a.(P^) . N J u ( r ) = r - [—]m m J f o r any J integer r and AT Now . notice recall , „ that t II a. (P..) j 3 1 is [^] m - 38 - p r i n c i p a l t o get the r e s u l t f o r P^ . I n g e n e r a l the r e l a t i o n s f o l l o w on c h o o s i n g an a p p r o p r i a t e P^ i n each i d e a l class. • Prop. 9: In case the following m = q i s prime, Mackenzie's r e l a t i o n s have form: a. (ft) ( 1 <_ s, t <_ q - 1 ) where i s principal JeY (T ) t s 1) Y T 2) T s 3) T g T 4) 1 {1,2,...,q-l} the permutation of sending has q - 1 ^ s elements i f s = q - 1 . I f 1 <_ s < q - 1 i f fq -j i T . j eT Without l o s s o f g e n e r a l i t y we may assume f(j,s,t) = 0 or 1 SL, f(j,£s,£t) = f(£j,s,t), t is t r i v i a l . To see 4 ) and hence 3 ) , f(q-j,8,i) = [ q i " ] - [£ia=il] 1) q {-§1 + } q =£!=!_(!_ q q/(s+l)j and q j s j = > s {x+n} = {x} { ( s + 1 ) q J}) [x+y] - [x] - [ y ] . we have Y ( T ) = {j | f (j , s, t) = 1} ± s = Y _•, (£) • T h i s g i v e s 1 ) and 2 ) . t (gtl)( l<_s,t<_q-l . s i n c e i t i s o f the form { j | f ( j , s , t ) = 1} = Y ( T _ ) • C o n v e r s e l y st also -1 t o k t """mod q has " ^ 2 ^ elements i f s ^ q - 1 S i n c e , f o r any for k = { j | f ( j , s , l ) = 1} . Proof: Also S f c £ s = q - 1, I n case 3) s e t {x} = x - [x] . Now -i t l . _ q { _Xs±I)l+ + ( i - {*!•}) . q = 1 - { ^ 1 } i f n e 2 . Therefore q ( s + 1 ) } + T h i s h o l d s because and {=f-} = 1 - {f-}; - 39 f<q-j,s,i> = 1 - 4 ! = - {-^} - _ ([iliyij _ e» [ILL]) = _ 1 f ( j , ,i), s so 4) i s done. • A s i m i l a r r e s u l t holds for m not prime (with i d e n t i c a l a l t h o u g h i t i s a l i t t l e more c o m p l i c a t e d to s t a t e . i d e a l c l a s s e s as r e p r e s e n t e d by primes P^ proof) I f we consider as above, what Prop. 9 e. says i s t h a t the r e l a t i o n s are a l l of the form p r i n c i p a l " where ^ except i n the t r i v i a l Now K . + a m v Prop. case • = 1 (j,m>=l J e. = 0 3 h a l f the 1 + . .. + a with 0 n time, for a l l j . reciprocity An E i s e n s t e i n p o l y n o m i a l i s one of the .X™" m-1 is J F i r s t we need some f a c t s about l o c a l e x t e n s i o n s . p-adic f i e l d E(X) = X ' h a l f the time and P. l e t us c o n s i d e r g l o b a l r o o t numbers f o r g l o b a l symbols. and e. = 1 3 II " ord ,(a.) > 1 K l — for T Fix a form 1 < i < m - 1 — — ord^Xag) = 1 . 10. ([CF]I§6 Th. 1 ) . irreducible. If r a m i f i e d and ord^ir = 1 . b) If L TT a) An E i s e n s t e i n p o l y n o m i a l i s i s a r o o t of E(X) i s t o t a l l y r a m i f i e d over then K L = K(TT) and ord is totally (ir) = 1, then JLi the minimal Prop. 11: unity with polynomial f o r ([ I ] IV §2.2). m TT i s E i s e n s t e i n and Suppose K contains prime to the c h a r a c t e r i s t i c of L = K(ir) , mth K . Let r o o t s of ord (TO = 1 . - 40 - Then t h e r e e x i s t two Kummer f i e l d s of exponent K(5ml,/'ATS~D\ IN ( f ) m over a n K( /rT) d K i s c o n t a i n e d i n t h e i r composite. f i e l d s i s unramified. Let K over such t h a t every Kummer f i e l d m -x) m The f i r s t The second i s t o t a l l y now be a number f i e l d K namely of exponent o f these ramified. containing mth • r o o t s of u n i t y . ft For b e K , d e f i n e the g l o b a l r e c i p r o c i t y character (b,x) = n ( b , x ) , t h e product taken over a l l primes, archimedean (x p p ones. i s any i d e l e here.) (b,-) by including (b,-) i s an i d e l e ft n(b,a) class character: P We w i l l choose (*) b F o r a l l p|m, Such a b such t h a t = 1 for a e K ([CF] Ex. 2.9). P to s a t i s f y a c e r t a i n condition: ord b = 1 . p .. i s found by f i r s t c h o o s i n g , f o r each o r d b(p) = 1 and i f p ' ^ p , p'|m, P Such a c h o i c e i s a v a i l a b l e by weak a p p r o x i m a t i o n . b = II b ( p ) . p.|m b satisfies Prop. 12: If b and w i l d l y ramified. c " iF o r any p m, Proof: — b = x , m so i f X m - b satisfies b (*) . (*) then f o r i s order r = r ord b = m ord x = > P P Kp(8) F o r any m b, Then s e t l e t B™ = b . K (B)/K P P * * m i n K /(K ) , p p m|r . 1 pIm ft p|m, b(p) e K o r d ,b(p) = 0 . P Hence i s totally for i f [K (B) : K ] = m, P P i s t o t a l l y ramified, i t i s wildly ramified. i s Eisenstein for K P But so t h e r e s u l t f o l l o w s from Prop. 10a) • - 41 - Prop. 13: For any b (not n e c e s s a r i l y s a t i s f y i n g (*)) a) If K (B)/K P P i s tame, b) If K (B)/K i s t o t a l l y and w i l d l y r a m i f i e d , p p (b,-) i s wildly Proof: a) has been done. so i s extension in that U (r) contained P i s contained in N *P F be the maximal p K (B) p p unramified By lemma 4, the l e a s t P * . . (K (B) p ^ ' p ) r such i s at least p + 1 = S- 1^^ HD G^(K (B),K ) For b ) , l e t g^ = o r d e r of Let K then ramified. (the r a m i f i c a t i o n g r o u p s ) . of (b,-) P > 1 • But (u£>) = U< p P p N by r ) P * lemma 3 a ) . Hence the l e a s t r such t h a t N (K (B) , ., K (B)/K p ) is p p (r) contained in U i s also P * N(K (g) ), > 1 . S i n c e the k e r n e l o f (b,-) K i s exactly Now P we have p suppose i s unramified, b satisfies 21 • p |fl((b,-) ) . p (*). If pjm p j b , then (b,-) K (B)/K i s . I f p|m (b,-) i s wildly P P P r a m i f i e d by Prop. 12 and 13. I f pjm, p|b, then we c l a i m K (B)/K P P b = u ir P P as i s t o t a l l y and tamely r a m i f i e d . with r by Prop. 11. u If P e IL, K and r 4 0 . For we may If r = 1, write we a r e done p r 4 1, then K (B) <= K ( / u IT , / T T P P P P P m M R _ 1 ) . But . By _ K P ( m /iT r P X ) Prop. 11, c - P P P K ( /u TT ) P P P i s contained K K (mZn~) hence i s t o t a l l y r a m i f i e d over i s t o t a l l y r a m i f i e d over K P . K P So K i n the composite of two f i e l d s t o t a l l y r a m i f i e d and i s t h e r e f o r e t o t a l l y r a m i f i e d over K P P (B) over - 42 - What has been done i s to i s o l a t e (b,-) i s unramified, Prop. 14: If r o o t number S(b) tame and = S, W((b,-)) e x a c t l y the p l a c e s where wild. i.e. p|b => p|m, then the global i s a r o o t of u n i t y . Proof: W((b,-)) = II W((b,-) ) P component (b,-) i s tame. which i s a r o o t of u n i t y i f no P • p Example: Let TT Let K = Q(E. = 1 - E, P p . ) and Consider o p = unique d i v i s o r (TT ,-) and (E. P a p p l i e s to the former and p f o r the l a t t e r , if ,-) of p^ . i s a Kummer p^-extension unramified w i l d l y ramified at characters b = p . ) P 0 s L = Q(£, n + outside p ^) = K( 0/E. ) , F r U and P totally 0 and Indeed the same h o l d s f o r ( s , t e Z) . ^ P 14 Hence the r o o t numbers of b o t h these are r o o t s of u n i t y . (1 - E Prop. K 0 P L in 0 • - 43 - §6. Behaviour of Gauss Sums With Respect to In t h i s s e c t i o n we c o n s i d e r subgroups Restriction ( u s u a l l y normal) of G a l o i s groups of l o c a l number f i e l d s and t r y to f i n d a relation between the Gauss sum of a c h a r a c t e r of the group and the Gauss sum of the restriction o f t h a t c h a r a c t e r to the subgroup. i s a theorem of Hasse-Davenport f i e l d with y e ft IF q elements and ([ 9 ] ) : N: IF . Then (ct*N) G q f a Theorem 6: of IF be the norm map. i s a c h a r a c t e r sum over IF ^, f o r any q ft q - G (a«N) = ( [ 9 ] §3.1). L/F Let q ( - G(a)) . f 7 let point be the f i n i t e y character Now W Let —> IF x q The s t a r t i n g • 7 be an u n r a m i f i e d e x t e n s i o n of l o c a l f i e l d s and w r i t e ft for a • N^/p" tame, so i s a w n e r e i s a c h a r a c t e r of a by lemma 3 a ) . Let p F If be the prime of a is F, P the J_i prime of y x = T r L . L/L We ^ L d n y = T^p.p L L/K ° 6 + n r F = L, n d T = a d G F (See § 4 ) . w i r d L K 6 we 1 + = Tr ° r d K K 6 T r K see t h a t — y e F . Finally, Cor. If n, then t h y Since L/F y . = d„ . K = Tr Since T a — — TL * F l a^(d^) = a(d^) ' i s tamely r a m i f i e d and T (a ) = ( - l ) P L N + 1 T p (a) .. ... L n F nr nr v ( d ^ p ) = Tr nr 1: with # L L L a T _ ( a ) = a (d )6 (a N) F l_i L Li p ^ ( ) (°0 i s u n r a m i f i e d we may , . l+ord 6 and by t r a n s i t i v i t y of d i f f e r e n t d = TT = L L TT = TT L r 1 + 0 r d TT (p (dp^p^). nr take l p have n . L . and (d^p ) = hr and we L/F have i s u n r a m i f i e d of degree • n - 44 - Cor. 2: If a,L and F a r e as i n Cor. 1 , W(a ) = ( - l ) n + 1 W(a) . n • L L e t us say t h a t L/F at most tame e x t e n s i o n Prop. 1 5 : x (a ) ^ p L Proof: If T L/F of i s t o t a l l y wildly ramified i f the maximal F itself. contained in L is F i s t o t a l l y w i l d l y r a m i f i e d and a i s tame, then ( ) • a p ([18] §2 lemma 2 ) . i s tame, we c l a i m a Since i s tame: L/F Let i s totally q = order r a m i f i e d and a of F . I f JLI X e Up - Up ^ automorphism of F x e U . Q F) , X = N L/p( ^ x a (x) 4 1 Hence T [L:F] x'' "^ = 1 and a Li I? T (a ) > x I—*- x^"^"' then, s i n c e (by t h e f a c t t h a t c h a r a c t e r i s t i c of some OJ(XQ) =f 1 and 1 a T i s an i s a power of the e u p ~ p"^ > U i s tame. f o r Now LI ^_a (x)g-(x) xeL L r 'v V / ^ l_a^(x)g-(x xeL V / [L:F]. ) , [L :FK . . [L:F] . ( s i n c e x t—> x i s an automorphism _ _ of L = F) , [L:F]. xeF 'v £_a(x)gp-(x) xeF ^ T (a) . • p Prop. 1 6 : and a Proof: Suppose i s tame. Case I : L/F Then a n i s totally n T ( a ) ^ T (a ) . P L p 11 T i s unramified: t o t a l l y r a m i f i e d , there and tamely r a m i f i e d o f degree exists Let y e Up x e U such t h a t . Since L/F is y = x mod U^"^ . - 45 - Therefore and y = N ^ C y ) = N ^ ( x ) mod I L ^ . 11 L a (U ) = 1 . L and a a (x) 4 1 and 11 n - 11^"^ J_i ) = a (d ) £ 4 Let a(x ) x e U J_i x^Ca = 1 n L P i s unramified: 1 « (x) = a (y) x (a ) a r e both r o o t s of u n i t y . 1? LI L Case I I : Then x (a ) So p so 0 f o r some x e U-.