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The behaviour of Galois Gauss sums with respect to restriction of characters Margolick, Michael William 1978

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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 ©Michael William Margolick, 1978 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s . u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f Mathematics T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e October 10, 1978 - i i -THE BEHAVIOUR OF GALOIS GAUSS SUMS WITH RESPECT TO RESTRICTION OF CHARACTERS Abstract: 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 relation-ship 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 Frohlich. - i i i -Table of Contents ACKNOWLEDGEMENTS IV INTRODUCTION V §1 Review of Tate's Thesis 1 A) Characters and Quasicharacters of a Local Field of Characteristic 0 1 B) Local ^-functions . 2 C) Global ?-functions 4 D) Comparison with Hecke's L-functions 5 §2 Non-abelian L-functions A) The Unramified Finite Primes - First Definition of L-functions 8 ' B) The Ramified Finite Primes - Final Definition of L-functions 9 C) The Infinite Primes and Exponential Factor; The Extended Function A; Analytic Continuation 11 D) Appendix 16 §3 Local Constants 19 §4 Structure of Local Root Numbers . 24 §5 Gauss Sums for Hilbert Symbols 29 §6 Behaviour of Gauss Sums With Respect to Restriction 43 §7 The Locally Free Class Group, Resolvents and Gauss Sums 52 Bibliography 64 - iv -Acknowledgements I would like 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 Hilton for advice and support. - V -Introduction Classical Gaussian sums, as defined in the nineteenth century, are certain sums of roots of unity whose properties have played an important role in the theory of cyclotomic fields. Of considerable importance i s their factorization as algebraic integers, f i r s t discovered by Stickelberger in 1890. This factorization led to relations in the ideal class group of cyclotomic fields, an old and largely open problem in classical number theory. More precisely, a classical Gaussian sum is a certain f i n i t e Fourier transform of a character x °f the multiplicative group of the (finite) residue f i e l d (say JF^) of a local number f i e l d with respect to the additive group 3F^  . Hence classical Gaussian sums are of the form x(x) = _ x(x)*(x) where * is a fixed character of x d F q 2F^ and x is a character of W_ . (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 local number f i e l d K but over a system of representatives chosen as follows: Suppose p i s the prime of K and U the units of K . Let U^n^ denote those units of K congruent to 1 mod p n . We take representatives of the cosets of U^n^ in U . * (n) Any character x of K is t r i v i a l on some U , so that these character sums can be defined for characters of K . Let us c a l l this generalization just a Gauss sum as distinct from the classical Gaussian sum, which arises as the case n = 1 above. In case n = 1 is the smallest integer such that x is t r i v i a l on U^n^ we say x is tamely ramified, for arithmetic reasons. When the least n such that - v i -X is t r i v i a l on U n is not 1, the associated Gauss sums divided by i t s complex norm is a root of unity. The local abelian root number may be defined as a Gauss sum divided by i t s complex norm. The local abelian root number appears as a constant in the functional equation of the local zeta function of Tate. This local zeta function is in turn a kind of Fourier transform on the multiplicative group of, a local number f i e l d . If we f i x a global number f i e l d and take the product of the local zeta functions at a l l the completions of that number f i e l d , we get a generalization of Hecke's classical L-function and in particular the functional equation of that L-function together with i t s constant, the global abelian root number. This turns out, not surprisingly, to be the product of the local abelian root numbers. It is more productive to view the Gauss sum in i t s role in the functional equation of zeta or L-functions because this theory was generalized to the global non-abelian case by Artin in the 1920's. In §2 we define the Artin L-function. This is defined for characters, not of the idele class group (Tate) or ideal group (Hecke) as is the abelian L-function, but for characters of Galois groups. The connection between Artin and Hecke L-functions in the abelian case is the fundamental construction of class f i e l d theory, the reciprocity 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 are by implication discussions about non-abelian class f i e l d theory and arithmetic. In §2 we prove the functional equation of the Artin L-- v i i -function. The constant appearing in this functional equation is the global non-abelian Artin root number. So far in this discussion we have global abelain and non-abelian L-functions, local zeta functions, local abelian root numbers and global abelian and non-abelian root numbers, but we don't have local non-abelian root numbers. This is the subject of §3. The local non-abelian root number was found by Langlands and Deligne and is character-ized by the facts that i t works well with respect to addition and induction of characters and agrees with the local abelian root number in case the given character is 1-dimensional (i.e. factors through an abelian quotient of the given Galois group and hence can be considered as a local f i e l d character via the local reciprocity map). The product of the Langlands-Deligne root numbers is the global non-abelian root number. In §4 we examine the structure of the Langlands-Deligne local root number and find that i t is a root of unity times a classical Gaussian sum divided by i t s complex norm. This is obvious in the abelian case, from above remarks. The contents of §5 are classical in nature and refer to classical Gaussian sums. We prove a few scattered statements on these sums and the two known theorems on relations in ideal class groups of cyclotomic fields arising from Stickelberger's factorization. The local Galois  Gauss sum is defined naturally from the Langlands-Deligne local root number in such a way that the local Galois Gauss sum divided by i t s complex norm is the Langlands-Deligne local root number. The actual subject of the thesis begins in §6. Let E ^ L K - v i i i -be local number fields with E/K Galois. We wish to find a relation between the Galois Gauss sum of a character x of Gal(E,K) and the Galois Gauss sum of the restriction, Res x> of x t o Gal(E,L) . This relation is primarily determined by the nature of the ramification of L/K . Let us suppose E/K is abelian and x is tame or unramified. This means that we are looking essentially at classical Gaussian sums. A theorm of Hasse-Davenport says that i f L/K is TL *Kl unramified, then r(Res x) = ±T(x) ' • Results relating;;: T(X ) to i(Res x) a r e given under two more assumptions on the ramification of L/K . The case L/K arbitrary may be derived on decomposing L/K into a tower of extensions of the 3 given types. The last theorem of the section indicates how the relation may be generalized to the case E/K non-abelian. However the result is too complicated to be pleasing. The global Galois Gauss sum is defined to be the product of the local Galois Gauss sums. Section §7 contains a summary of some recent work of Frohlich, relating the following three constructions: 1) The locally free class group of the group ring of a Galois group of a tame extension of number fields over the integer ring of the ground f i e l d . E/K is a tame extension of number fields i f for every prime p of K, the ramification degree of p in E is prime to 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 yields information about how much a tame extension of number fields f a i l s to have a normal - ix -integral basis. E/K has a normal integral basis i f f the integer ring of E is free as a module over the group ring of Gal(E,K) over the integers of K . The second construction is a generalization of the classical resolvent, going back to Lagrange. The connection between these three constructions is forged by a characterization of the locally free class group as a quotient of a certain group of Galois homomorphisms defined on the free abelian group on the set of irreducible 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 factorizations. We prove a theorem generalizing the above result of Hasse-Davenport to the global non-abelian case: Let E/K be a tame extension of number fields. Let E £ L £ K with L/K normal and unramified. Let x be a character of Gal(E,K) and Res x i t s restriction to Gal(E,L) . TL *Kl Then x (Res x) a n < i T ( x ) * have the same factorization as algebraic integers. - 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 classical functional equation can be seen in the success of Jacquet-Godement's reworking [8 ] of the theory of L-functions attached to automorphic represent-ations 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 Field of Characteristic 0. Let k be the completion of a number f i e l d , R the completion of Q at the rational prime below that of k and suppose these primes are p n and p respectively. If k i s archimedean let A: IR —>-R/Z by A(x) = -x mod Z . If k is non-archimedean let X: Q — y Q./Z by setting A(x) = unique rational (mod Z) with only P0 a p_-power in i t s denominator and such that A(x) - x e "E . Let P0 A: k —> S 1 be defined by A(£) = X(Tr^R(0) . A is called the canonical character, of (the additive group of) k because Theorem: k is canonically isomorphic to i t s dual group under the correspondence n «—> [5 — y exp(27ri A(£n))] . If k is non-archimedean, let 5 be the absolute different of k, so that n e 6 i f f "^k/R-^^k^ — ^p ' I t : i s e a s Y t o s e e t n a t t n e character associated to n is t r i v i a l on 0 k i f f n e 6 L . - 2 -For k , one considers continuous multiplicative maps k —*• <E called quasicharacters. A quasicharacter i s unramified i f i t is t r i v i a l on the units U of k . Two quasicharacters are equivalent i f their quotient is unramified. Remark. Let k be the algebraic closure of k, JL =Gal(k ,k) sep ° k sep ab * ab and ft^. l t s maximal abelian quotient. Let 0: k —>• ft be the local reciprocity (Artin) map and x : ^ —• S 1 a character. Weil ([W ] (XII §2)) defines x a s unramified i f the cyclic extension L (i.e. Ker x = Gal(k ,L )) is unramified. However, by local .X sep x class f i e l d theory, x(9(°0) is t r i v i a l for a l l a e U i f f X is unramified and so the two definitions are "essentially" the same for characters. (See [W ] XII §3 Prop. 6). Theorem: Quasicharacters are maps of the form a —• c (a) = c(a)|a|"" for s e C where c i s a character of U, a = unit part of a (write a = aa with a > 0 i f k is archimedean) and |a| s = exp(s log |a|p) • c and s are uniquely determined. Re s i s called the exponent of c . As a result of this theorem, the problem of quasicharacters reduces to finding characters of U . Define the conductor of c as follows: take the least positive n such that c is t r i v i a l on 1 + p n and set fj(c) = conductor of c = p n . n is f i n i t e by the "no small subgroup" theorem. B) Local g-functions * Let us select a Haar measure on k which gives the units measure 1 . - 3 -Definition: If f is a complex function on k (satisfying certain integral convergence and continuity properties) and c a quasi-* f character of k of exponent > 0, let ?