ON THE GALOIS STRUCTURE OF ARTIN S L-FUNCTION AT ITS CRITICAL POINTS KENNETH WARD

Transcription

1 ON THE GALOIS STRUCTURE OF ARTIN S L-FUNCTION AT ITS CRITICAL POINTS By KENNETH WARD Bachelor of Arts of mathematics The University of Chicago Chicago, Illinois, United States of America 2004 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of Master of Science May 2009

2 COPYRIGHT c By KENNETH WARD May 2009

3 ON THE GALOIS STRUCTURE OF ARTIN S L-FUNCTION AT ITS CRITICAL POINTS Thesis Approved: Dr. Anantharam Raghuram Thesis Advisor Dr. Dale Alspach Dr. Alan Noell Dr. A. Gordon Emslie Dean of the Graduate College iii

4 ACKNOWLEDGMENTS I thank Anantharam Raghuram for introducing me to the problem in number theory that is the heart of this thesis. iv

5 TABLE OF CONTENTS Chapter Page 1 Introduction 1 2 Representation theory Basic notions Induction Brauer s theorem on induction of characters Artin s L-function Artin s L-function The basic properties of this function Class field theory Idèles The Artin symbol The functional equation Hecke s L-function The Artin conductor The functional equation The result of Coates and Lichtenbaum Polyhedric cones Shintani s unit theorem Rewriting a particular kind of L-function v

6 6.4 Siegel and Klingen s result on special values of L-functions The value of Artin s L-function at negative integers The main result: Behavior of special values under twisting 100 BIBLIOGRAPHY 109 vi

7 CHAPTER 1 Introduction This work provides a relation between an Artin L-function for a character χ ρ corresponding to a representation ρ of a Galois group of a Galois extension F Q of number fields, and an Artin L-function for a character χ ρ χ = χ ρ χ of the same group, where χ is an even Dirichlet character. In particular, it is shown that, with τ(χ) denoting the Gauss sum of χ, the value of L(F Q, χ ρ χ, s) is, up to an element of Q(χ ρ, χ), equal to τ(χ) dim(ρ) L(F Q, χ ρ, s), so long as s is an integer, at least 2, where the Euler factors at infinity for the L-function of χ ρ are regular at 1 s. This naturally extends a work of Coates and Lichtenbaum [CL], which states that L(F K, χ, s) Q(χ) for a character χ of a Galois extension F K of algebraic number fields if s is a negative integer. The work of Coates and Lichtenbaum employs a result of Siegel and Klingen [Neu, VII], which states that L(F Q, χ, s) Q(χ) if s is a negative integer and χ is a Dirichlet character. This work, as well as those mentioned above, employ the functional equation for the given L-function. It is of some importance that regularity is involved with the main result of this work, since poles emerge in the functional equation at other integer points that limit its use in determining the value of the L-function. Preparation is given to the basic topics necessary for understanding the main result. Proofs are provided, with exceptions made for class field theory and the theory of Lubin-Tate extensions. These omissions are for brevity, and references are provided where omissions have been made. The progression of this work is as follows. Chapter 2 proves Brauer s theorem 1

8 on induction of characters. Chapter 3 introduces Artin s L-function and its basic properties. Chapter 4 surveys the main results of class field theory. Chapter 5 establishes the functional equation for Artin s L-function. Chapter 6 proves the above mentioned result of Coates and Lichtenbaum. Chapter 7 yields the main result. 2

9 CHAPTER 2 Representation theory 2.1 Basic notions This section introduces the basic notions of representation theory. Here, G will denote a finite group. A representation is a homomorphism ρ : G GL(V ) where V denotes a vector space over C of finite dimension. Here, the notation ρ g will be adopted instead of ρ(g). The dimension of V will be called the dimension of the representation ρ. A subspace W V will be called stable if ρ g W W for all g G. Define a representation ρ to be irreducible if ρ may not be written as ρ 1 ρ2 : G G(V 1 ) G(V 2 ) where ρ i : G G(V i ) for each i {1, 2}, where V = V 1 V2. Theorem 2.1 Given a representation ρ of a finite group G, there is a decomposition and likewise where V = ρ = r V i, i=1 r ρ i, i=1 ρ i : G GL(V i ) and is irreducible, for each i {1, 2,..., r}. 3