-U^^ . r r i s r a m i f i e d . We have a J_I JLJ a(x )i(j (d "^"x), s i n c e a s e t or r e p r e s e n t - U xeUp/Up^ a t i v e s of Tr T U.f^ L (d/St) and in U may be taken from L = Tr„(x T r , ^ ( d 7 ) ) ([CF]I§5 Th. 2 i i i ) ) ord (6 ^ ) L L p - 1) . l+ord 6 ... l + o r d 6 L L L = that i f = e(L/F) T , T T V = ^L + o r / d 6 T L x ( a ) = a(d ) L = d T L n+n ord ,6„ T r r \ r F * for x e F . £ p ^ u F a (x)^ (d F / u F i s an automorphism of 1 Theorem 7: Let a K L/K and l e t g(L/K) and 1 p Hence nx) . Since (q,n) = 1 , x K n x } TT /U^"^ h i Let r 1 r = n - 1 . i s tame, p r i s totally ord^(6^^) T r ^ d ^ x ) = Tr (x T r ^ C d " ) ) = Tr^nxd" ) \b ( d "*"x) = ^ ( d - ^ n x ) p L/F F ^F T Since and ,_+ord 6„ L = F Therefore L/F Now We have L 1 L T T ^ = Tr^ . r for x e F . 1 T tamely r a m i f i e d , we may take (Recall U and so x_,(a ) a, x (a ) . 1 T P L P rj be an a t most tame c h a r a c t e r of t h e p - a d i c field be any f i n i t e normal e x t e n s i o n w i t h G a l o i s group be t h e index of the w i l d r a m i f i c a t i o n group of G G . in its - 46 i n e r t i a group. Proof: field L/K Let has f (L/K) = a filtration G . Then r a m i f i e d and E/K g(L/K) unity. If a Prop. 15, E F/E a n < j T_(OL.) P L <v, x ( a where F [E:K] and a T (a^^^) p ( L / K ) ) f ( L / fixed field of = f(L/K) i s normal, t o t a l l y and If g i s the i s the f i x e d and i s normal, u n r a m i f i e d . L t f c O L £ F £ E £ K [F:E] = g(L/K) i s totally wildly ramified, of . of the w i l d r a m i f i c a t i o n group and the i n e r t i a group of a [L:K] - . L/F tamely i s unramified so i s a r e both r o o t s of i s tame, the r e s u l t f o l l o w s by s u c c e s s i v e a p p l i c a t i o n 16 and Theorem 6. • Corollary: x e R-^(K), Let x r e s t r i c t i o n of tt to x c Q L Proof: Since X Res and x T(X) a a . ab tt K K 0 > a L* — ± i . l o c a l A r t i n maps. Any T ( X (see > the ^ ^ ) F ^ L / / R ^ may consider at) tt respectively. L [AT] Ch. 14 §5) It i s that ab K nf i s induced The 6 be p and * R(K) ^ remains i n v a r i a n t under l i f t i n g , we K imbedded i n x Res Ir a p r o p e r t y of c l a s s f o r m a t i o n s commutes, where Let x_,(Res x) Then K c h a r a c t e r s of s most tame. t b R by e s X : s i i n c l u s i o n and 6 and r e s u l t f o l l o w s on r e c a l l i n g how 'K R-^(K) are is . • attempt to g e n e r a l i z e t h i s r e s u l t i n v o l v e l o o k i n g a t the r e s t r i c t i o n of an to R(K) induced d i r e c t l y must representation. For K ) - 47 - t h i s we need the s o - c a l l e d a t i o n theory: Let H and subgroup theorem of elementary r e p r e s e n t T be subgroups of G and a a one- (s) d i m e n s i o n a l c h a r a c t e r of H . For s e G, let a be the char n T d e f i n e d by a^ \t) = a ( s t s "*") . W r i t e T a c t e r of sHs Res„ f o r r e s t r i c t i o n of r e p r e s e n t a t i o n s from G to T . The (j 1 S f o l l o w i n g i s a s p e c i a l case of the subgroup Theorem 8: ([S2]§7.4 Prop. 1 5 ) . T T (s) Res p = # Ind _^ (a ) s sHs . nT representatives {s} p If = theorem. IndS* then where the sum i s taken over a s e t of of the double c o s e t s H\G/T . • Now suppose Lemma 8 : Proof: T(a Let H i s a b e l i a n and ) = x(Res ri T i s normal i n a) . K(resp.K' resp.K" resp.K"') H ( r e s p . sHs \ be the f i x e d f i e l d 1 \ L/K"' L/K. L/K', are a l l abelian. and and We have since sHs nT = s(HnT)s , _ 1 By c l a s s f i e l d L/K" K" K' = sK K of r e s p . sHs n T , r e s p . HnT). Note t h a t / K"' G = Gal(L,F) K' theory _ 1 K" = sK"' . - 48 - Gal(L,K) K - 7 Yi 1 Y > Gal(L,K') - s G a l ( L , K ) s K' 4 K" commutes, where of a Gal(L,K") Y-^(t) = s t ^ s a n -1 (s) Y ( x ) = sx . d c l a s s f o r m a t i o n s - [AT] Ch. 14 §5 Th. 6 ) . (s) that a Since T N „^ ,(x)) K c o n s i d e r e d as l o c a l f i e l d K i s normal and L/F i s normal, I f T i s a s e t of r e p r e s e n t a t i v e s of i s a s e t of r e p r e s e n t a t i v e s of b) From t h i s i t i s c l e a r -1 (x) = a ( s a) (This i s a property o sHs "*"nT i s an imbedding of i s an imbedding of K"' into K" in in HnT sHs (Q ) ^Pgsep in characters. H, sTs -1 . over F i f f s "*"os (Q ) over F . P sep for a l l x e K ' " . 0 c) ord „sx = ord „,x d) A l l r a m i f i c a t i o n subgroups of K K sHs "''nT and HnT a r e s- conjugate and i n p a r t i c u l a r of the same o r d e r , (s) e) a By a) Tr , K for are I / F s ^.^.