(f,c) = ^f(a)c(a)da, Jk the local £-function. Remark: Each equivalence class of quasicharacters consists of those of the form c(a) = CQ(a)(a) s for a fixed representative c n . So each class can be considered as a Riemann surface, parametrized by s . In the discrete case, this gives the complex plane in which 27Ti points differing by an integral multiple of N are identified, log N(p) l i2iri/log N(p) . ,. 1 n „ , since a =1 for a l l a . N denotes absolute norm. Theorem: A ^-function has an analytic 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(?) = c , \0-2TTiA(ri?) , i ,. . f(n)e dn = additive k A i i -1 Fourier transform of f, and c(a) = |a|c (a) . (Exponent c = 1 - exponent c). p(c) is independent of f and is a meromorphic function on the surface C on which c l i e s . • k/Q s In the p-adic case p (c ) = N , (<5f)(c)) £W(c), where W(c) P0 is the local root number of the character c . We w i l l discuss i t s form later. The theorem uses a Lemma: For 0 < exponent c < 1 and (any nice) f and g, C(f,cK(g,c) = C(f,c)?(g,c) . Q The theorem is then proved by finding, for each equivalence class of - 4 -quasicharacters C a function f^, such that ?(f^,c)/?(f^,,c) is a meromorphic function of c (i.e. has not identically zero denominator) for 0 < exponent c < 1 . This quotient p(c) w i l l then easily be seen to have an analytic continuation to the entire surface C, i.e. to c of any exponent. Since, for any c, either exponent c or exponent c is positive, ^(f^,,c) has an analytic continuation. Finally the lemma allows us to replace f^ by any suitable f ( f i r s t for 0 < exponent c < 1, then for a l l c by analytic continuation). C. Global ^-functions Let k now be a number f i e l d . We consider quasicharacters of * (1) the idele group J t r i v i a l on k . Let be the subgroup of ideles of norm 1 . If c is a quasicharacter t r i v i a l on i t depends only on |a| and so yields a continuous multiplicative + A * . * homomorphism ~R —>• C . (Imbed E. or <E in the idele group in the usual way and c a l l image elements x a r c j 1 • ^ o w c ( a) = c(|a| ,) for c t r i v i a l on ; x —> c(x ) is the map), arcn K. arcn g Any such map is of the form t —> t for s e <E . Therefore a quasi-character is t r i v i a l on i f f i t is of the form c(ct) = |a|S . Since, for any c, a —> |c(a)| is*quasicharacter t r i v i a l on , we have |c(a)| = |a| S for some s e (C . Indeed s is real, because |c(a)| > 0 for a l l a . This s i s the exponent of c. Now we define the global C function for quasicharacters of exponent > 1 by £(f,c) = f(a)c(a)da, where f is a complex-Jk valued function on the adele group satisfying certain integral convergence - 5 -and continuity conditions. The 'exponent > 1' restriction is forced -s -1 by the fact that II (1 - Np ) converges only for Re s > 1 (see p<oo D) below). The global analytic continuation theorem i s : Theorem: By analytic continuation we may extend the ^-function to the domain of a l l quasicharacters. The extended function is regular except at c.(a) = 1 and c(a) = |a|, where i t has simple poles with residues -Kf(0) and xf(0) respectively. The equation £(f,c) = £(f,c) is satisfied. In the above f = Fourier transform with respect to adele group (i.e. f(£) = f ( n ) e _ 2 ^ i A ( n 5 ) d n where A(x) = £ A (x )) •'A P P P c(a) = |a|c ^(a) and r r 1 2 - 2 (2TT) hR . w/| a | the residue of the classical zeta function of k at 1 . D) Comparison with Hecke's L-functions Let S be a fixed set of primes containing a l l archimedean primes. Let us consider idele characters unramified outside S . Such a character is a product c(a) = II c (a ) of local characters p P P * where c is unramified i f p i S and II c (a) = 1 for a e k P P P S Let I be the part of the ideal group generated by primes outside g ord a S and let <b : J —> I be the usual map a —> II p P P . Set b k P^ S * c (a) = II c (a ) . Then there is a unique ideal character Y P^ S P P • given by x ( < r g ( a ) ) = c ( a) • This i s well defined, since ord a = 0 => c (a ) = 1, for p I S . Write c(a) = II c (a )x(<rn(oO) P P P P P E S P P s For p e S, let f be any non-trivial function for which the local function £(f ,c ) is defined, and for p / S let f be the P P P characteristic function of the integers 0 of k . P P Theorem : For exponent c > 1 (c any quasicharacter) and for f p ' s sufficiently nice (e.g. as above), £(f,c) = 11 ? (f ,c ) where . P P P P f (a) = n f (a ) . P P P Sketch of proof: 1) For p i S U f (a )c (a )da = P P P P P (char. fn. of 0 ) ( t r i v i a l character) da = 1 P 2) f (a)c(a)da = lim , .. ,- / x , . , J j k T # i n k * x n u p f P ( a p ) c p ( a P ) d a p ' w h e r e peT pj!T the limit i s taken over f i n i t e sets of primes (approaching the set of a l l primes and) containing S . (This uses the important fact that II p<oo f f (a ) I I a I da < 0 0 for P P 1 1 P1 P a > 1 . See remark in section C .) 1) and 2) imply 3) ?(f,c) f (a)c(a)da = lim, II ? p(f »c ) (Fubini-type lemma) T£S peT = n C ( f , c ) p P P P By analytic continuation the theorem holds for c of any exponent, - 7 -in particular for characters. If c(a) = II c (a )x(<J>C (a)) then peS P P S one shows by a direct computation of the integral that for p i S, N6" 1 / 2  s^ P C (f ,c • ) = — P P P " P 1 - X ( P ) N ( P ) - S for Re s > 1 . Therefore 5(f,c|| s)= n ? (f , C ) n N6 _ 1 / 2 n ( i - X ( P ) N P " S ) _ 1 peS P P P p^S p^S - 1 /2 = n c (f ,c ) n N6 X / Z L ( S , X ) peS P P P p^S P S where L(s , x ) is 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 this equation for f and "cf"p\ and using the fact that for p e S the choice of f 's is immaterial (local P functional equation), the global functional equation of Tate expresses exactly the classical functional equation of Hecke (see §2D)). - 8 -§2. Non-abelian L-functions A) The Unramified Finite Primes - Fir s t Definition of L-functions Let E/K be a f i n i t e normal extension of number fields with Galois group G . Let V be a finite-dimensional complex vector space and p a V-representation with character x • L e t P be a prime ideal of K, P/p in E and assume p is unramified in E . Let E/K Op = [ ] be a Frobenius of P over p . Then the complex number -s -1 (Euler factor) det^(l - p(o~p)Np ) doesn't depend on the choice -s -1 of P and we set L(s , x ) = n d e t v ( x ~ p(op)Np ) . The p unramified series converges for Re s > 1. Right away one sees the change from one-dimensional (continuous) representations of the idele class group or ideal group to higher-dimensional representations of the (not necessarily abelian) Galois group. The link in the abelian case i s class f i e l d theory: If G is abelian and p irreducible then p is a character into S 1 . g Let 6 : I —>• G be the global Artin map (S = a l l archimedean primes s •and those ramified in E; I is the ideal group outside S) . Then p • 6 is a Hecke character"^, 6(p) = and (Hecke)L(s,p*6) = (Artin)L(s ,p) . We have the following properties: a) L(s ,x 1+X 2) = L( s»X 1)L(s,X 2) (Additivity) A Hecke character is a character of I R t r i v i a l on some (congruence) subgroup of principal ideals generated by elements of K congruent to 1 mod p p e S . - 9 -b) Let H be a normal subgroup of G . Let p be a representation of G/H with character x and p' i t s l i f t i n g to G with character x' • Then L(s,x') and L(s,x) differ by a f i n i t e number of Euler factors. (Lifting). c) Let H be a subgroup of G, x the character of a represent-ation and Ind x the induced character on G . Then L(s,Ind x) and L(s,x) differ by a f i n i t e number of Euler factors. (Induction). Remark: b) is clear from properties of the Frobenius. For c) see, for example, Heilbronn in [CF] . For both b) and c), local factors agree wherever both are defined but there may be primes in K ramified (respectively unramified) in the intermediate f i e l d F whose divisors in F are unramified (respectively ramified) in E . , Before going on to other facts about L-functions derived basically from these properties and the Braiier induction formula, we give the fi n a l definition of L-functions. B) The Ramified Finite Primes - Final Definition of L-functions The problem i s to define local factors at ramified prime ideals such that a), b) and c) above hold with equality. Consider any prime p and let D^  (respectively I p) be the decomposition (respectively inertial) group of a divisor P of p in E . Let 0 be the Frobenius in B /I . G acts on the representation space V of p and i f we write V for the set of vectors fixed by Ip, V is a representation space for ~D^/IR i.e. there exists a p., making - 10 -D p • p(Dp) c GL(V) T h p i h Dp/Ip • GL(V r) commutative, where h is restriction of automorphisms to the p-invariant subspace V . That V is p-invariant is immediate from the normality of I p in D p . Now set -s -1 L(s , x ) = n det T (1 - p (a )Np ) , independent of choice of P P<" V p Theorem: a), b) and c) above hold with equality. • Remark: This is made plausible anyway by the fact that reducing p to p^ "replaces" each local extension Ep/ Kp by an unramified one, with Galois group Dp/ip • It i s clear from the definition that LCs,!^) = £ (s), the G K zeta function of the number f i e l d K . Further, i f r„ is the G regular representation of G and 1 the t r i v i a l representation of the t r i v i a l subgroup of G then Ind 1 = r has character 1 G I X(l ) x a n d we have: X irreducible Corollary: 5 £(s) = II L ( s , x ) X ( 1 ) . X irreducible • Note that i f dim V = 1 and p = X is f a i t h f u l , then this definition coincides with that of section A). (For i f 1 ^ 1 , then V = 0,) An outstanding problem in L-functions is the Artin Conjecture: It w i l l be shown that L-functions have an analytic continuation to - l i -the whole plane. Artin conjectured, in analogy with the one-dimensional case, that i f x does not contain the t r i v i a l character, then L(s , x ) is entire. It seems that the only characters for which a direct approach is effective are those x = I a- ind X. i with x^ one-dimensional and a^ rational and positive. In this case properties a) and c) show that some power of L(s , x ) is entire. However, a much more powerful approach is available through the theory of L-functions of automorphic representations (see e.g. [7 ]). L-functions live most naturally in the following setting: Let p: Q —• GL(V) be a continuous representation with open kernel. K b) shows that we may define an L-function for p and a) allows us 2) to extend the definition to any virtual representation of Q, . K C) The Infinite Primes and Exponential Factor; The Extended Function  A; Analytic Continuation s/2 A w i l l be of the form A(s , x ) = A(x) Y (s)L(s , x ) for some X functions A and y and where L is as in B), and w i l l satisfy the functional equation A(s , x ) = W( x)A(l-s , x ) for some constant W(x) of absolute value 1 . x is the conjugate of x : i f X is the character of p: G —>• GL(V), then x is the character of -1, 1' : G —> GL(V') (V = dual of V), defined by p'(f)(x) = f(p ""(x)) s s 2) The group of virtual representations of ft i s , by definition, K the free abelian group on the irreducible continuous representations of ttv . K - 12 -Definitions of y (s), A(x): X •" S / 2^ S) and y v(s) = " - v ' - 1 v v i n f i n i t e as follows: Set y(s) = TT ' r ( j )  (s II y v(s) where y is v complex: y V(s) = [y(s)y(s+l)J^ 1) . C X v real: Write V = V + « V , where i f wlv and a generates v v 1 w Gal(E ,K ), V + = eigenspace of p(a ) corresponding to +1 and W V V o r w V = eigenspace of p(o ) corresponding to -1 . Set V w dim V dim V Y v(s) = Y(S) y(s+l) X In order to define A(x) we need the notion of the Artin oo conductor: Fix a prime ideal p and let {G.