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