(x) = N „ K (x) = sTr „ xeK". y i s tame, s i n c e K p r e s e r v e s the u n i t s ( s x) (s~ x) = Tr „, 1 I / F By c) and d) . X I / K K / F (s _ 1 By filtration. b) x ) => ip „(x) = ^ „ , ( s K K o r d ( 6 ^ „ ) = ord (<5 ^ „,) L L K L L K 1 x) (differents determined by the o r d e r s of r a m i f i c a t i o n groups; see e.g. [CF]I §9 Prop. 4) and so by e) we may take d „ , = s "'"d^.,, . K" K We have - 49 - r(a^ S ; ) = a (d „) w K £_a(s \, xeK" / K (x))^,(<i „x) R = a ( s N „ , , (d „)) I _ a ( \ n t / ( s x ) ) i J ; xeK' 1 1 K / K K K K l I I (s "'"(d^x)) a(N „, , (d „,)) I a ( N „ , / ( ) ) ^ . M (d ?i,x) xeK"' x K K T(a-N , K x(Res Lemma 9: K discriminant of p = Ind^a . L/K, Then since We have • L e t G = Gal(E,K) with fixed f i e l d a G 2 x(Ind l) = for D H and TT N K / Since / Q 0 ( L/K^ D p t h e A r t i n conductor be t h e G G —1 (a-1)) = x (p) x ( I n d l ) ri H However, f o r any c h a r a c t e r H P L e t j_,/K 0 G (j$(Ind l)) = H be an R ([14] Prop. 4.1). u L . H (D^, )^x(a) . R / x(a) =' x ( a - l ) = x(Ind N(f}(0))det. (-1) and l e t a one-dimensional c h a r a c t e r of x(p) ^ N dim(a-1) = 0 . R R P Proof: k a) ri G k ^ ) It ( c f . Lemma 2 ) . a b e l i a n subgroup of K (see §1C)) 0, x(0)x(0) = Ind^l ri b y t h has r e a l e character, induction formula 0 . • Indeed the q u o t i e n t for i s c l e a r l y a 4 t h r o o t of u n i t y . Let N stand a b s o l u t e norm as u s u a l . Prop. 17: Q L e t p = Ind a H be normal i n G . L e t with H a b e l i a n and a tame. Let T - 50 - g = index of w i l d r a m i f i c a t i o n group of f = r e l a t i v e degree o f L = fixed f i e l d of HnT K = fixed f i e l d of T . T f R e s ^ p ) ^ N(n ,„) P G L/K Proof: in i t s inertia H\G/T m / 2 x (a ) g . f m Q p T T Res p = $ Ind s sHs By Theorem 8, (. s} (a ) where t h e sum nT i s taken over a s e t o f r e p r e s e n t a t i v e s o f H\G/T . N(D ^ s L / K = N(D ) L / R with ) m / 2 N(D L / K group H/HnT m = number of double c o s e t s Then H/HnT ) and so x (ResJp) p n x ( R e s | J a ) <\- N ( D nT p L / K ) m / 2 (a ) g T We may r e p l a c e n[x (a(s))N(D p f m ) ] % L / K by Theorem 7 and lemmas 8 and 9. • Of c o u r s e Prop. 17 and Brauer i n d u c t i o n g i v e us some k i n d o f statement r e l a t i n g x(Res p) to x(a)'s induced r e p r e s e n t a t i o n s a r e components of induction i s n o t simply r e l a t e d to f i n d a r e l a t i o n of t h e form of) x(p)", difficulty unless g = 1 "x(Res a result whose However, s i n c e p) ^ (some simple f u n c t i o n ( f o r the v a r i o u s p g ' s ) . Another be a r e p r e s e n t a t i o n of an (Prop. 17 a p p l i e s i n t h i s case f o r H w i l l be a t most tame and so w i l l yields p . a's t o p r o d u c t s i t would be d i f f i c u l t i s t h e requirement t h a t at most tame G a l o i s group. f o r various a .) However a d i f f e r e n t approach i n the u n r a m i f i e d case w i t h o u t e x p l i c i t use of Brauer - 51 - induction. The requirement of tameness can be seen to come from a different direction. Definition: K Let a number f i e l d . product II T ( X p ) , p the r e s t r i c t i o n of x We w i l l take t h i s up i n §7. he the c h a r a c t e r of a r e p r e s e n t a t i o n of The g l o b a l G a l o i s Gauss sum taken over a l l f i n i t e primes. x to Q, . T(X) 0. K i s the Here, xp is - 52 §7. The L o c a l l y Free Class This section i s a brief Group, R e s o l v e n t s , and us to i n f e r constructions. from i n f o r m a t i o n Gauss sums of c h a r a c t e r s of i t s Galois some of t h i s m a t e r i a l These theorems group, and to prove a g e n e r a l i z a t i o n to the g l o b a l , n o n - a b e l i a n c a s e . Let be a finite, CL, N throughout. i s l o c a l l y f r e e rank n for a l l 0*' N,p if of of number T = Gal(N,K) . Let We fields. We say 0„ (r) K,p i s i s o m o r p h i c to 17 n p . Theorem 8 . free at most tame e x t e n s i o n and vice-versa. 1, T h i s assumption w i l l h o l d integer about the r e s o l v e n t s Theorem 6 Cor. N/K the f a c t s about G a l o i s module s t r u c t u r e of the r i n g of a tame e x t e n s i o n w i l l use Gauss Sums summary of some work of F r o h l i c h on r e l a t i o n s h i p between these t h r e e allow - (E. N o e t h e r ) . 0 (rank 1) (T) N/K 0^ i s a t most tame i f f is a locally module. • K Let finitely K 0 (0 K (T)) be the G r o t h e n d i e c k group of the 0 generated l o c a l l y f r e e (T) modules. This category of i s the free 3) a b e l i a n group on modules, mod stable the isomorphism c l a s s e s of l o c a l l y f r e e subgroup generated by stands f o r s t a b l e i s o c l a s s of KQ(o (r)) K M). The (M#N) - (M) notion - (N) 0 (T) ((M) of rank extends to . 3) M 3') i s s t a b l y isomorphic to M <& M - N <& M 0N, demotes p 1 N i f there i s an such that . sefv,i- loccxV comfleVio^- 0 , p K ^ "°"^ e s co^pk+.on. - 53 - Definition: The l o c a l l y f r e e c l a s s group, subgroup of K (0 (T)) c o n s i s t i n g U K C£(0 ( T ) ) i s the of those elements l o c a l l y free of rank 0 . • The l o c a l l y f r e e c l a s s group i s s i m i l a r l y d e f i n e d f o r any order i n a f i n i t e d i m e n s i o n a l semi-simple K - a l g e b r a 0 K order i n A we mean a s u b r i n g of generated t o r s i o n - f r e e v e c t o r space maximal map Z A . A, A . 