}. . be the l 1=0 ramification sequence of G(E,K) = G, with respect to p . Let g. be the number of elements of G. and set i x 0 0 -, G. G. v —x i i n(x) = 2. 8n 8- codim V , where V is the subspace of a l l i=0 1 v in V fixed by G. . x (Remark: This is a purely local construction and can be related to conductors of class fields as follows: Suppose K (just for now) is local and let x ^ 1 be a (just for now) one-dimensional character of £2 . Let L be the cyclic extension attached to x is. (r) r (see §1A)). Let U be the units of K congruent to 1 mod p L/K (indeed any f i n i t e normal extension of K) has a conductor f (r) p , where f is the least r such that U is contained in NL/K^I, ) . It is well-known (see §5 lemma 4) that f = <f>(c) + 1, where <J> is the Herbrand function and c is the largest integer - 13 -such that G f 1 . But <j)(c) + 1 = J g.g- . Let x n be the ° i=0 1 character of Gal(L,K) arising from x • We have dim V = 1 and G. X^ 4- 1 so codim V =0 i f G^ = 1 and 1 i f G ^ 1, (since cannot be t r i v i a l on any non-trivial subgroup of Gal(L,K) without x also being t r i v i a l ) . The field-theoretic conductor of L/K is the same as the Artin conductor of x • Of course this i s also clear i f x = 1 •) Now let |$(x) = n p n^ x' p^; we have n(x,p) = 0 i f f p i s p<°o unramified in E . Finally, we define A(x) = I d I . (j{ (x) ) » K K/Q where d is the absolute discriminant of K . That n(x,p) is an integer follows from the formula $(x) = <t>(c) + 1 for x a one-dimensional character and for some integer c such that G 4- G ,., , c c+1 the theorem of Hasse-Arf (which says that in this case $(c) is integral), a formula relating ($(x) to j[(Ind x) (see immediately below), and the Brauer induction formula. The Functional Equation: The method of proof is 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). Finally one shows that the one-dimensional case of the desired functional equation is exactly the functional equation of Hecke-Tate, using property b). First one must verify properties a), b) and c) for A, i.e. for A(x) and y (s) . The only troublesome X one is induction for the Artin conductor: ([16] §4.3 Prop. 4) j$(Ind x) = dp / j ^ 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 verified for A . Write x = I n ^ I n a f ° r X^ a degree one character of some abelian subgroup R\ of G . Let F. = fixed f i e l d of H., Hi the kernel of Y- and Fl the x 1 1 1 1 fixed f i e l d of H! . F!/F. is cyclic with Galois group H./Hl . i i i i i Let G. = H./H! - Image x . • Finally, let x! be the character i l l l l of G_^  arising from x^ • BY property b), A(s , x ^ ) = A(s , x | ) • Let us work with a x^ at a time and suppress the " i " in the notation. Let S be the set of primes of F that ramify in F', plus a l l archimedean primes. Now x ' defines an idele character c via the Artin map 8: J_ —> G . Let *' be the ideal character s arising from c as follows: If § : J — I - r - i - s the usual map, *' is the unique ideal character such that ip' (<j>„ (a) ) = II c (a ) s P^ S P P for a e J . r We know from §2A) and the remark in §2B) that (Hecke)L(s,*') = U. (1 - *'(p)Np" S) _ 1 = (Artin)L(s ,X') . Now we ps^ S w i l l write down the Hecke-Tate functional equation for *' and compare the exponential factors "A" and factors "y" corresponding to i n f i n i t e primes in i t , to those factors as defined above. The superscript '"" refers to a factor in the Hecke-Tate equation. s/2 Set A'(s,*') = A ' ) y ^ i(s)L(s,* 1) . Here, we let A* (*') = [ | D KS N K/Q(<5 Cc) ) ] where ^(c) is the product of the local conductors of the Cp's> a s defined in §1A) and y',(s) = n Y(S)Y(S+1) II y(s) n Y(S+1), with y as above, v complex v real v real c = 1 c 4 1 V V From the last line of Tate's thesis (see D) below) we have A'(1-S,TJJ') = W(ip')A' (s,<p') for some constant W(ip') of absolute - 15 -value 1 . We must show that the factors A and A' and the factors y and j 1 coincide. To show A = A' we must prove the equality of the Artin conductor $ ( x ' ) and the conductor (5(c) . (r) The p-part of the latter i s the least r such that c (U ) = 1 . P P As was shown above, the p-part of the former is the conductor of Lp/Fp, where L p is defined by: Gal(F^,Lp) i s the kernel of the restriction (to Gal(F',F )) of x ' • This holds, since F'/F is P P cyclic. More precisely, i f we consider the map f which makes res x Gal(F',F ) P P commute, then L R is the cyclic extension attached to the composition res f 1 *V * Gal(L_,F ) • S . Recalling that the conductor of K r p p fr) * L „ / F is the least r such that I P ' c Kernel(0 : F P p P - P P Gal(L p,F ) ) , where 0^  is the local Artin map, the equality of conductors then follows from the commutativity of the following diagram and the injectivity of f . 1. ft F >-P 0 Y ' 1 — ^ — G a l ( F ' ,F) -X- >- S r F p — P — , Gal(L p,F p) -> S To show Y = Y'» w e need only show that for v real, c = sgn (i.e. c ^1) i f f dim V = 1 . Now dim V =1 i f f X*(a ) = -1 where a generates Gal(F',F ) for w v . The local A v v w v - 16 -Artin map 8 satisfies 6 (-1) = a for a l l real v and so r v v v Y'(O ) = -1 i f f a is order 2 i f f c (-1) is order 2 in A v v v v c (-1) = -1 i f f c = sgn. v v Finally then, i f x 1 S the character of the orginal represent-n. n. ation p we have A(s , x ) = n A(s , i p ! ) 1 = II W(ip!) 1 i i _ n. _ n. A(l-s , i p ' . ) 1 = W( x)A(l-s , x ) with W(x) = II W-(ip!) 1 . Since each i W(ip^) has absolute value 1, W(x) has absolute value 1 . D) Appendix Let us verify in f u l l the Hecke-Tate functional equation and compute the local constants in i t in the one-dimensional case. Let us write ip (the ip' in section C)) for the unique ideal character satisfying ip(<f> (a)) = II c (a ) for a l l a £ J b P^ S P P The last line of Tate's thesis i s : ? ( i-s , i p 1 ) = n P (c ||s) n N ( 6 ) s %<p x(6 K ( s , i p ) , peS P P p^S P P where the functions p are defined as follows: P I | ( 2 ^ ) 1 _ s r ( s + M ) M s n ^ ) = (-i) 1 1 1—r where * * (2,) s r ( i-s + - I f L ) , iO. in 6 „ Cp(e ) = e , n e Z . p p ( H S ) vvoSy , iisv (-i)y(s+l)  P p ( S S n | 1 } = Y(2-s) - 17 -p f i n i t e : p (c II s) = N(S &(c ) ) S \(j$(c )) ^x (c ) where x (c ) P P P U P ° P P P P P is the local Gauss sum £ c (XTT m)exp(2Tri A(XTT m ) ) . Here x P m = ord (6 |(c ) ) , TT (not to be confused with the IT in P P P exp ( 2 T r i . . .) !) i s any element of order 1, and the sum is taken over a set of representatives of 1 + $(c p) in U p . The A here is that of §1A) of course, not an extended L-function. It is readily verified that the sum x p is independent of choice of uniformizer ir . Now we are ready to compute: c d - s , ^ " 1 ) _ n p ( c I I s ) n y ( s ) n T(s+D-(-i) ? ( S ' ^ v complex P P v real y ( 1 " S ) V real y ( 2 " s ) c = 1 c 4 1 V V n N ( 6 p J f ( c p ) ) S _ % N ( ^ ( c p ) ) " % x p(c p) n N ( 6 p ) S " % ^ _ 1 ( 6 p ) . peS If v is complex, c =1 because c arises from the (t r i v i a l ) r ' v v local Artin map. For the complex quasicharacter ||S, rl+s, l - 2 s I, s. = (2TT)-1- / ST(s) = U 2 m 2 ; i T = Y(s)y(l+s) V " J T(l-s) r ( l - s ) r ( 2 - S ) Y ( l - s ) Y ( 2 - s ) by the Legendre duplication formula. Furthermore IT N(6 ) S 2 = |d„|S p r p«x> r Now define constants W(c^) as follows: v complex: W(c^) = 1 . rl i f c v = 1 v real: W(c ) = i V l - i i f c 4 1 . v v f i n i t e ; v = p: W(c ) = N()J(c )) 2 T p ( c p ) • (Note that i f p I S, -ord 5 _^ c is unramified and W(c ) = c (ir P P) = ib (6 ) . P P P P - 18 -We get: S(1-S,T[) 1) _ y(s) Y(1+S) Y(S) C(S,I|I) . Y(l-s) Y((1-S)+D 1 Y(l-s) v complex v real c = 1 V ^ t f l | d F | N ( < ( c ) ) , S ~ * n w ( c v ) V v real a l l v c 4 l w W1-) A ' C * ) ' 1 - " 2 where W(ip) = n W(c ) . Finally then, A'(1-S,IJJ) = W(i|>)A' ( s , \ b ) , or v A'(l-s , i p ) = W(ijj)A' (s , !p) as was to be shown. - 19 -§3. Local Constants The following theorem is due to Langlands and Deligne. Theorem: Let K be a local f i e l d of characteristic zero with valuation v . Let J2T, = Gal(K ,K) and let p be a virtual K sep representation of Q . Then there exists a unique function W K • from the group of virtual representations of Q, to such that K i) W ( P l + p 2 ) = W(p1)W(p2) for a l l P - ^ p ^ . i i ) Let p be a virt u a l representation of degree 0 and Ind p an induced representation. Then W(Ind p ) = W(p) . i i i ) If p is irreducible and of degree 1, hence a character of ab * fiTr and i f c is the character of K defined by p through the K v ° local Artin map, then W(p) is the local constant for c in the functional equation of the zeta-function (i.e. the W(cv) of §2D)). • Note (after Deligne [ l ] ) that we may also write, in case i i i ) above, c ( f v , < l ? ) -W(p) = 5— , for any f such that £(f ,c ) is defined. S(f ,c ||%) V V A good framework in which to prove this theorem is due to Tate ([18]): If K is global or local characteristic 0, let R(K) denote the set of pairs (L ,p ) such that L/K is f i n i t e and p is a virt u a l representation of 9. . If E/K is f i n i t e , let R(E/K) denote those pairs (L ,p ) such that E £ L £ K and p is a virtual representation of Gal(E,L) . R(E/K) may be considered a subset of R(K) and we have R(K) = u R(E/K) . How do one-E=>K - 20 -dimensional representations f i t in? Let R^(K) denote the set of pairs (L,x) where L/K is f i n i t e and x is a character of L i f K i s local (respectively of C , the idele class group, i f J-4 K is global). R^(K) imbeds in R(K) as follows: A Any character x of L (resp. C^) factors through an open sub-group. By the existence theorem of class f i e l d theory, this can it be taken to be N(L' ) (resp. N(C ,)) for some f i n i t e extension L* L'/L . Considering x to have domain L /N(L' ) (resp. C /N(C ,)) J_i Li and recalling that the local (resp. global) Artin map 6 is an isomorphism onto Gal(L',L), we can imbed R-^ (K) into R(K) by taking (L,x) to (L,xe - 1) . Finally, set R^(E/K) = R±(K) n R(E/K) Definition: Suppose F is a function defined on R-^ (K) , taking values in some abelian group. Say F is extendible i f i t can be extended to R(K) satisfying: a) F ( L , P ] + p 2 ) = F(L,p 1)F(L,p 2) for (L,p ±) e R ( K ) . b) If (L , p ) e R(K) with dimension p = 0 and L £ L' £ K, L' L' then F(L , p ) = F(L',Ind p ) , where Ind is the virtual represent-Li -LI ation of Q , induced from p . If E/K is f i n i t e , Galois, say F Li i s extendible in E/K i f i t can be extended from R^(E/K) to R(E/K) satisfying (the appropriate modifications of) a) and b). Say F is strongly extendible i f b) holds for p of any dimension. • If F i s extendible, then the extension is unique, essentially by Brauer induction. As a consequence F is extendible in R(K) i f f i t i s extendible in E/K for every E . We have shown in §2 that i f K is global, (L,x) >-»- A(s,x) and (L,x) I—y W.