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Galois Module Structure and Artin L-Functions Astérisque, Tome 24-25 (1975), P Astérisque ALBRECHT FRÖHLICH Galois module structure and Artin L-functions Astérisque, tome 24-25 (1975), p. 9-13 <http://www.numdam.org/item?id=AST_1975__24-25__9_0> © Société mathématique de France, 1975, tous droits réservés. L’accès aux archives de la collection « Astérisque » (http://smf4.emath.fr/ Publications/Asterisque/) implique l’accord avec les conditions générales d’uti- lisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Société Mathématique de France Astérisque 24-25 (1975) GALOIS MODULE STRUCTURE AND ARTIN L-FUNCTIONS by Albrecht FRÖHLICH 1) Let L and K be number fields, always of finite degree over Q , and L normal over K with Galois group Gal(L/K) = T . Let x De a character of r . We shall write A(s,x) for the enlarged Artin L-function, to include gam­ ma and exponential factor-s. It satisfies a functional équation W(x) A(s,X) = A(l-s, X) , where x" is tne complex conjugate of x anc^ W(x) » the Artin root number, is of absolute value 1 . We write 1/2 W(L/K,x) - w(x) - T(X) wa(x) /Nf(x) , where W (x) is a power of i = ^-1 , depending an ramification at infinity, 1/2 Nf(x) is the positive square root of the absolute norm of the conductor f(x) , and t(x) is the "Galois Gauss-sum" (Hasse). If x is real valued then W(x) = ± 1 . If actually x cornes from a real 9 A. FRÔHLICH représentation then W(x) = 1 . Example : Let T = H be the generalised quaternion group of order 4lr r 4t » where £ is an odd prime and r ^ 1 . For each j = 1, ... , r there are faithful irreducible représentations (of degree 2) of the quotient H . of H , which have real valued characters \|f. , but which cannot be realised over the J fields of real numbers. If L/K is tame then the root numbers of the distinct charcters \|j = i|f. , with the same j , ail coincide (cf. [1] theorem 5"). We shall write W(L/K, il = W (L/K) , (j=l r) . 2) Let again L/K be normal and tame, with Gal(L/K) = T , and let r : r GL(n, E) be a représentation of T over some number field E with character x • Let aç L , define the résolvent (a|x) = det( S aY T(y)'1) . Y Let K(x) be the field obtained by adjoining the values of x to K and let o , ~ be its ring of algebraic integers. Let & be the ring of algebraic integers in L . Then the (a|x) » with a£&, generate a finitely generated rank one module over 0 / . , containe.d in L(x) , which we dénote by (& |x) . This module is basic for the détermination of the Galois module structure of & . Let f rom now on K = Q . As a spécial case of a gênerai theorem one has 10 GALOIS MODULE STRUCTURE THEOREM 1. - (G|X) = 0Q( ^ T(X) (cf. [3]). Let (&) be the class in the class-group Cl(Z(r)) of the intégral group- ring Z(r) , given by the Z(r)-module Let D(Z(I7) be the kernel group, i. e. the kernel of Cl(Z(r)) -* Cl(3Jl) , SDR being a maximal order of Q(V) . Form Theorem 1, one deduces (cf. [3]") : THEOREM 2. - (&) ç D(Z(r)) • (Martinet's conjecture). 3) "With L/Q as above, let again T = H , whith each j = 1, ... , r (or rather with the corresponding class of characters = .) there is associated a surjection : 0 : D(Z(H ))) -» ±1 , and thèse are independent (cf. [2] ). Write e.((&)) = U.(L) . Then one has (cf. [2]) : THEOREM 3. - (i) _If_ l s -1 (mod 4) then U.(L) = W (L/Q) (j = l,...,r) , (ii) If_ = 1 (mod 4) then U.(L) = 1 (j = l,...,r) . One can show that given f : [l,...,r] -> [±1] 11 A. FRÔHLICH there are infinitely many fields L with W.(L") = f(j) . 4) Case (ii) of Theorem 3 shows that the Galois module structure of & will not suffice to détermine the root numbers. One needs additional structure, and has to consider in the gênerai situation of tame extensions Ô as a "Hermitian Galois module". There is indeed a Hermitian form of L over Q(T) , induced by the trace. In the new situation one has to construct a Hermitian Class-group HCl(Z(r~)"), which classifies locally free rank one Z(T")-module s M with a non- singular Hermitian form on the Q(F) -module spanned by M . The élément cor- responding to & will be denoted by [&] . Now return to the case T = H . Then [&] belong to a certain sub- 4£r group G of H C1(Z(H )) which under the map H C1(Z(H )) -+ C1(Z(H )) falls into D(Z(H . Let IF be the field of l élément, IF its multipli- 4lr 1 ^ cative group. There are then canonical surjections uu. : G IF , so that j r t the diagram uu. # Gr J—' Wt 2 (Dl [x .x^-^ =(f)2] * e. D(Z(H ))")—J-» ±1 4lr commutes. Write (JUJ(^) = Vj(L) • In place of Theorem 3 one now gets the better THEOREM 4. - V^L) = W^L/Q") (j = 1,... ,r) . 12 GALOIS MODULE STRUCTURE Note that the distinctions of cases in Theorem 3 now becomes obvious from diagram (D) . For other surveys of this gênerai topic see Martinet's Bourbaki report (cf. [4]) and the report of my address to the Vancouver congress (cf. [5]). LITERATURE [1] A. FRÖHLICH. - Artin Root numbers, Conductors, and Representations for Generalized Quaternion groups. Proc. London Math. Soc. 28 (1974) 402-438. [2] A. FRÖHLICH. - Module Invariants and root numbers for quaternion fields of degree 4lr. Proc. Camb. Phil. Soc. 76 (1974) 393-399. [3] A. FRÖHLICH. - Arithmetic and Galois module structure for tame ex­ tensions, to be published. [4] J. MARTINET. - Bases normales et constante de l'équation fonctionnelle des fonctions L d'Artin. Sém. Bourbaki 450, 1973/74. [5] Proceeding of the International Congress of Mathematicians Vancouver, 1974. Albrecht FRÖHLICH University of London King's Collège Department of Mathematics Strand London WC 2R 2LS 13 .
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