0 By an which i s a f i n i t e l y 0 -module c o n t a i n i n g K a basis of the K- In p a r t i c u l a r we may c o n s i d e r , f o r K = Q, o r d e r i n Q(r) which we w i l l c a l l C£(Z(r)) —> C£(M) given b y (M) —> (M ® M . the There i s a Q(T)) . The k e r n e l z ( r ) of t h i s map i s c a l l e d D(Z(T)) . L e t us w r i t e ( ) u rkM c l a s s of M - 0 Theorem 9: R ([3] (r) C£(0 (r)) in Th. 3 ) . D ( Z ( D ) , when e n K(r) . R ( < V z ( r ) f o r the M K = Q . • L e t us s k e t c h a proof of t h i s theorem as i t seems t o r e q u i r e a l l the concepts and major theorems of the t h e o r y . in C£(0 (r)) R i n t e g r a l basis i s f r e e over Let measures how f a r away over 0 K 0 R (r) . 0__ N 0 The l o c a t i o n o f i s from having a normal N has such a b a s i s over i . e . (° )Q ( r ) K n = 1 ' 4) The l o c a l l y f r e e c l a s s group has an a l t e r n a t i v e R„ 1 be the r i n g of v i r t u a l c h a r a c t e r s of ^This t h a t doesn't h o l d <Va(D " for arbitrary 1 • 0 N T . 0 K 17 i f f 0. N T description. This i s an Q K N/K tame, a l t h o u g h i t i s c o n j e c t u r e d - 54 module v i a X^CY) X(Y) = a l g e b r a i c numbers. r L, L , and a l l values as an E - r e p r e s e n t a t i o n . write f o r the i d e l e s of II 0 p ' K represent- E and i t s be a " d e p o s i t o r y " f o r v a r i o u s f u n c t i o n s on For any number f i e l d for are f o r a l l , an a b s o l u t e l y Indeed l e t us suppose t h a t any can be c o n s i d e r e d a d e l e group w i l l J(L) . X(Y) since E, l a r g e enough to c o n t a i n of a l l c h a r a c t e r s of T m e ft , y e E, L e t us f i x , once and normal number f i e l d a t i o n of for W - L, the product Ad(L) U(L) R^ f o r the a d e l e s of f o r the u n i t i d e l e s taken over a l l primes. . L, and Let p ft U(0„.(r)) = n 0„ p ' K elements of any If to a T: (T) T —»- GL(E) det X det A = det T(A) X a map GL(E x T . invertible in T: K(T) Ad(K)(T) = Ad(K) into c h a r a c t e r of i s a representation, homomorphism r e p r e s e n t a t i o n of Ad(K)) always denotes the invertible ring). K-algebra over (where "*" p Ad(E) ® K(T), K i s a mapping from A U(0 d e t : U ( O ( D —> Hom ir (the group r i n g of T Let x be and we may In p a r t i c u l a r det Ad(K)(T) . (T)) Now K- to r J(E) . by sending we may A the define is invertible in (R ,J(E)) 0 extended or even to a ® Ad(K)) = GL(Ad(E)) . K A e Ad(K)(T) . if can be —>• GL(E) extends along w i t h for T X A is So define to K X i—> det A . Definition: C£(0K(r)) (det ,(A) i s d e f i n e d as (det A ) ( d e t , A ) - 1 (-Theorem): = Hom^ (R ,J(E))/Hom r K (Rp,E*) .• det U ( 0 R ( r ) ) fi K . .) V - 55 For a proof ([ 4 ] t h a t t h i s c o i n c i d e s w i t h the e a r l i e r d e f i n i t i o n we must d e f i n e g l o b a l and a e N generating adelic resolvents. a normal b a s i s Normal B a s i s Theorem) l (a|x) define {a } of v = det( l X = character resolvent. of the r e p r e s e n t a t i o n T For a d e l i c r e s o l v e n t s , l e t by Ad(K)(T) —> GL(Ad(E)) = ( a l x ) N / K = det( (a|x) I a T T(Y) _ 1 (Hilbert i s the a e Ad(N) and Rj. e — l o c a l normal i n t e g r a l b a s i s everywhere, i . e . to a map N/K (a|x) . x For Y 1 a T(Y) ) yeT T see Appendix I ) . Now and - where global generating aA^ = A^ . a Extend d e f i n e the a d e l i c r e s o l v e n t ) e Ad(E) . If a e N yeT then t h i s i s the same as the g l o b a l r e s o l v e n t . see (a|x) t h a t r e s o l v e n t s are i n v e r t i b l e elements e J(E) Next we f e Horn ([4] Prop. 1.2 Cor. . If K _£ k i s a map tt^, in from Horn (R define ,J(E)) and {a} (\/ k f Mx) e E and function i s a set of -1 = n f(x ) represent. \ / k a to Horn (R K ,J(E)) . In p a r t i c u l a r k the f u n c t i o n s x ( I x) the c h o i c e {a} changes the d e f i n i t i o n by a of to 1). "K a t i v e s of (a|x) i.e. need the n o t i o n of the norm of a (R_,J(E)) I t i s not hard a a n d X *~*" ( | x ) a have norms. Changing homomorphism X t~*• ( r o o t of u n i t y ) . The next n o t i o n i s t h a t of a f a m i l y of i n v a r i a n t s . s p e a k i n g , these are maps such t h a t f e C£(0 K. b(x) (T)) even c r i t i c a l : b from R^, to the i d e a l group of i s generated by the image . Roughly f(x) E f o r some f u n c t i o n However, the f o l l o w i n g m o d i f i c a t i o n i s n e c e s s a r y , Let J(E,r) be the subgroup of i d e l e s of E whose - 56 - components a t a l l i n f i n i t e primes of E r a t i o n a l prime d i v i d i n g t h e order o f T, = U(E) n J ( E , D U(E,r) Hom . J(E,D and a t a l l d i v i s o r s of anyare 1 . > J(E) induces (R ,J(E,T)) — y Hom_ (R ,J(E)). —*• C£(.0 (D) t h i s i s s u r j e c t i v e and one can show t h a t det U(0 (T)) . K Horn U ft By a p p r o x i m a t i o n , (R , U ( E , r ) ) maps i n t o On t h e other hand we have an isomorphism Horn (R ,J(E,T))/Horn (R ,U(E,T)) - I "K K K , r of . V 0 Let homomorphisms b from R r where . I °K,r i s the set t o t h e f r a c t i o n a l i d e a l s of E such t h a t b(x) i) has numerator and denominator prime t o the o r d e r of i i ) b(x) i