(x) are - 21 -extendible, even strongly. Other familiar extendible functions are: i) (L, X) ^> N L / K ( ^ ( X ) ) ( K 8 l o b a l o r P-adic). i i ) (L ,x) v - > x ( ° ) f° r c a fixed element of C (if K is global) K ft or K (if K is local). i i i ) (L ,x) I -* a(L) where a is a function of L only. Now the above theorem looks li k e : Theorem: If K is local characteristic 0 then (L ,x) —• W(x), the local root number, is extendible. • Remark: It is not hard to see that W(x) cannot be strongly extendible. The condition "dim p = 0" is therefore significant; i t shows the necessity of working with the entire representation ring. For general X, one has W(Ind x ) / w ( x ) i s a fourth root of unity (see §4). The key lemma for this theorem is roughly as follows: Let a * be a character of K (K non-archimedean) and let 8 be a character ft of L with L/K f i n i t e . Let = a • . If a is "sufficiently highly ramified" and 8 is "sufficiently lowly ramified relative to a", ft then there exists a c e K such that W(8a ) = 8(c)W(a ) . The theorem JLi LJ can be proved in several steps: 1) Fix E/K f i n i t e . It suffices to show (L ,x) l ~ > W(x) is extendible in E/K . One may also assume K non-archimedean. 2) Find a global extension e/k with k totally complex, and a place v n of k with only one divisor u n in e such that Gal(e,k) - Gal(e ,k ) is canonically isomorphic to Gal(E,K) . Hence u0 v0 - 22 -there exists an isomorphism R(E/K) - R(e,k) given by ( £ , p ) < — y (H ,p ) where wn is the unique divisor of v n in % wQ wQ 0 0 and p is the restriction of p to Gal(E,£ ) . One must show w w 0 0 that (&,x) 1— y W^^ w ^' t' i e x o c a ± r o o t number is extendible in e/k . 3) One may (by the Griinwald-Wang Theorem ([AT] Ch. 10)) choose a single idele class character a of C^ such that: a) For every v 4 v n, the conditions of the key lemma are satisfied for a v and every (all) (F,B) in R^(eu,k ) . This i s possible because the conductors of a l l possible B's for a l l possible v's are bounded above. b) a = 1 . V0 4) Choose, for v 4 v», c as in the key lemma. Let a. be the ' 0 v J I idele class character of C„ given by (a„) = a • N„ „ I I w v &„/k_ w v We have, for each ( & , x ) e R-^(e/k), X T T( C„ )w((°Ott) w non arch., w 4 wn w v x, w u W(x ) w = w w q 0 w arch. The f i r s t case is the lemma. The second is from a = 1 . The third is from k totally complex. v0 5) In view of 4) write W ( x a T ) = W(Y )x(c)a(£) with L *w0 a(£) = II W ( ( a . ) ), independent of x • B Y i i ) a n d i i i ) above , X , W w f w q w non-arch. - 23 -X(c) and a(I) are extendible. By extendibility of the global root number and Frobenius reciprocity, (&>x) w ( x a ^ ) is extendible. Hence (&,x) l—*" W(x ) is extendible and we are done. w0 • One interesting feature of this proof i s the global-to-local (!) approach, using the known extendibility of the Artin A function. This may indicate the d i f f i c u l t y of approaching Artin L-functions in a purely local way. Another feature i s the examination of highly ramified local characters. This allows us to replace a character sum over U /U^r^ by a similar sum over U^X^/U^r^ . As a consequence, P P P P one can show that for a one-dimensional character a, W(a) is a root of unity i f a is unramified or p |$(a) (a is wildly ramified). In the remaining case, a tamely ramified, W(a) is essentially a character sum (finite Fourier transform) of a multiplicative character over the additive group of the residue f i e l d (divided by i t s complex norm). These character sums or Gaussian sums are ultra classical objects and have played an important role in the theory of cyclotomic fields (see, for example, [19]). In the next section we w i l l examine this relation more closely. References for §1,2,3: Tate's thesis can be found in i t s entirety in Cassels-Frohlich ( [CF ] ) . Our discussion of non-abelian L-functions follows that of Martinet's article in the Durham notes ([14]). The discussion of local root numbers is due to Tate in the same series. For Deligne's proof of the extendibility of local root numbers, one is referred to [ 1 ]. Any well-known facts about arithmetic of local fields (Herbrand function, induction formula for Artin conductor etc.) can be found in Corps Locaux ([SI]). - 24 -§4. Structure of Local Root Numbers Throughout l e t F be a l o c a l p-adic 1 f i e l d , over Q , say. Let P0 F be i t s residue f i e l d , p the prime of F, TT an element of r l + o r d F 6 F order 1, d_ = TL , where 6^ i s the absolute d i f f e r e n t r r r of F . We denote the valuation r i n g of F by 0^  and the units r by T7 . Let i p , , be the canonical character of F as i n 11A) . r r We have ^ ( x ) = exp(2iTi X(Tr x)) where Tr_ i s the absolute r r r trace 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 that X(x) - x e & . The U p0 character group of F i s canonically isomorphic to F . If g- = g — 2Tri i s the character of F given by g(x) = exp( Tr=(x)) then the P 0 F isomorphism i s given by y v-*- g^ where gy(x) = g(xy) . F i n a l l y , l e t F be the maximal unramified extension of Q contained Po-rn F . Let us make the convention that "tame" or "tamely ramified" r e f e r s to a character or f i e l d extension that i s ramified and tamely so, reserving "at most tame" for "unramified or tamely ramified". The aim of t h i s section i s to describe the structure of l o c a l root numbers. If a i s a tamely ramified character of F , W(a) i s a c l a s s i c a l Gauss sum, or character sum over F, divided by i t s complex norm. If a i s w i l d l y ramified ( i . e . ord^^(a)• >^  2) or unramified, W(a) i s a root of unity. From t h i s i t i s possible to f i n d W(x) for x a n y character of Q^, up to roots of unity. Theorem 1: (Dwork [ 2 ] ) . Let m(x) = o r d p ( ^ ( x ) ) / x ( l ) • If m(x) 4 1 then W(x) i s a root of unity. ^ - 25 -If a = x i s one-dimensional, W(a) i s a root of unity unless a i s tame. Lemma 1: a) Suppose F i s absolutely unramified. Then ippCpQ^x) = g(x) for a l l x e 0 F . b) For a r b i t r a r y f i n i t e F/Q , l e t y = T r p / F ( a , ~ 1 p 0 ) P 0 nr Then I|J (d 1x) = g p(xy) for a l l x e 0 p nr Proof: a) Choose, for x e 0 p , a(x) e Z such that a(x) = Tr-(x) mod p_ 2, and b(x) e Q with only a p„-power i n i t s denominator such that b(x) = A(Tr^Cp^^x))mod 2 . We want to r U 2 Tri show that e x p ( — a(x)) = exp(2iTi b(x)) . This i s the same as P 0 a(x) - Pgb(x) e v^TZ . Now, since F i s absolutely unramified Tr (x) = I <j(x) = I a(x) = Tr-(x) , so aeGal(F,q) aeGal(F,Z/p Q) a(x) E Tr„(x)mod vJZ . Hence v 0 P Q a(x) - p Qb(x) = T r p ( x ) - p ^ C T r p C p ^ x ) ) - T r p ( x ) - p ^ r ^ p ^ x ) (by d e f i n i t i o n of = 0 mod p„Z 0 P Q Therefore a(x) - p Qb(x) e PQ%^ n • T o s e e t n a t t n i s i s a c t u a l l y an integer, write b(x) = e(x)/p n with e(x) e 2 and, without c(x) loss of g e n e r a l i t y , r >_ 1 . Write a(x) - Ppb(x) = p A ~[7^ y r o r c(x), d(x) integers and p n prime to d(x). We see that P Q - 1 a(x) - e(x) = p^ => d(x)|c(x) and a) i s proved. - 2 6 -For x e 0_ , Fnr V ^ x ) = *F ^ r F / P ( d / x ) ) nr ' r , -1 ^1 % <P 0 x T r F / F ^ ( d F p n ) ; ) ( b y l i n e a r i t y of trace) nr m nr = g-(xy) by a) (for we have F = F ) . n r • ft Let us consider a tame character a of F as defined on -ft (1) F - U_/UJ, , and set a(0) = 0 . The d e f i n i t i o n of the l o c a l Gauss r r V -1 -1 sum x (a) (= ). a(d„ x)<pT,(d„ x), where the sum i s taken over a p r r r X system of representatives of U^ m^ i n TT and m i s the order of |J(a)) and lemma 1 y i e l d ft Prop. 1: Let a be a tame character of F . Then _1 _ft x (a) = a(d„ T r w , _ . (d„ p r i ) ) G 1 ( a ) , where for y e F , G Y ( a ) i s the p v k / \ - n r b u l ' character sum over F given by G (a) = £_ a(x)g(xy) . 7 xeF •. Proof: Note that a set of representatives f o r U^"^ i n U p may be taken from TT and that a ( y ) G n (a) = G (a) . £ 1 y n nr J • • • Now we can give the structure of l o c a l root numbers, up to roots of unity. Let us write a ^ b i f ab X i s a root of unity. Let E/K be f i n i t e normal with Galois group G and l e t G^ be the wild r a m i f i c a t i o n group of G . Let L be the f i x e d f i e l d of G^, so that L i s the maximal at most tame extension of K contained i n E . The structure of Gal(L,K) has been determined by Hasse: - 27 -Prop. 2: ([H ] §16). Let L/K be a l o c a l tame Galois extension q = order of K, f = r e l a t i v e degree of L/K = [L:K], and e = e(L/K) = r a m i f i c a t i o n index of L/K . Then Gal(L,K) i s meta-e f r c y c l i c , generated by a and x such that x =1, a = x for some r such that e [ r ( q - l ) , and axa X = x^ . • If p i s an i r r e d u c i b l e representation of G on the vector space G l V, p gives r i s e to a V representation p ^ of G/G^ = Gal(L,K) . Since p i s i r r e d u c i b l e , p ^ = 1, or p ^ = p . Dwork [2 ] has shown that i n the f i r s t case W(p) ^ 1 so that we may assume ( i n determining structure up to roots of unity) that p i s a represent-at i o n of Gal(L,K) . Now Gal(L,K) i s metacyclic hence super-solvable and so by ([S2] Th. 20), i t i s a monomial group. By d e f i n i t i o n t h i s means that every i r r e d u c i b l e representation i s one-dimensional or induced by a one-dimensional representation. If p i s not one-Q dimensional, l e t H c Gal(L,K) with Ind a = p for a character r i i< a of K n, where K n i s the fixed f i e l d of H . Lemma 2: ([18] §2 lemma 1). If ( L , p ) e R(K) and L £ L' £ K L' then W(p) ^ W(Ind p ) . Indeed the quotient i s a 4th root i f unity. • Remark: A s i m i l a r statement holds f o r l o c a l Galois Gauss sums which are defined i n §5. Lemma 2 may be proved j u s t as lemma 9 of §6. By lemma 2, W(p) 'v W(a) . We w i l l see i n §5 that any one-dimensional character of a tame Galois group i s at most tame when considered as a l o c a l f i e l d character. Putting t h i s together with - 28 -Prop. 1 we can describe W(p) . Prop. 3. Let p be an i r r e d u c i b l e representation of G = Gal(E,K) . Then W(p) ^ W(a) where a i s an at most tame character of some f i e l d KQ, at most tame over K and contained i n E . W(a) i s either a root of unity or a Gauss sum divided by i t s complex norm. • No uniqueness i s possible i n t h i s decomposition i n general. However i f f = f(E/K) i s 1 or a prime, Gal(L,K) has the property that every i r r e d u c i b l e representation i s one-dimensional or induced by a character of the subgroup of order e generated by x ([CR] §47.14). This subgroup i s exactly the Galois group of L over the maximal unramified extension of K contained i n E, say E nr Prop. 4: If 1) the r e l a t i v e degree f(E/K) i s prime or 1, and G l 2) p = p i s i r r e d u c i b l e and not 1-dimensional, then there i s a tame character a of E with I n d a = p nr E nr and W(p) = l^ia) . a i s uniquely determined by p (mod 4). Proof: A l l that remains to be shown i s uniqueness of a . If K K p = Ind a = Ind a', then W(a) = W(a'), since a - a 1 i s E E nr nr O-dimensional. So a i s uniquely determined (mod 4). • - 29 -§5. Gauss Sums for H i l b e r t Symbols Gauss sums for H i l b e r t symbols are some of the most important algebraic numbers of nineteenth century number theory. They l i e at the core of cyclotomic f i e l d s and hence ultimately cl a s s f i e l d theory. In modern number theory they are to be found not only i n the func t i o n a l equation of the A r t i n L-function, but also i n the theory of p-adic L-functions where generalizations of Stickelberger's Theorem (see below) are sought. Let us assume that L/K i s a f i n i t e normal extension of p-adic (r) r f i e l d s and write for the units of K, =l.mod p 7 . K Lemma 3: Let L/K be an at most tame extension. a) I f L/K i s unramified N^^(U^ r^) = u £ r ) for a l l r . b) I f L/K i s ramified, \/K0Jh'^ = V ^ • Proof: The f i r s t statement i s ([SI] V §2 Prop. 3a)). The second i s ([CF]I§8 Prop. 2). Lemma 4: Let L/K be any t o t a l l y ramified extension. Let ^ i ^ - o be the sequence of r a m i f i c a t i o n subgroups of G = Gal(L,K) . Let <j> be the Herbrand function ([SI] IV §3). Let c be the largest (r) integer such that G 4 1 . Then the l e a s t r such that U i s C K. contained i n N ^ ^ ( L ) i s <)>(c) + 1 • Proof: ([SI ] XV §2) . • - 30 -Prop. 5. Let L/K be an at most tame l o c a l extension and a a one-dimensional character of Gal(L,K), considered as a f i e l d character v i a the l o c a l A r t i n map. a) If L/K i s unramified so i s a . b) If L/K i s ramified, a i s at most tame. Proof: If L/K i s unramified, c_ N ^ ^ ( L ) by lemma 3a). I f (1) * L/K i s t o t a l l y ramified, then • _c N ^ ^ ( L ) by lemma 4, as we have c|>(0) = 0 . The case L/K ramified, but not t o t a l l y , reduces A to the t o t a l l y ramified case by lemma 3b) . Since N ^ ^ ( L ) i s exactly the kernel of the A r t i n map, the r e s u l t follows. • Now suppose K contains mth roots of unity. Let p 0 c h a r a c t e r i s t i c of K and assume (HI,PQ) = 1 . Let q = order of — A K . For a e K there i s the H i l b e r t character x = (a,-) defined by the equation ( a , x ) m v ^ = 0(x ) m v ^ , where 9: K —> Gal(K( /a),K) i s the l o c a l A r t i n map. An important property of x i s that i t s kernel i s exactly N m T- . (K(mv^a) ) . Since ,a K( vaj/K K(m/a) i s always at most tame, so i s x • W e also have x SL 3. unramified i f and only i f K( m/a)/K i s unramified, for i f i n ?— *k X (U ) = 1, U c N(K( /a) ) and so by l o c a l class f i e l d theory, a K K K(m>/'a)/K i s unramified. w(x a)w(x b) Prop. 6. —r-( r — =/^(a.iJCxP) i f X a unramified. W U a X b J ( a b , T i ) X b ( - D i f X a» X b ramified and x a b unramified. 7 -ord,a -ord^b , .,. , q " % I [x (1-x) ] ( q 4 ) / m xeK x^0,l i f x a , X b » X a b a l l ramified. - 31 -Proof: If x i s unramified the r e s u l t follows from ([18] §2 a Cor. 5). (In general i f x i s unramified and one-dimensional and p i s any representation, W(x ® p) = x(&(p))W(x)dimPW(p)) . In W ( x a b ) W ( x b 1 } -1 -1 the second case , . = X a b ( t f ( x b )) = (ab,7rK) by the f i r s t x a case. However ([14] Prop. 2.2) WXx^WXx^) = x b ( _ l ) a n d s o the second case follows. In the t h i r d case we are dealing with a property of character sums; ( i f they are considered as Fourier transforms, then t h i s i s a statement about convolution): G (a)G (B) = l_ a(x)BU-x) G ( a B ) . (See [L ] IV §3, GS4). 7 7 xeK 7 V X a > V X b > r In our s i t u a t i o n t h i s looks l i k e — . . = £ (a,x)(b,l-x) . Tp t Xab' ) xeK x^0,l Since x i s evaluated at elements -x i n 0 not i n p, the a K t h i r d c l a s s follows immediately from the evaluation of the H i l b e r t symbol i n the tame case which we state as lemma 5. • Lemma 5: ([SI] XIV §3 Prop. 8, Cor.). (a,b) = c ^ q where * . . , z -. x (ord a) (ord b) ord b -ord a c e K i s given by c = (-1) a b In case we are dealing with quadratic symbols we have (for more subtle reasons involving root numbers of orthogonal representations) w( xjw(x h) — - 7 r ^ - = (a,b) (see [18] §3 Cor. 2). W ( x a b } Recall that the map x f)(x)> the A r t i n conductor, i s extendible i n R(K) . D e f i n i t i o n : Let x be the character of a representation of ft^ • - 32 -Define the l o c a l Galois Gauss sum by x(x) = T (X) = N K/Q ^ ( X ) ) *W(x) p0 dimx2 dimx-j^  tCx^ T ( x 2 ) Define the Jacobi sum J ( x L 5 X 2 ) bV J ( x 1 » X 2 ) = — T ( X 1 X 2 ) • Local Galois Gauss sums s a t i s f y a l l the main properties of l o c a l root numbers: a d d i t i v i t y and invariance under l i f t i n g and under induction of characters of dimension 0 . Before going on to discuss Gauss sums, i n the one-dimensional case i n t h i s section and i n the higher dimensional case i n § 6 , l e t us mention one property of Jacobi sums i n the one-dimensional case, which we w i l l express i n terms of root numbers. Let K be the cyclotomic f i e l d of mth roots of unity. Let i ' " be the part of the i d e a l group of K generated by primes not K d i v i d i n g m . D e f i n i t i o n : Let K be a number f i e l d and m an i d e a l of 0 . A is. Yft * Grossencharakter ( i n the sense of Hecke) i s a function J : I —> (E K s a t i s f y i n g J(fi)J(v) = J(fiv) and such that i f x e 0 and x E l mod m then and x = x^,...,x^ are the conjugates of x, d e c J(xO „) = II x I x | . f o r some r a t i o n a l integers e and complex K , n 1 1 n n=l numbers (independent of x ) . ^ Two Grossencharakters are said to be equivalent i f they agree on each i d e a l m. outside of which they are defined. These equivalence classes are i n 1-1 correspondence with quasi-characters of C , the K - 33 -the i d e l e c l a s s group. One can show that Grb'ssencharakters of f i n i t e order are exactly those for which Hecke L-functions are defined, i . e . they are t r i v i a l on some subgroup of I of p r i n c i p a l i deals generated K . by elements of K, = 1 mod m . Assume K i s the f i e l d of mth roots of unity again, and f i x integers a,c t 0(m) . Define a map Ja,c: lv — • (C by w(x J w ( x ) Ja,c(p) = % X a + c ( _ 1 ) • W(X a X c ) " P TT IT P P Weil has proved the following seemingly i s o l a t e d but i n t e r e s t i n g f a c t . Theorem 2: ([20]) J i s a Grossencharakter i n the sense of a, c Hecke. ^ Note that J w i l l not be of f i n i t e order i n general. I t would be a,c i n t e r e s t i n g to extend t h i s theorem (somehow) to non-abelian Jacobi sums. Relations i n Ideal Class Groups For t h i s section, l e t E be a p r i m i t i v e mth root of unity, m > 3 m J — and l e t K = 0.(5 ) . Let S be the set of archimedean primes of K m plus those d i v i d i n g m . For a e K , l e t S(a) be S plus a l l primes d i v i d i n g a . Let (—) be the power residue symbol for K, defined by e^K 'Va) = (j) ("Va) , (0 i s the global Ar t i n map 9: I ^ ( a ) -*• Gal(K ( m v/a) ,K)) . (f) i s defined only for P I S(a) . - 34 -ord b Prop. 7: ([CF] Exercise 2.8). If P i S(a), (|) Y = (a , b ) p , where (a,b) p i s the H i l b e r t symbol for Kp . Now f i x P i S and choose an element TT p of Kp with o r d i r = 1 and a set of representatives {x} of U /U^ X\ that • a l l l i e i n K . For such x, P I S(x) and t \"p»^/p \">"p/ \p/ \p t -1 t x t x X ^OO = ( ^ ' X ) ^ = (x,^) = (-) = (—) . This holds f o r any ^P /m t e Z . Since Kp(m/rTp) i s t o t a l l y ramified, by l o c a l Kummer theory, K p ^ / i T p 1 " ) i s ( t o t a l l y ) ramified i f t $ 0(m) and so x _ t i s ramified (and tame) i f f t t 0(m) . F i n a l l y , note that ^P Tr>(x <_) = ('^•J'> d )G (x .) for a sui t a b l e y . Let us write r —t r Jv_, y — t TT p "P TT p G t for G (x _ t ) • Let p A be the r a t i o n a l prime under P . We 7 ^P can give 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 . F i r s t , Lemma 6: (Generalized Euler C r i t e r i o n : [CF] Exercise 1.4). For P i S(a), (—) i s the unique mth root of unity such that N(P)-1 (~) = a m mod P . It i s not hard to see that only prime d i v i s o r s of P i n K(£ ) P 0 appear i n the f a c t o r i z a t i o n ( i n K(£ )) of G , for i t s absolute P 0 • norm xs a p^-power. Prop. 8. If p n = l(m) and t £ 0(m) then factors the same as y 1 fc i n d p 0 j 3 , > E E where ind i i s the index of i i n (Z/p.Z) j = l p0 P 0 0 and E. i s a p A t h p r i m i t i v e root of unity. Pn 0 Proof: If p n = l(m), p n s p l i t s completely i n K so that {1,...,PQ-1} i s a system of representatives for Kp Hence V 1 G = \ (—) t g(jy) • But g i s a non t r i v i a l character of t j = l P K_, - 2/pn2? so g(y) = 5 some P n - t h p r i m i t i v e root of unity. r U PQ U _ * By ([SI] XIV §3 lemma 1) the order of i n (K^) i s the same as i t s order i n K^, m . Therefore i f z i s a generator of (K^) PQ 1 i n d p j E = z m mod P . Obviously j = z ^ mod P and so m ind j PO" 1 P 0 " 1 Pn • E. = j m mod P . By Prop. 7, (^ ) = j m mod P . Putting m r a l l of t h i s together we get the r e s u l t . • For p n = l(m) the f a c t o r i z a t i o n of Gfc i s due to Stickelberger and i s the source of the only two theorems on r e l a t i o n s i n i d e a l class groups of cyclotomic f i e l d s - the Stickelberger Theorem ([12] §4.2) and that of Mackenzie ([13]). We w i l l examine these now. Let a. be the element of G = Gal(K,Q) such that a.(£,J) = E . 3 j m m Suppose p„ = l(m) and l e t {P.},. v , -, ^ . ^  be the prime d i v i s o r s v v F0 3 (3,m;=l,l<j<m of PQ i n K, ordered so that a (P ] L) = P . Let P = P 1 . The V1 P.'s ramify completely i n K ( E . ); P.0 W_ . = P. for some P. 3 Pn 1 ) J 3 u p0 Let a)(n) be the l e a s t non-negative integer congruent to n mod m - 36 -Theorem 3: (Stickelberger [15] p. 355). G factors as (p n-l)(l-a)(jt)/m) n P- U i n K(E ) . j J P 0 • The group r i n g Z[G] acts on K and on the i d e a l class group of K i n the obvious way and we write t h i s action exponentially. Let 0 e Q[G] be - £ u>(jt)o\ . (j,m)=l 2 l<j<m Lemma 7: Let 6 e Z[G] such that 60 e Z[G] . Then G e K Theorem 4: (Stickelbergers theorem). Every element of 0 Z[G] n Z[G] anni h i l a t e s the i d e a l c l a s s group of K , Proof: Choose a prime P = P^ i n each i d e a l class so that the r a t i o n a l prime p n under i t i s =1 mod m . I t s u f f i c e s to show 66 that P i s p r i n c i p a l . Claim: G^  e K . By Prop. 6, ^t^2t ± S a S U m °^ m t ^ r o o t s °^ unity. By induction ^ t / ^ t e K ^ o r a n y k . But 2 P 0 _ 1 G = E, +5 +...+5 = - l > since x i s of exponent m m t P 0 P 0 P 0 ir" The claim i s proved. By theorem 3, G™ factors as ~~m0 „ T 1m(l - c j(jt)/m) „ _m - t d ( t j ) m t II P = I I P . n P . = p„ 0TrP.. m K . Now suppose 1 3 . j 0 K 1 1 J 3 3 60 t £ Z[G] . Since s by lemma 7 4 Gfc e K we have G t P 0 °K = P l Gt P0°K = P l 1 3 P r i n c i P a l -• - 37 -Proof of lemma 7: Let L = K ( G ) . Let [L:K] = d . Now d|m since L = K ( m / G ™ ^ ) i s of exponent m over K . Since G ^ -m<50t m g f a c t o r s as P p 0 and 69 e Z[G], G factors as the mth 1 U K t t power of an i d e a l of K . Hence, l o c a l l y , f or any prime q of K and Q of L d i v i d i n g q, L„ = K (mJu~) where u e U . By x Q q q q q Local Kummer theory ([AT] Ch. 6 §2 Th. 4), i f q i s ramified i n L i t must divide m, i n which case q | p n - l , since m|p n-l . However L c K(£ ) and K(£ ) i s t o t a l l y ramified over K at each prime ~ P 0 P 0 P of K over p n and unramified elsewhere. So for any P|PQ> p|m|p -1, a contradiction unless d = 1 . 0 N Theorem 5: (Mackenzie [13]). Let [r] denote the greatest integer < r , and for any s , t , j e 1 l e t f ( j , s , t ) = [ j ( s + t ) ] - [^] - [^] — m m m Then II a C O f ^ ' S ' ^ i s p r i n c i p a l f or a l l s,t e 2 . 3 Proof: We may assume neither s nor t = 0 mod m, for i n that G G s t case the theorem i s t r i v i a l . We have p e K by Prop. 6 . s+t The f a c t o r i z a t i o n of t h i s quotient i n K(£ ) i s P 0 (p -1) (l+w(j (s+t))/m-w(js)/m-o)(jt)/m) I P. . Hence i n K i t i s j 3 Nl+o)(j (s+t))/m- ( i)(js)/m-a)(jt)/n A T . t, „ II a.(P^) J J J . Now notice that u(r) = r - [—]m for any integer r and r e c a l l II a. (P..) i s m j 3 1 - 38 -p r i n c i p a l to get the r e s u l t f o r P^ . In general the r e l a t i o n s follow on choosing an appropriate P^ i n each i d e a l c l a s s . Prop. 9: In case m = q i s prime, Mackenzie's r e l a t i o n s have the following form: • J e Y t ( T s ) a. (ft) i s p r i n c i p a l (1 <_ s, t <_ q - 1 ) where - 1 1) Y T 1 S the permutation of { 1 , 2,...,q-l} sending k to kt """mod q 2) T s = { j | f ( j , s , l ) = 1} . 