s a f r a c t i o n a l i d e a l of T and 0 , . MX) (K(x) of always denotes the l e a s t f i e l d c o n t a i n i n g X )• T h i s isomorphism i s g i v e n by f r—»• b^ f r a c t i o n a l i d e a l generated by the image of get a s u r j e c t i o n I —>->• C£(0 (T)) °K,r k e r n e l , an isomorphism f and i f K and a l l where b^(x) i n J(E,T) . H values i s the So we denotes i t s °K,r I /H ~ C£(0 (T)) . A f a m i l y of K,T K,T • i n v a r i a n t s f o r an element (M) o f C£(0 (T)) i s by d e f i n i t i o n a K r e p r e s e n t a t i v e f o r (M) i n I Prop. 18. Let U) a) £ K ( G a l o i s a c t i o n on Gauss sums and r e s o l v e n t s , v e r . : ft^ —>• ft^ be t h e t r a n s f e r map. K/ Q 1} K ft , vi. n 0) x(x -1 ) = x(x)det ( v e r ^ ( u ) ) . K Then f o r [ 4 ] , [14]). x e R,,, 1 - 57 - b) If a e N satisfies -1 (a|xW ) i) aK(T). = N = ( a | x ) d e t co A. -1 ii) N (a|xU ) = N W K / q (a| )det (ver X ( a | ) = (a|x)det X iii) 0 K / Q for X e A X If aA =A K x (o )) K M ) K(D* N -1 Calx" i) ii) N K / q ) = (oi|x)det a) W A. (a|xW V = N ^(a| )det .(ver K / X x K / q (a>)) • (0 : x ) Now d e f i n e the r e s o l v e n t module (a|x) module generated by a l l as the -'Q(^) T h i s i s the 0 , N ^(a|x) Define r e s o l v e n t modules a r e a c t u a l l y f r a c t i o n a l i d e a l s . (0Tr:x) w i t h r e s o l v e n t s i s g i v e n by: K N K / Q ( ° K : x p ) = N K / q ( a | x p ) A / = (a x) p a A K = K/Q.(°K X) : These The c o n n e c t i o n 0„, and N K(x),p P % X ) , P ' N a e 0^ . with R N KAX-> a e 0^ . with module generated by < • K ^ • Theorem 10 ( [ 3 ] Th. 8 ) . i) that (N Let a e N aA^ = A^ . K / q f)( ) = N X The c l a s s o f W i ii) n C ^ With Let ^(a|x)N K / f r such t h a t > ) a f ( a ) = (a|x)(a|x) K / % (a| )- . 1 X i n C£(0^(r)) K is aK(T) = N . (0 ) N z ( r ) is 1 Then (0. ). N 0 L e t a e Ad(N) such whence f e Hom^(R ,J(E)) r . T The c l a s s of K ( r ) . as above, assume t h a t f o r a l l prime d i v i s o r s p - 58 - of order Then r , we have (r) = 0 . L e t b ( ) = (C> : ) ( a | x ) ,P JN , y Ik. i s a f a m i l y of i n v a r i a n t s f o r (0 ) and K(D (b(x)} aO X 1 X N {N J}(x)} i s a f a m i l y of i n v a r i a n t s f o r w Theorem 11: i) x X K / q (a|x) x(x)0 ii) A 1 / T n , . ([ 3 ] Th. 9 ) . L e t a e Ad(N) i-> T ( ) N (0 )_ such t h a t aA^ = i s i n Hom _ 1 = N K / Q (0 : ) K Then the map (R ,U(E)) fi q ( x ) . r • X q That the map i n Theorem 11 i s i n Horn (R ,J(E)) i s immediate from Prop. 18. The depth o f t h e theorem i s i n t h e ( f a c t o r i z a t i o n ) statement t h a t i t lands i n U(E) . the f a m i l y of i n v a r i a n t s the f a m i l y {x(x)N y Theorem 12 ( [ 3 ] Horn (a|x) R (Rr-oE ) . Th. 1 ) . If a From theorem 11 i i ) we see t h a t b ( x ) of Theorem 10 may be r e p l a c e d by ^} • Hence The map x T (x)N ^(a | ) R i s chosen so t h a t i f _ is in 1 X p T, d i v i d e s order \ a0 R (r) = 0 N p , then the i d e a l s a f a m i l y of i n v a r i a n t s f o r b(x) = (x( )N X R / ^(a| ) X f e Horn I (R^E) . ) define (0 ) . . . To prove Theorem 9, we need a c h a r a c t e r i z a t i o n of a subgroup o f _ 1 + L e t Horn such t h a t i f (R_,E x * ) D(Z'(r)) denote t h e set of i s symplectic (i.e. x is real \ valued and the r e p r e s e n t a t i o n of x as preserves a skew-symmetric - 59 f(x) b i l i n e a r form), i s t o t a l l y p o s i t i v e at a l l i n f i n i t e i n t h i s d e f i n i t i o n we must be r e a l and Theorem 13: - i n s i s t t h a t even i f p o s i t i v e at (Frohlich). p p . P(Z(r)) Let Now the map to see t h a t x H - y D(2(D) Then r ^iq^Z(T) ( ( x ) N ^ ^ ( a | x') c o n s i s t of those f a m i l i e s 6 D » P^/H^ ^ ^ ^ ' ^) T c h a r a c t e r s to elements of f(x) i s complex, f(x) of i n v a r i a n t s generated by the images of f u n c t i o n s f e Horn* (R ,E*) . places; w . e n e e of Theorem 12 P(Z(T)) i.e. for d o n l y show t h a t takes symplectic to f r a c t i o n a l i d e a l s generated by t o t a l l y p o s i t i v e elements. x then ' T ( ) N ^ (a|x) i s t o t a l l y r e a l , e i t h e r t o t a l l y p o s i t i v e or t o t a l l y n e g a t i v e , with Theorem 14 ( [ 3 ] Th. 2). If i t s s i g n independent of a sign(x(x)N W( ) Remark: K / ( ^(a|x) In g e n e r a l c h a r a c t e r has = i s symplectic, and given X R by: X W(x+x) = • d e (-1) X t t r i v i a l determinant, W(x) • Since any symplectic 2 = 1 for X symplectic. From Theorem 14, Theorem 9 i s complete (modulo a l l the of t h i s s e c t i o n ! ) . One a p p l i c a t i o n of t h i s theory S t i c k e l b e r g e r ' s theorem without S p e i s e r Theorem). See field r e f e r e n c e to Gaussian sums. [5 ] . of mth proofs i s a proof theorem i s seen to r e s u l t from the e x i s t e n c e of a normal b a s i s f o r the c y c l o t o m i c 1 r o o t s of u n i t y of The integral (Hilbert- - 60 - In order to prove the " g l o b a l Hasse-Davenport" we w i l l draw