3) T g has " ^ 2 ^ elements i f s ^  q - 1 T has q - 1 elements i f s = q - 1 . s ^ 4) If 1 <_ s < q - 1 j e T i f f q - j i T . Proof: Without loss of generality we may assume l < _ s , t < _ q - l . Also f ( j , s , t ) = 0 or 1 since i t i s of the form [x+y] - [x] - [y]. Since, for any SL, f(j,£s,£t) = f(£j,s,t), we have {j|f ( j , s , t ) = 1} = Y t ( T _ ±) • Conversely Y f c(T £) = {j | f (j , s, t) = 1} st f o r s = Y _•, (£) • This gives 1) and 2 ) . In case s = q - 1 , 3) t i s t r i v i a l . To see 4) and hence 3), set {x} = x - [x] . Now f(q-j,8 , i ) = [ ( g t l ) (i" 1 )] - [£ia=il] - i t l . _ { _ X s ± I ) l + ( s + 1 ) } + q q q q {-§1 + s} = £ ! = ! _ ( ! _ { ( s + 1 )J}) + ( i - {*!•}) . This holds because q q q q q/(s+l)j and qjs j => = 1 - { ^ 1 } and {=f-} = 1 - {f-}; also {x+n} = {x} i f n e 2 . Therefore - 39 -f<q-j,s,i> = 1 - 4 - { - ^ } - e » = ! _ ( [ i l i y i j _ [ILL]) = 1 _ f ( j , s , i ) , 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 proof) although i t i s a l i t t l e more complicated to state. If we consider i d e a l classes as represented by primes P^ as above, what Prop. 9 e. says i s that the r e l a t i o n s are a l l of the form " II P.J i s (j,m>=l J p r i n c i p a l " where e. = 1 half the time and e. = 0 h a l f the time, ^ 3 3 except i n the t r i v i a l case • = 1 for a l l j . Now l e t us consider global root numbers for global r e c i p r o c i t y symbols. F i r s t we need some fa c t s about l o c a l extensions. F i x a p-adic f i e l d K . An E i s e n s t e i n polynomial i s one of the form E(X) = X m + a .X™" 1 + . .. + a n with ord T,(a.) > 1 for 1 < i < m - 1 v ' m-1 0 K l — — — and ord^Xag) = 1 . Prop. 10. ([CF]I§6 Th. 1). a) An E i s e n s t e i n polynomial i s i r r e d u c i b l e . If TT i s a root of E(X) then L = K(TT) i s t o t a l l y ramified and ord^ir = 1 . b) If L i s t o t a l l y ramified over K and ord (ir) = 1, then JLi the minimal polynomial for TT i s E i s e n s t e i n and L = K(ir) , Prop. 11: ([ I ] IV §2.2). Suppose K contains mth roots of unity with m prime to the c h a r a c t e r i s t i c of K . Let ord (TO = 1 . - 40 -Then there e x i s t two Kummer f i e l d s of exponent m over K namely K(5 /'ATS~D\ a n d K( m/rT) such that every Kummer f i e l d of exponent ml, IN ( f ) - x ) m over K i s contained i n t h e i r composite. The f i r s t of these f i e l d s i s unramified. The second i s t o t a l l y ramified. • Let K now be a number f i e l d containing mth roots of unity. ft For b e K , define the global r e c i p r o c i t y character (b,-) by (b,x) = n(b,x p) p, the product taken over a l l primes, including archimedean ones. (x i s any i d e l e here.) (b,-) i s an i d e l e ft c l a s s character: n(b,a) = 1 for a e K ([CF] Ex. 2.9). P P We w i l l choose b to s a t i s f y a c e r t a i n condition: (*) For a l l p|m, ord pb = 1 . . . ft Such a b i s found by f i r s t choosing, for each p|m, b(p) e K such that ord b(p) = 1 and i f p' ^  p, p'|m, ord ,b(p) = 0 . P P Such a choice i s a v a i l a b l e by weak approximation. Then set b = II b(p) . b s a t i s f i e s (*) . For any b, l e t B™ = b . p.|m Prop. 12: If b s a t i s f i e s (*) then for pIm K (B)/K i s t o t a l l y c P P and w i l d l y ramified. " i- * * m Proof: For any p m, b i s order m i n K /(K ) , for i f — p p b = x m, r = r ord b = m ord x => m|r . Hence [K (B) : K ] = m, P P 1 P P so i f Kp ( 8 ) i s t o t a l l y ramified, i t i s w i l d l y ramified. But X m - b i s Eisenstein for K so the r e s u l t follows from Prop. 10a) P • - 41 -Prop. 13: For any b (not ne c e s s a r i l y s a t i s f y i n g (*)) a) If K (B)/K i s tame, so i s (b,-) P P P b) If K p(B)/K p i s t o t a l l y and w i l d l y ramified, then (b,-) i s w i l d l y ramified. Proof: a) has been done. For b), l e t g^ = order of G^(K p (B),K p) (the r a m i f i c a t i o n groups). Let F p be the maximal unramified extension of K contained i n K (B) By lemma 4, the l e a s t r such P P (r) * that U i s contained i n N . . (K (B) ) i s at l e a s t *P p ^ ' p p HD + 1 = S- 1^^ > 1 • But N ( u £ > ) = U < r ) by P p P p * lemma 3a). Hence the l e a s t r such that N , . , (K (B) ) i s K p (B)/K p p (r) contained i n U i s also > 1 . Since the kernel of (b,-) K P P * 21 i s exactly N(K p(g) ), we have p |fl((b,-) p) . Now suppose b s a t i s f i e s (*). If pjm pjb, then (b,-) i s unramified, as K (B)/K i s . If p|m (b,-) i s w i l d l y P P P ramified by Prop. 12 and 13. If pjm, p|b, then we claim K (B)/K i s t o t a l l y and tamely ramified. For we may write P P b = u i r r with u e IL, and r 4 0 . If r = 1, we are done P P P K p by Prop. 11. If r 4 1, then K (B) <= K ( m/u IT , M/TT R _ 1) . But P _ P P P P K ( m / i T r X ) c K (mZn~) hence i s t o t a l l y ramified over K . By P P - P P P Prop. 11, K ( /u TT ) i s t o t a l l y ramified over K . So K (B) P P P P P i s contained i n the composite of two f i e l d s t o t a l l y ramified over K and i s therefore t o t a l l y ramified over K P P • - 42 -What has been done i s to i s o l a t e exactly the places where (b,-) i s unramified, tame and wild. Prop. 14: If S(b) = S, i . e . p|b => p|m, then the global root number W((b,-)) i s a root of unity. Proof: W((b,-)) = II W((b,-) ) which i s a root of unity i f no P P component (b,-) i s tame. p • Example: Let K = Q(E. ) and p = unique d i v i s o r of p^ i n K Let TT = 1 - E, . Consider (TT ,-) and (E. ,-) . Prop. 14 P p o P p0 applies to the former and for the l a t t e r , i f L = Q(£, n + ^ ) = K(F0/E. r ) , PU P 0 L i s a Kummer p^-extension unramified outside p and t o t a l l y and w i l d l y ramified at p . Hence the root numbers of both these characters are roots of unity. Indeed the same holds for b = (1 - E ) s ^ ( s , t e Z) . P0 P 0 • - 43 -§6. Behaviour of Gauss Sums With Respect to R e s t r i c t i o n In t h i s section we consider subgroups (usually normal) of Galois groups of l o c a l number f i e l d s and t r y to f i n d a r e l a t i o n between the Gauss sum of a character of the group and the Gauss sum of the r e s t r i c t i o n of that character to the subgroup. The s t a r t i n g point i s a theorem of Hasse-Davenport ([ 9 ] ) : Let W be the f i n i t e f i e l d with q elements and N: IF x —> IF be the norm map. Let q f q ft y e IF . Then G (ct*N) i s a character sum over IF ^, for any q y q ft character a of IF q Theorem 6: ( [ 9 ] §3.1). - G (a«N) = ( - G ( a ) ) f . 7 7 • Now l e t L/F be an unramified extension of l o c a l f i e l d s and write ft for a • N^/p" w n e r e a i s a character of F If a i s tame, so i s a by lemma 3a). Let p be the prime of F, P the J_i prime of L . We have T_(a ) = a (d )6 (a #N) with F l_i L Li y x = T r L / L n ^ dL l p ( p a n d T p ^ = a ( d F ) G y ( ° 0 w i t h y = T^p.p (dp^p^). (See §4). Since L/F i s unramified we may nr , . l + o r d L 6 L take TT = TT 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 r L L 1 + 0 r d L 6 L / K + ° r d L 6 K 1 + ° r d K 6 K . T .. v... . TT = T r T r = d„ . Since L n F and L K K nr nr L n r F = L, we see that y = Tr ( d ^ p ) = Tr ( d ^ p ) = nr hr — — — TL *Fl y e F . F i n a l l y , a^(d^) = a(d^) ' and we have Cor. 1: If a i s tamely ramified and L/F i s unramified of degree n, then T (a ) = ( - l ) N + 1 T ( a ) n . n P L p • - 44 -Cor. 2: If a,L and F are as i n Cor. 1, W(a ) = ( - l ) n + 1 W ( a ) n . L • Let us say that L/F i s t o t a l l y w i l d l y ramified i f the maximal at most tame extension of F contained i n L i s F i t s e l f . Prop. 15: If L/F i s t o t a l l y w i l d l y ramified and a i s tame, then x p ( a L ) ^ T p ( a ) • Proof: ([18] §2 lemma 2). Since L/F i s t o t a l l y ramified and a i s tame, we claim a i s tame: Let q = order of F . If JLI X a e Up - Up1^ and OJ(XQ) =f 1 then, since x I—*- x^"^"' i s an automorphism of F (by the f a c t that [L:F] i s a power of the c h a r a c t e r i s t i c of F) , XQ = N L / p ( x ^ = x'' 1"^ e u p ~ Up"^ > f o r some x e U . Hence a T(x) 4 1 and a T i s tame. Now I? Li LI T (a ) > ^_a L(x)g-(x) r xeL V / ^ / [L:F]. . . [L:F] . 'v l_a^(x)g-(x ) (since x t—> x i s an automorphism xeL _ _ of L = F) V , [L :FK , [L:F]. xeF 'v £_a(x)gp-(x) xeF ^ T (a) . p • Prop. 16: Suppose L/F i s t o t a l l y and tamely ramified of degree n and a i s tame. Then T (a T) ^  T (a 1 1) . P L p Proof: Case I: a n i s unramified: Let x e U . Since L/F i s t o t a l l y ramified, there e x i s t s y e Up such that y = x mod U^"^ . - 45 -Therefore y 1 1 = N ^ C y ) = N L^ p(x) mod I L ^ . So « L(x) = a n(y) = 1 and a (U ) = 1 . x (a ) and x (a 1 1) are both roots of unity. L L 1? LI P Case I I : a1 i s unramified: Let a(x n) 4 0 for some x e U-.-U^^ . r r Then a (x) 4 1 and x e U - 11^ "^  so a i s ramified. We have J_i J_i JLJ J_I x^Ca ) = a (d ) £ a(x U)i(j (d "^"x), since a set or represent-xeUp/Up^ atives of U.f^ i n U may be taken from U . Now L L r T r T (d/St) = Tr„(x T r T , ^ ( d 7 1 ) ) f o r x e F . Since L/F i s t o t a l l y and tamely ramified, we may take TT^ = Tr^  and ord^(6^^) = n - 1 . (Recall ([CF]I§5 Th. 2 i i i ) ) that i f L/F i s tame, o r d L ( 6 L ^ p ) = e(L/F) - 1) . We have l+ord T 6T ... l+ord T6 ,_+ordT 6„ n+n ordT,6„ , L L L L / r L r r r L = ^L = V = \ 1 + o r dF6 F ^F = d F * Therefore T r ^ d ^ x ) = T r p ( x T r ^ C d " 1 ) ) = T r ^ n x d " 1 ) and \bT (d T "*"x) = ^(d-^nx) for x e F . Hence L L r r x p ( a L ) = a(d p) £ a ( x ) ^ F ( d p nx) . Since (q,n) =1, x K n x ^ u F / u F 1 } i s an automorphism of TT /U^"^ and so x_,(aT) a, x (a1) . h i P L P rj Theorem 7: Let a be an at most tame character of the p-adic f i e l d K and l e t L/K be any f i n i t e normal extension with Galois group G . Let g(L/K) be the index of the wild r a m i f i c a t i o n group of G i n i t s - 46 -i n e r t i a group. Let f (L/K) = [L:K] . Then t f c O <v, x ( a g ( L / K ) ) f ( L / K ) Proof: L/K has a f i l t r a t i o n L £ F £ E £ K where F i s the f i x e d f i e l d of the wild r a m i f i c a t i o n group and E i s the f i x e d f i e l d of the i n e r t i a group of G . [F:E] = g(L/K) and [E:K] = f(L/K) . L/F i s t o t a l l y w i l d l y ramified, F/E i s normal, t o t a l l y and tamely ramified and E/K i s normal, unramified. If a i s unramified so i s ag(L/K) a n < j T_(OL.) and T ( a ^ ^ ^ ) are both roots of L P L p unity. I f a i s tame, the r e s u l t follows by successive a p p l i c a t i o n of Prop. 15, 16 and Theorem 6. • C o r o l l a r y : Let x e R-^(K), x a t most tame. Let Res x be the r e s t r i c t i o n of x to tt c Q . Then x_,(Res x) ^ T ( X 6 ^ ^ ) F ^ L / / R ^ L K Ir p Proof: Since T (X ) remains inv a r i a n t under l i f t i n g , we may consider ab at) X and Res x a s characters of tt and tt r e s p e c t i v e l y . I t i s K L a property of c l a s s formations (see [AT] Ch. 14 §5) that K K 0ab > a K * L * — ± > nfb R e s X : s i commutes, where i . i s induced by i n c l u s i o n and 6 and 'K are l o c a l A r t i n maps. The r e s u l t follows on r e c a l l i n g how R-^(K) i s imbedded i n R(K) . • Any attempt to generalize t h i s r e s u l t to R(K) d i r e c t l y must involve looking at the r e s t r i c t i o n of an induced representation. For - 47 -t h i s we need the so-called subgroup theorem of elementary represent-at i o n theory: Let H and T be subgroups of G and a a one-(s) dimensional character of H . For s e G, l e t a be the character of sHs 1 n T defined by a^S\t) = a(sts "*") . Write T Res„ for r e s t r i c t i o n of representations from G to T . The (j following i s a s p e c i a l case of the subgroup theorem. Theorem 8: ([S2]§7.4 Prop. 15). If p = I n d S * then T T (s) Res p = # Ind _^ (a ) where the sum i s taken over a set of s sHs . nT representatives {s} of the double cosets H\G/T . Now suppose H i s abelian and T i s normal i n G = Gal(L,F) Lemma 8 : T ( a ) = x(Res a) . r i Proof: Let K(resp.K' resp.K" resp.K"') be the fixed f i e l d of H(resp. sHs \ resp. sHs 1nT, resp. HnT). Note that L/K. L/K', L/K" and / \ L/K"' are a l l abelian. We have K' = sK and since sHs _ 1nT = s(HnT)s _ 1, K" = sK"' . • K"' K" K K' By cla s s f i e l d theory - 48 -K -1 Y K' 4 K" Gal(L,K) 7 Yi > Gal(L,K') - sGal(L,K)s (s) Gal(L,K") -1 a commutes, where Y-^(t) = s t s ^ a n d Y(x) = sx . (This i s a property of c l a s s formations - [AT] Ch. 14 §5 Th. 6). From t h i s i t i s clear (s) -1 that a (x) = a(s N K„^ K,(x)) considered as l o c a l f i e l d characters. Since T i s normal and L/F i s normal, a) If T i s a set of representatives of HnT i n H, sTs i s a set of representatives of sHs "*"nT i n sHs . -1 b) o i s an imbedding of K" i n (Q ) over F i f f s "*"os ^ P g s e p i s an imbedding of K"' into (Q ) over F . P 0 sep c) ord „sx = ord „,x for a l l x e K ' " . K K d) A l l r a m i f i c a t i o n subgroups of sHs "''nT and HnT are s-conjugate and i n p a r t i c u l a r of the same order, (s) e) a i s tame, since y preserves the units f i l t r a t i o n . By a) s ^ . ^ . ( x ) = N K „ I / K ( s Xx) . By b) T r K , I / F ( x ) = s T r K „ I / F ( s ~ 1 x ) = T r K „ , / F ( s _ 1 x ) => ipK„(x) = ^ K„,(s 1x) for x e K " . By c) and d) ord L(6 L^ K„) = ord L(<5 L^ K„,) ( d i f f e r e n t s are determined by the orders 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 K „ , = s "'"d^.,, . We have K" - 49 -r ( a ^ S ; ) = a w ( d K „ ) £_a(s \ , / K ( x ) ) ^ , ( < i R „ x ) xeK" = a(s 1 N K „ / , K , (d K„)) I _ a ( \ n t / K ( s 1 x ) ) i J ; K l I I (s "'"(d^x)) xeK' a(N K„, , K(d K„,)) I a(N K„, / k ( x ) ) ^ k . M (d R?i,x) xeK"' T(a-N K, I t ^ R) x(Res a) ri • Lemma 9: (cf. Lemma 2). Let G = Gal(E,K) and l e t H be an abelian subgroup of G with fixed f i e l d L . Let Dj_,/K be the discriminant of L/K, a a one-dimensional character of H and p = Ind^a . Then x(p) ^ N R / (D^, R)^x(a) . P 0 G G —1 Proof: We have x(a) =' x(a-l) = x(Ind (a-1)) = x (p) x (Ind T Tl) ri H since dim(a-1) = 0 . However, for any character 0, x(0)x(0) = N(f}(0))det. (-1) ([14] Prop. 4.1). Since Ind^l has r e a l character, u ri G 2 G x( I n d H l ) = (j$(Ind Hl)) = N K / / Q ( D L / K ^ b y t h e induction formula P0 p0 for the A r t i n conductor (see §1C)) . • Indeed the quotient i s c l e a r l y a 4th root of unity. Let N stand for absolute norm as usual. Q Prop. 17: Let p = Ind a with H abelian and a tame. Let T H be normal i n G . Let - 50 -g = index of wild r a m i f i c a t i o n group of H/HnT i n i t s i n e r t i a group f = r e l a t i v e degree of H/HnT m = number of double cosets H\G/T L = fi x e d f i e l d of HnT K = fixed f i e l d of T . Then T f R e s ^ p ) ^  N ( n ,„) m / 2x ( a g ) f m . P G L/K p Q T T (. s} Proof: By Theorem 8, Res p = $ Ind (a ) where the sum s sHs nT i s taken over a set of representatives of H\G/T . We may replace N ( D s L / K ) with N(D L / K) and so x p(ResJp) ^ n [ x p ( a ( s ) ) N ( D L / K ) % ] = N ( D L / R ) m / 2 n x p(Res|J n Ta) <\- N ( D L / K ) m / 2 T ( a g ) f m by Theorem 7 and lemmas 8 and 9. • Of course Prop. 17 and Brauer induction give us some kind of statement r e l a t i n g x(Res p) to x(a)'s f o r various a's whose induced representations are components of p . However, since induction i s not simply rel a t e d to products i t would be d i f f i c u l t to f i n d a r e l a t i o n of the form "x(Res p) ^  (some simple function of) x(p)", unless g = 1 (for the various g's). Another d i f f i c u l t y i s the requirement that p be a representation of an at most tame Galois group. (Prop. 17 applies i n t h i s case for H w i l l be at most tame and so w i l l a .) However a d i f f e r e n t approach y i e l d s a r e s u l t i n the unramified case without e x p l i c i t use of Brauer - 51 -induction. The requirement of tameness can be seen to come from a d i f f e r e n t d i r e c t i o n . We w i l l take t h i s up i n §7. D e f i n i t i o n : Let x he the character of a representation of 0. K K a number f i e l d . The global Galois Gauss sum T(X) i s the product II T p(Xp), taken over a l l f i n i t e primes. Here, x p i s the r e s t r i c t i o n of x to Q, . - 52 -§7. The L o c a l l y Free Class Group, Resolvents, and Gauss Sums This section i s a b r i e f summary of some work of F r o h l i c h on the r e l a t i o n s h i p between these three constructions. These theorems allow us to i n f e r f a c t s about Galois module structure of the integer r i n g of a tame extension from information about the resolvents and Gauss sums of characters of i t s Galois group, and vice-versa. We w i l l use some of t h i s material to prove a generalization of Theorem 6 Cor. 1, to the g l o b a l , non-abelian case. Let N/K be a f i n i t e , at most tame extension of number f i e l d s . This assumption w i l l hold throughout. Let T = Gal(N,K) . We say C L , i s l o c a l l y f r ee rank n i f 0 * ' i s isomorphic to 0 „ ( r ) n N N,p 1 7 K,p for a l l p . Theorem 8 . (E. Noether). N/K i s at most tame i f f 0 ^ i s a l o c a l l y free (rank 1) 0 (T) module. K • Let K ( 0 (T)) be the Grothendieck group of the category of 0 K f i n i t e l y generated l o c a l l y free 0 (T) modules. This i s the free 3) abelian group on stable isomorphism classes of l o c a l l y free 0 (T) modules, mod the subgroup generated by (M#N) - (M) - (N) ((M) stands for stable iso c l a s s of M). The notion of rank extends to K Q ( o K ( r ) ) . 3) M i s stably isomorphic to N i f there i s an such that M <& M - N <& M 1 . 3') 0N, p d e m o t e s s e f v , i - l o c c x V c o m f l e V i o ^ - 0 K , p ^ e " ° " ^ s c o ^ p k + . o n . - 53 -D e f i n i t i o n : The l o c a l l y free class group, C£(0 (T)) i s the subgroup of K (0 (T)) cons i s t i n g of those elements l o c a l l y free U K of rank 0 . The l o c a l l y free c l a s s group i s s i m i l a r l y defined for any 0 -order i n a f i n i t e dimensional semi-simple K-algebra A . By an 0 order i n A we mean a subring of A, which i s a f i n i t e l y K generated t o r s i o n - f r e e 0 -module containing a basis of the K-K vector space A . In p a r t i c u l a r we may consider, for K = Q, the maximal Z order i n Q(r) which we w i l l c a l l M . There i s a map C£(Z(r)) —> C£(M) given b y (M) —> (M ® Q(T)) . The kernel z ( r ) of t h i s map i s c a l l e d D(Z(T)) . Let us write ( M ) n for the u K ( r ) rkM c l a s s of M - 0 R ( r ) i n C £ ( 0 R ( r ) ) . Theorem 9: ( [ 3 ] Th. 3). ( < V z ( r ) e D(Z(D) , when K = Q . Let us sketch a proof of t h i s theorem as i t seems to require a l l the concepts and major theorems of the theory. The l o c a t i o n of 0 i n C £ ( 0 R ( r ) ) measures how far away 0 N i s from having a normal i n t e g r a l basis over 0 . 0__ has such a basis over 017 i f f 0.T K N K N 4) i s free over 0 R ( r ) i . e . (° n)Q ( r ) = 1 ' K The l o c a l l y free c l a s s group has an a l t e r n a t i v e d e s c r i p t i o n . Let R„ be the r i n g of v i r t u a l characters of T . This i s an Q 1 K • • N ^ T h i s doesn't hold f o r a r b i t r a r y N/K tame, although i t i s conjectured t h a t <Va(D " 1 • - 54 -module v i a X^CY) = X ( Y ) W for y e E, m e ft , since X (Y) are algebraic numbers. Let us f i x , once and for a l l , an absolutely normal number f i e l d E, large enough to contain K and a l l values of a l l characters of r . Indeed l e t us suppose that any represent-ation of T can be considered as an E-representation. E and i t s adele group w i l l be a "depository" for various functions on R^ . For any number f i e l d L, write Ad(L) for the adeles of L, J(L) for the i d e l e s of L, U(L) for the unit i d e l e s and fo r II 0 , the product taken over a l l primes. Let p L ' p ft U(0 „ . ( r ) ) = n 0„ (T) (where "*" always denotes the i n v e r t i b l e p K ' p elements of any r i n g ) . If T: T —»- GL(E) i s a representation, T can be extended to a K-algebra homomorphism T: K(T) —>• GL(E) or even to a representation of Ad(K)(T) = Ad(K) ® K(T), (the group r i n g of V K over Ad(K)) into GL(E ® Ad(K)) = GL(Ad(E)) . Let x be the K character of T . x extends along with T and we may define det A = det T(A) for A e Ad(K)(T) . In p a r t i c u l a r det A i s X X i n v e r t i b l e i n Ad(E) i f A i s i n v e r t i b l e i n Ad(K)(T) . So det i s a mapping from U(0 (T)) to J(E) . Now we may define X K-a map det: U(O i r(D —> Hom0 (R r,J(E)) by sending A to K X i—> det A . (det ,(A) i s defined as (det A ) ( d e t , A ) - 1 .) D e f i n i t i o n : (-Theorem): C£(0 K ( r ) ) = Hom^ (R r,J(E))/Hom f i (Rp,E*) .• det U ( 0 R ( r ) ) . K K - 55 -For a proof that t h i s coincides with the e a r l i e r d e f i n i t i o n see ([ 4] Appendix I ) . Now we must define global and a d e l i c resolvents. For x e Rj. and a e N generating a normal basis {a } of N/K (Hilbert l v Y — 1 Normal Basis Theorem) define (a | x ) = det( l a T(Y) ) where yeT X = character of the representation T . (a | x ) i s the global  resolvent. For a d e l i c resolvents, l e t a e Ad(N) generating a l o c a l normal i n t e g r a l basis everywhere, i . e . aA^ = A^ . Extend T to a map Ad(K)(T) —> GL(Ad(E)) and define the a d e l i c resolvent by (a | x ) = ( a l x ) N / K = det( I a T T ( Y ) _ 1 ) e Ad(E) . If a e N yeT then t h i s i s the same as the global resolvent. I t i s not hard to see that resolvents are i n v e r t i b l e elements i . e . (a | x ) e E and (a | x ) e J(E) ( [ 4 ] Prop. 1.2 Cor. 1). Next we need the notion of the norm of a function f e Horn (R_,J(E)) . If K _£ k and {a} i s a set of represent-" K -1 atives of i n tt^, define ( \ / k f M x ) = n f ( x ) . \ / k a i s a map from Horn (R ,J(E)) to Horn (R ,J(E)) . In p a r t i c u l a r K k the functions x ( a I x) a n d X *~*" ( a | x ) have norms. Changing the choice of {a} changes the d e f i n i t i o n by a homomorphism X t~*• (root of unity) . The next notion i s that of a family of i n v a r i a n t s . Roughly speaking, these are maps b from R^ , to the i d e a l group of E such that b( x ) i s generated by the image f ( x ) for some function f e C£(0 (T)) . However, the following modification i s necessary, K. even c r i t i c a l : Let J(E,r) be the subgroup of i d e l e s of E whose - 56 -components at a l l i n f i n i t e primes of E and at a l l d i v i s o r s of any-r a t i o n a l prime d i v i d i n g the order of T, are 1 . Let U(E , r ) = U(E) n J ( E , D . J ( E , D > J(E) induces Hom0 (R ,J(E,T)) — y Hom_ (R ,J(E)). —*• C£(.0 V(D) . By approximation, t h i s i s s u r j e c t i v e and one can show that Horn (R ,U(E , r ) ) maps into det U(0 (T)) . On the other hand we have an isomorphism K Horn (R ,J(E,T))/Horn (R ,U(E,T)) - I where I i s the set " K K U K , r . ° K , r of ft homomorphisms b from R r to the f r a c t i o n a l ideals of E such that i ) b (x) has numerator and denominator prime to the order of T and i i ) b (x) i s a f r a c t i o n a l i d e a l of 0 , . M X ) (K(x) always denotes the lea s t f i e l d containing K and a l l values of X )• This isomorphism i s given by f r—»• b^ where b^(x) i s the f r a c t i o n a l i d e a l generated by the image of f i n J(E,T) . So we get a s u r j e c t i o n I —>->• C£(0 (T)) and i f H denotes i t s ° K , r K ° K , r kernel, an isomorphism I /H ~ C£(0 (T)) . A family of K,T K,T • invariants f o r an element (M) of C£(0 (T)) i s by d e f i n i t i o n a K representative for (M) i n I Prop. 18. (Galois a c t i o n on Gauss sums and resolvents, [ 4 ] , [14]). Let ver . : ft^ —>• ft^ be the transfer map. Then f o r x e R,,, K/ Q 1} K 1 U) £ ftn, vi. -1 0) a) x(x ) = x (x)det ( v e r K ^ ( u ) ) . - 57 -b) If a e N s a t i s f i e s aK(T). = N -1 i ) ( a | x W ) = (a | x)det co A . -1 i i ) N K / Q ( a | x U ) W = N K / q ( a | X ) d e t x ( v e r K M ( o ) ) ) i i i ) ( a A | X ) = (a | x)det X for X e K ( D * 0 If a A K = A N -1 i ) C a l x " ) W = (oi | x)det a) A . i i ) N K / q ( a | x W V = N K /^(a| X)det x.