o n l y on Theorem 11 i ) and Prop. 18 t o g e t h e r w i t h the f o l l o w i n g theorem on r e s t r i c t i o n o f r e s o l v e n t s : Theorem 15 ([ 3 ] Th. 11). with fixed f i e l d 3 e A L exists Res x A and assume aA^(V) such t h a t N Let X e A^(T) be a normal subgroup of L/K = A^ i s unramified. and such t h a t BA^A) = A^ r, Let . a e A^, Then there ( B | R e s x ) ^ ^ = (a | x ) ^ ^ d e t ^ X , where R i s the r e s t r i c t i o n of x e R t A . o r • Theorem 16: L and assume l—h x Let xfResv) A A T be a normal subgroup of L/K i s u n r a m i f i e d of degree i s in Horn * (R ,0 ), where with fixed n . Res x field Then the map i s the r e s t r i c t i o n T(X) of x to Proof: A . (1) In p a r t i c u l a r The map x ^ x(Resx)0 ^ K T(x)T(det ) n. = x(x) ° ( ) K is in _ 1 Horn if \ e R with (R , J ( E ) ) : \ X From Prop. 18 a ) , • X det = 1, then X T (x) X is in Horn (R ,J(E)) . r % U (One checks det = (det ) ) . W There i s a X e A^(T) v-f- x ( R e s x ) T ( x ) ~ N ^ ( a | d e t ) N Horn (R , U ( E ) ) : R e s ( x ) = (Resx)^ is in n K / U Horn det - w n X x X x (2) (1) f o l l o w s on a p p l y i n g t h i s t o 1 Let x a, B and such t h a t ^(a|det )" N ^(det X)" 1 L / ( X x x be as i n Theorem 15. and so by Theorem 11, (R ,U(E)) . L / L e t us w r i t e x l — > 1 is in Now x ( R e s x ) N ^ ( B |Resx) L 1 - 61 - T(Res )N ^(3|Res ) X N L/ = T(Res ) 1 X ,-(det A) W<i X -1 by Theorem 15. x T n /Q[ / N K X L ( ( A K I*^ I d the map t } } ]N x L/Q. (Norm i s indeed t r a n s i t i v e ) . c o n s i d e r the e x p r e s s i o n i n square b r a c k e t s . Prop. 18 c i ) , e X ^ I (A d e t v> 1 X L e t us J u s t as i n ( 1 ) , or by (ct|x)(ot|det ) ^ is in X Horn (R , J ( E ) ) £ Horn (R , J ( E ) ) £ Horn ^ K f e Rom ( R , J ( E ) ) , N Q (R , J ( E ) ) f ( x ) = f(x) Now f o r any and so n r . L L / R K. N L / K (a| ) X X V is _ 1 (a|det )) x x(Res )N X in Horn \ Hom^ K / q = (a | ) ~ ( a | d e t ^ n (a | ) ~ \ / ^ a| d e t ^ N X (R ,U(E)) Q. . r (R JU(E)) . 1 1 X x However L / q Therefore (a |d e t ^ ^ ^ ( d e t ^ ) " ^/q^ aI x)" (x) U T by Theorem 11 a g a i n . l s i X n M u l t i p l y i n g these l a s t two functions yields (2). (3) x(Resx) f ( d e t x The map \ Now (R , U ( E ) ) : T(Res d e t ) A T h i s f o l l o w s from (2) on r e p l a c i n g X by x - det let X f p = [ L ^ K ] = [L„:K ] . P p P p -f ) ) ) x (det p P is in 1 x (Res (det X n ~— T(X) Horn ) P i s r a m i f i e d and By l o c a l theory, f +1 P = e (x) where e (x) = (-1) P if p 1 if x P i s unramified. are e x p l a i n e d by the f o l l o w i n g diagram: The v a r i o u s symbols - 62 det * K P Artin „ , „ ab > Gal(NT,,K ) P p AT N (inclusion)* Res„(det ) P X Artin ab •y Gal(N ,L .) v * p Now T (Res det ) = II of e(x) X X there are = n e ( ) n . X P P|p We have T ( and ) = n x (det P ) , R P s i n c e by n e s X } e p . x(x) W = £ ) P normality (x) _ 1 Set e Horn (R T(X) e(x ) X X dividing P However i t i s obvious t h a t x(Resx) x(det X P L ^ X and x (det P P ^ r j primes i n )) P x (Res (det )) = II e (x) p P|p II P|p A x (Res (det P P|P X But II p (x) U(E)) . \ = £ (x) » a n W d since Q(x)> are a l g e b r a i c integers i n the theorem i s proved. • Corollary. X N Under the c o n d i t i o n s of the theorem, the map (a| ) N n K / Q X L / ( ^(B|Resx) is in 1 Hom fi (R ,U(E)) . p x(Resx)N ^(B|Res ) L/ Proof: By Theorem x 11, -1 X is i n l — T(x) n N K / ( ^(a|x)~ n Horn^ ( R , U ( E ) ) r • Remark: The G a l o i s module c o m p a t i b i l i t y of Theorem v e r i f i e d without r e f e r e n c e to Theorem x(Resx ') a Kx") T ( R e s x ) x( ) X a i d e t Resx ( v e r det (ver x K / Q L/Q ( a i ) ) C (a))) 11. For we Let 16 i s e a s i l y have \b = det and - 63 - CJ / N = ver ^(w) k K /T . Since to t h e n t h power is really d e t Re8 whence ( v e r X 0 K H ^ ^ . Z ) —»- H-^Q^Z) L/K ( u K ) ) =^ T(Resv) Y \—> — — (or even Coniecture: L R ( [ S I ] V I I I §2 Prop. 4 and n o t e t h i s sequence T(X) L/K ver , , (xncl) . , ^ab L/K a b * _ab . .. ft • ft >- fi i s r a i s i n g H^ft^Z)) M v e r ^ ) ) = * • ( i n d ) * • v e r ^ C i ^ ) = *(o * (R„,E ) i s i n Hoiri we have w i t h no c o n d i t i o n s on Q N/K) . ^- Under the c o n d i t i o n s o f Theorem 16, r \ is a n T(X) r o o t of u n i t y . Indeed i t i s p r o b a b l y a f o u r t h r o o t of u n i t y . - 64 - Bibliography Books [AT] A r t i n , E. and N.Y., [CF] Tate, J . C l a s s F i e l d Theory, W.A. 1967. C a s s e l s , J.W.S. and F r o h l i c h , A., Academic P r e s s , London, N.Y., [CR] C u r t i s , C.W. ed. and R e i n e r , I . F r o h l i c h , A., ed. A l g e b r a i c Number Theory, 1967. R e p r e s e n t a t i o n Theory of Groups and A s s o c i a t i v e A l g e b r a s , Wiley [F] Benjamin, I n t e r s c i e n c e , N.Y., Hasse, H. 1962. A l g e b r a i c Number F i e l d s , L f u n c t i o n s and G a l o i s P r o p e r t i e s , Academic P r e s s , London, ¥.Y., [H] Finite 1977. Z a h l e n t h e o r i e I I , A u f l . B e r l i n Akademie V e r l a g , 1963. [I[ Iyanga, S., ed. The Theory of Numbers, North H o l l a n d L i b r a r y #8, Amsterdam, N.Y., 1975. [L] Lang, S. [SI] S e r r e , J.