(ver K / q(a>)) • Now define the resolvent module (0 : x ) • This i s the 0 , N K KAX-> module generated by a l l (a | x ) with a e 0^ . Define N K/Q.(°K : X ) as the <-'Q(^ ) module generated by N R ^ ( a | x ) with a e 0^ . These resolvent modules are a c t u a l l y f r a c t i o n a l i d e a l s . The connection with resolvents i s given by: (0 T r : x ) = (a x) 0„, N and K A / p P K ( x ),p N K / Q ( ° K : x ) p = N K / q ( a | x ) p % X ) , P ' a A K = ^ • Theorem 10 ([3 ] Th. 8). i ) Let a e N such that aK(T) = N . Let a e Ad(N) such that aA^ = A^ . Let f ( a ) = ( a | x ) ( a | x ) 1 whence ( N K / q f ) ( X ) = N K / ^ ( a | x ) N K / % ( a | X ) - 1 . Then f e Hom^(R r,J(E)) The cl a s s of f i n C£(0^(r)) i s (0. T). . The class of K N 0 K ( r ) W i n C ^ r > ) i s ( 0 N ) z ( r ) . i i ) With a as above, assume that for a l l prime d i v i s o r s p - 58 -of order r, we have aO (r) = 0 . Let b ( X ) = (C> : X)(a|x) 1 , P JN , y Ik. Then (b(x)} i s a family of invari a n t s f o r (0 ) and N K(D {N w J } ( x ) } i s a family of invariants for ( 0 A 1 ) _ / T n , . Theorem 11: ([ 3 ] Th. 9). i ) Let a e Ad(N) such that aA^ = . Then the map x i-> T ( X ) N K / q ( a | x ) _ 1 i s i n Homfi (R r,U(E)) i i ) x ( x ) 0 q ( x ) = N K / Q ( 0 K : 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 of the theorem i s i n the (f a c t o r i z a t i o n ) statement that i t lands i n U(E) . From theorem 11 i i ) we see that the family of invariants b(x) of Theorem 10 may be replaced by the family {x(x)N Ry (a|x) }^ • Hence Theorem 12 ( [ 3 ] Th. 1). The map x T ( x ) N R ^ ( a | X ) _ 1 i s i n Horn (Rr-oE ) . If a i s chosen so that i f p divides order T, \ a0 R (r) = 0 N p , then the ide a l s b(x) = ( x ( X ) N R / ^ ( a | X ) _ 1 ) define a family of invariants for (0 ) . . . To prove Theorem 9, we need a chara c t e r i z a t i o n of D ( Z ' ( r ) ) as + * a subgroup of I . Let Horn (R_,E ) denote the set of f e Horn ( R ^ E ) such that i f x i s symplectic ( i . e . x i s r e a l \ valued and the representation of x preserves a skew-symmetric - 59 -b i l i n e a r form), f ( x ) 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 places; i n t h i s d e f i n i t i o n we i n s i s t that even i f p i s complex, f ( x ) must be r e a l and p o s i t i v e at p . Theorem 13: ( F r o h l i c h ) . Let P(Z ( r ) ) consist of those f a m i l i e s of i n v a r i a n t s generated by the images of functions f ( x ) for f e Horn* (R r,E*) . Then D(2 ( D ) » P ^ / H ^ . Now to see that ^iq^Z(T) 6 D ^ ^ ^ ' w e n e e d only show that the map x H - y ( T ( x)N^^(a | x') ^) of Theorem 12 takes symplectic characters to elements of P(Z(T)) i . e . 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. Theorem 14 ([3 ] Th. 2). If x i s symplectic, then ' T ( X ) N R ^ (a|x) 1 i s t o t a l l y r e a l , either t o t a l l y p o s i t i v e or t o t a l l y negative, with i t s sign independent of a and given by: sign(x ( x)N K / (^(a | x ) = W( X) Remark: In general W(x+x) = d e t (-1) • Since any symplectic X 2 character has t r i v i a l determinant, W(x) = 1 for X symplectic. From Theorem 14, Theorem 9 i s complete (modulo a l l the proofs 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 i s a proof of Stickelberger's theorem without reference to Gaussian sums. The theorem i s seen to r e s u l t from the existence of a normal i n t e g r a l basis for the cyclotomic f i e l d of mth roots of unity ( H i l b e r t -Speiser Theorem). See [5 ]. • - 60 -In order to prove the "global Hasse-Davenport" we w i l l draw only on Theorem 11 i ) and Prop. 18 together with the following theorem on r e s t r i c t i o n of resolvents: Theorem 15 ([ 3 ] Th. 11). Let A be a normal subgroup of r, with fixed f i e l d L and assume L/K i s unramified. Let a e A^, 3 e A N such that aA^(V) = A^ and BA^A) = A^ . Then there e x i s t s X e A^(T) such that (B|Resx)^^ = (a | x)^^ Rdet^X, where Res x i s the r e s t r i c t i o n of x e R r t o A . • Theorem 16: Let A be a normal subgroup of T with fixed f i e l d L and assume L/K i s unramified of degree n . Then the map xfResv) * x l—h A i s i n Horn (R ,0 ), where Res x i s the r e s t r i c t i o n T(X) n. of x to A . In p a r t i c u l a r x ( R e s x ) 0 K ^ = x(x) ° K( X) • Proof: (1) The map x ^ T ( x ) T ( d e t _ 1 ) i s i n Horn (R ,J (E)): X \ From Prop. 18 a), i f \ e R with det = 1, then X T(x) X i s i n Horn (R r,J(E)) . (1) follows on applying t h i s to x - det U% 1 X (One checks det = (det ) W ) . x w (2) There i s a X e A^(T) such that X v-f- x ( R e s x ) T ( x ) ~ n N K / ^ ( a | d e t x ) n N L / ( ^ ( a | d e t x ) " 1 N L / ^ ( d e t x X ) " 1 i s i n Horn (R ,U(E)): Let a, B and X be as i n Theorem 15. Now Res(x U) = (Resx)^ and so by Theorem 11, x l — > x (Resx)N L^(B |Resx) 1 i s i n Horn (R ,U(E)) . Let us write - 61 -T(Res X)N L /^(3|Res X) 1 = T(Res X) nK / Q [ N L / K ( ( A I * ^  I d e t x } } ] NL/Q. ( A I d e t v > 1 -1 X NT ,-(det A) x by Theorem 15. (Norm i s indeed t r a n s i t i v e ) . Let us W<i X consider the expression i n square brackets. Just as i n (1), or by Prop. 18 c i ) , the map X ^ (ct|x)(ot|det ) ^ i s i n X Horn (R , J ( E ) ) £ Horn (R ,J ( E ) ) £ Horn (R ,J ( E ) ) . Now for any ^ K L f e RomQ ( R r , J ( E ) ) , N L / R f ( x ) = f(x)n and so K. N L / K ( a | X ) _ 1 ( a | d e t x ) ) = (a | X ) ~ n ( a | d e t ^ 1 1 . Therefore X V x ( R e s X ) N K / q ( a | X) ~ \ / ^ a | d e t ^ N L / q ( a | d e t ^ ^ ^ ( d e t ^ ) " X i s i n Hom^ (R r,U(E)) . However x T(x)  U^/q^aI x)" l s i n Q. Horn (R JU(E)) by Theorem 11 again. M u l t i p l y i n g these l a s t two \ functions y i e l d s (2). x(Resx) f(det ) n (3) The map x ~— i s i n T(X) T(Res det ) A Horn (R , U ( E ) ) : This follows from (2) on replacing X by x - det \ 1 X Now l e t f = [ L ^ K ] = [L„:K ] . By l o c a l theory, p P p P p -f f +1 x (Res (det ))x (det ) P = e (x) where e (x) = (-1) P i f p P p X i s ramified and 1 i f x i s unramified. The various symbols P P are explained by the following diagram: - 62 -det * A r t i n „ ,AT „ Nab K > Gal(NT,,K ) P P p ( i n c l u s i o n ) * Res„(det ) * A r t i n •y Gal(N p,L p.) ab P v X Now T (Res det ) = II II x (Res (det )) and x(det ) = n x (det ) X P P|P XP X P P XP But II x (Res (det )) = II e (x) x (det ) n , since by normality P|p Xp P | p P P XP of A there are ^ r j primes i n L d i v i d i n g p . Set e(x) = n n e ( X) . We have X ^ T ( R e s X } e ( x ) _ 1 e Horn (R U(E)) . P P|p P T(X) \ However i t i s obvious that e ( x W ) = £ ( x ) = £ ( x ) W » a n d since x(Resx) and x (x) are algebraic integers i n Q(x)> the theorem i s proved. • C o r o l l a r y . Under the conditions of the theorem, the map X N K / Q(a| X) nN L / (^(B|Resx) 1 i s i n Homfi (R p,U(E)) . Proof: By Theorem 11, x l — x(Resx)N L /^(B|Res X) T ( x ) n N K / ( ^ ( a | x ) ~ n -1 i s i n Horn^ (R r,U(E)) • Remark: The Galois module compa t i b i l i t y of Theorem 16 i s e a s i l y v e r i f i e d without reference to Theorem 11. For we have x(Resxa') T ( R e s x ) a i d e t R e s x ( v e r L / Q ( a i ) ) C K x " ) x ( X ) d e t x ( v e r K / Q ( a ) ) ) Let \b = det and - 63 -ver /T, , (xncl) . , / N ^ab L/K 0ab * _ab . . . C J k = v e r K ^ ( w ) . Since ftK • ftL >- fiR i s r a i s i n g to the nth power ([SI] VIII §2 Prop. 4 and note t h i s sequence i s r e a l l y H^^.Z) —»- H-^Q^Z) H ^ f t ^ Z ) ) we have d e t R e 8 X ( v e r L / K ( u K ) ) = ^  M v e r ^ ) ) = * • ( i n d ) * • v e r ^ C i ^ ) = *(o T(Resv) * whence Y \—> — — i s i n Hoiri (R„,E ) with no conditions on T(X) Q L/K (or even N/K) . Coniecture: Under the conditions of Theorem 16, ^- i s a r \ n T(X) root of unity. Indeed i t i s probably a fourth root of unity. - 64 -Bibliography Books [AT] A r t i n , E. and Tate, J . Class F i e l d Theory, W.A. Benjamin, N.Y., 1967. [CF] Cassels, J.W.S. and F r o h l i c h , A., ed. Algebraic Number Theory, Academic Press, London, N.Y., 1967. [CR] C u r t i s , C.W. and Reiner, I. Representation Theory of F i n i t e Groups and Associative Algebras, Wiley Interscience, N.Y., 1962. [F] F r o h l i c h , A., ed. Algebraic Number F i e l d s , L functions and Galois Properties, Academic Press, London, ¥.Y., 1977. [H] Hasse, H. Zahlentheorie I I , A u f l . B e r l i n Akademie Verlag, 1963. [I[ Iyanga, S., ed. The Theory of Numbers, North Holland Mathematical L i b r a r y #8, Amsterdam, N.Y., 1975. [L] Lang, S. Algebraic Number Theory, Addison Wesley, N.Y., 1970. [SI] Serre, J.-P. Corps Locaux, Hermann, P a r i s , 1962. [S2] Serre, J.-P. Representations Lindares des Groupes F i n i s , P a r i s , Hermann, 1971. [W] Weil, A. Basic Number Theory, Springer Verlag, B e r l i n , N.Y., 1974. Papers, Lecture Notes [I] Deligne, P. Les Constantes des Equations Functionelles des  Functions L, Springer Lecture 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. Galois Module Structure, i n [F], p. 133-192. [4] F r o h l i c h , A. Arithmetic and Galois Module Structure for Tame Extensions, J. Reine und Angew. Math. #286/287, p. 380-440, 1976. [5] F r o h l i c h , A. Stickelberger 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 Free Modules Over Arithmetic Orders, J . Reine und Angew. Math. #274/275, p. 112-124, 1975. [7] Gelbart, S. Automorphic Forms and A r t i n ' s Conjecture ( i n International Summer School on Modular Functions-, Bonn, 1976) Springer Lecture Notes #627, p. 243-276. [8] Godement, R. and Jacquet, H. Zeta Functions of Simple Algebras, Springer Lecture Notes #260, 1972. [9] Hasse, H. and Davenport, H. Die N u l l s t e l l e n der Kongruenzzeta-funktionen i n Gewissen Zyklischen F a l l e n , J . Reine und Angew. Math. #172, p. 151-182, 1934. [10] Jacquet, H. and Langlands, R.P. Automorphic Forms on GL(2), I, Springer Lecture Notes #114, 1970. [11] Jacquet, H. Automorphic Forms on GL(2), I I , Springer Lecture Notes #278, 1972. [12] Leopoldt, H.W. Zur Arithmetik i n Abelschen Zahlkorpern, J . Reine und Angew. Math. #209, p. 54-71, 1962. [13] Mackenzie, R. Class Group Relations i n Cyclotomic F i e l d s , Am. J. Math. #74, p. 759-763, 1952. - 66 -[14] Martinet, J. Character Theory and A r t i n L Functions, i n [F], p. 1-88. [15] Stickelberger, L. Ueber eine Verallgemeinerung der Krei s t h e i l u n g , Math. Ann. #37, p. 321-367, 1890. [16] Serre, J.-P. Local Class F i e l d Theory i n [CF], p. 128-161. [17] Tate, J. Fourier Analysis i n Number F i e l d s and Hecke's Zeta-functions, i n [CF], p. 305-347. [18] Tate, J. Local Constants, i n [F], p. 89-132. [19] Weil, A. La Cyclotomie Jadis et Naguere,Seminaire Bourbaki #452, 1974. [20] Weil, A. Jacobi Sums as "Grossencharaktere", Trans. Am. Math. Soc. #73, p. 487-495, 1952. 

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