-P. Corps Locaux, Hermann, P a r i s , [S2] S e r r e , J.-P. Representations [W] Mathematical A l g e b r a i c Number Theory, Addison Wesley, N.Y., 1970. 1962. L i n d a r e s des Groupes F i n i s , Hermann, 1971. W e i l , A. B a s i c Number Theory, S p r i n g e r V e r l a g , B e r l i n , Paris, N.Y., 1974. Papers, [I] L e c t u r e Notes D e l i g n e , P. Les Constantes des Equations F u n c t i o n e l l e s des F u n c t i o n s L, S p r i n g e r L e c t u r e Notes #349, p. 501-594, 1974. - 65 - [2] Dwork, B. On the A r t i n Root Number, Amer. J . Math. #78, p. 444-472, 1956. [3] F r o h l i c h , A. G a l o i s Module S t r u c t u r e , i n [ F ] , p. 133-192. [4] F r o h l i c h , A. A r i t h m e t i c and G a l o i s Module S t r u c t u r e f o r Tame E x t e n s i o n s , J . Reine und Angew. Math. #286/287, p. 380-440, 1976. [5] F r o h l i c h , A. S t i c k e l b e r g e r Without Gauss Sums, i n [ F ] , p. 589- 608. [6] F r o h l i c h , A. L o c a l l y F r e e Modules Over A r i t h m e t i c Orders, J . Reine und Angew. Math. #274/275, p. 112-124, 1975. [7] G e l b a r t , S. Automorphic Forms and A r t i n ' s C o n j e c t u r e ( i n I n t e r n a t i o n a l Summer School on Modular Functions-, Bonn, 1976) S p r i n g e r L e c t u r e Notes #627, p. 243-276. [8] Godement, R. and J a c q u e t , H. Zeta F u n c t i o n s of Simple Algebras, S p r i n g e r L e c t u r e Notes #260, 1972. [9] Hasse, H. and Davenport, H. D i e N u l l s t e l l e n der Kongruenzzeta- f u n k t i o n e n i n Gewissen Z y k l i s c h e n F a l l e n , J . Reine und Angew. Math. #172, p. 151-182, 1934. [10] J a c q u e t , H. and Langlands, R.P. S p r i n g e r L e c t u r e Notes #114, [11] Automorphic Forms on GL(2), I , 1970. Jacquet, H. Automorphic Forms on GL(2), I I , Springer Lecture Notes #278, 1972. [12] L e o p o l d t , H.W. Zur A r i t h m e t i k i n Abelschen Zahlkorpern, J . Reine und Angew. Math. #209, p. 54-71, 1962. [13] Mackenzie, R. C l a s s Group R e l a t i o n s i n Cyclotomic J . Math. #74, p. 759-763, 1952. Fields, Am. - 66 - [14] Martinet, J . C h a r a c t e r Theory and A r t i n L F u n c t i o n s , i n [ F ] , p. 1-88. [15] S t i c k e l b e r g e r , L. Ueber e i n e V e r a l l g e m e i n e r u n g der K r e i s t h e i l u n g , Math. Ann. #37, p. 321-367, 1890. [16] S e r r e , J.-P. L o c a l C l a s s F i e l d Theory [17] Tate, J . i n [ C F ] , p. 128-161. F o u r i e r A n a l y s i s i n Number F i e l d s and Hecke's Z e t a - f u n c t i o n s , i n [ C F ] , p. 305-347. [18] Tate, J . L o c a l C o n s t a n t s , i n [ F ] , p. 89-132. [19] W e i l , A. La C y c l o t o m i e J a d i s e t Naguere,Seminaire Bourbaki #452, 1974. [20] W e i l , A. J a c o b i Sums as " G r o s s e n c h a r a k t e r e " , T r a n s . Am. Math. Soc. #73, p. 487-495, 1952.
- Library Home /
- Search Collections /
- Open Collections /
- Browse Collections /
- UBC Theses and Dissertations /
- The behaviour of Galois Gauss sums with respect to...
Open Collections
UBC Theses and Dissertations
Featured Collection
UBC Theses and Dissertations
The behaviour of Galois Gauss sums with respect to restriction of characters Margolick, Michael William 1978
pdf
Page Metadata
Item Metadata
Title | The behaviour of Galois Gauss sums with respect to restriction of characters |
Creator |
Margolick, Michael William |
Date Issued | 1978 |
Description | The theory of abelian and non-abelian L-functions is developed with a view to providing an understanding of the Langlands-Deligne local root number and local Galois Gauss sum. The relationship between the Galois Gauss sum of a character of a group and the Galois Gauss sum of the restriction of that character to a subgroup is examined. In particular a generalization of a theorem of Hasse-Davenport (1934) to the global, non-abelian case is seen to result from the relation between Galois Gauss sums and the adelic resolvents of Fröhlich. |
Subject |
Gaussian processes Galois theory |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-03-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080167 |
URI | http://hdl.handle.net/2429/21635 |
Degree |
Doctor of Philosophy - PhD |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Unknown |
Aggregated Source Repository | DSpace |
Download
- Media
- 831-UBC_1978_A1 M36.pdf [ 3.36MB ]
- Metadata
- JSON: 831-1.0080167.json
- JSON-LD: 831-1.0080167-ld.json
- RDF/XML (Pretty): 831-1.0080167-rdf.xml
- RDF/JSON: 831-1.0080167-rdf.json
- Turtle: 831-1.0080167-turtle.txt
- N-Triples: 831-1.0080167-rdf-ntriples.txt
- Original Record: 831-1.0080167-source.json
- Full Text
- 831-1.0080167-fulltext.txt
- Citation
- 831-1.0080167.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.831.1-0080167/manifest