10 Proof. First, consider a subspace W V that is stable and proper. Adding elements to a basis for W to obtain a basis for V, one may obtain a subspace W satisfying W W = V. Consider the projection p of V onto W via this basis for V, and define the average p 0 = 1 G g G ρ g pρ 1 g. One notes that because W is stable, one must have p 0 mapping V into W. Also, because W is stable, one must have for every x W. Therefore ρ g pρ 1 g x = x, p 0 x = x for every x W, and so p 0 is a projection of V onto W. Its kernel W 0 satisfies W W 0 = V. But also, one has ρ h p 0 ρ 1 h = 1 G = 1 G g G g G ρ h ρ g pρ 1 g ρ 1 h ρ hg pρ 1 hg = p0 for all h G, and therefore x W 0 satisfies p 0 x = 0, and thus p 0 p g x = 0. Thus p g x W 0, and W 0 is a complement of W in V that is stable. By the definition of irreducibility, the result then follows by induction on the dimension of V. For a representation ρ, define the character χ as χ(g) = T r(ρ g ). A few basic properties of this character function are as follows: 4

11 Theorem 2.2 (i) χ(1) = dim(ρ); (ii) χ is a class function from G to C; (iii) χ(g 1 ) = χ(g). Proof. Properties (i) and (ii) are obvious. Property (iii) follows the observation that χ(g) = dim(ρ) where each λ i is a root of unity, and, in fact, an eigenvalue for ρ g. Since ρ g has finite order, one has that ρ g is conjugate to a matrix with nonzero entries only along the i=1 λ i diagonal, with each diagonal entry an eigenvalue for ρ g. But then ρ g 1 is conjugate to the inverse of this matrix of eigenvalues. Thus χ(ρ g 1) = dim(ρ) i=1 dim(ρ) λ 1 i = with the last equality holding because each λ i is a root of unity, as G has finite order. i=1 λ i, As the result follows. dim(ρ) i=1 λ i = χ(g), The following will now demonstrate various aspects of character theory that will be of use to us. In order to do this, one important definition must be established. Two representations ρ 1 : G GL(V 1 ) and ρ 2 : G GL(V 2 ) are called isomorphic if there is a isomorphism of vector spaces f : V 1 V 2 such that f ρ 1,g = ρ 2,g f for all g G. This definition motivates the following. Theorem 2.3 (Schur s lemma) Suppose that ρ 1 : G GL(V 1 ) 5

12 and ρ 2 : G GL(V 2 ) are irreducible representations, and that f : V 1 V 2 is a linear map satisfying f ρ 1,g = ρ 2,g f for all g G. (i) If ρ 1 and ρ 2 are not isomorphic, then f 0; (ii) if V 1 = V 2 and ρ 1 = ρ 2, then f(v) = λv for some λ C. Proof. For the first part, note that the kernel of f must be stable, as one must have 0 = f ρ 1,g x = ρ 2,g fx = ρ 2,g 0 = 0. As V 1 is irreducible, this kernel must be zero. Likewise, the image of f is stable. Therefore f is surjective and injective, hence an isomorphism, which contradicts the fact that ρ 1 and ρ 2 are isomorphic. (ii) As a linear map, f has an eigenvalue, say, f(v) = λv for some nonzero element v V = V 1 = V 2. With f λ = f λ1 V as a map from V to V, one has that f λ ρ 1,g = ρ 2,g f λ for all g G, and therefore the nonzero kernel of this map must equal all of V, as V is irreducible. Therefore f λ. as If χ 1 and χ 2 are two complex-valued functions on G, one defines their inner product χ 1, χ 2 = 1 G The following result is of importance here. χ 1 (g)χ 2 (g). g G Theorem 2.4 (Schur s orthogonality of characters) 6

13 (i) If χ 1 and χ 2 correspond to representations ρ 1 : G GL(V 1 ) and ρ 2 : G GL(V 2 ), respectively, that are irreducible and not isomorphic, then χ 1, χ 2 = 0; (ii) If ρ 1 and ρ 2 are irreducible and isomorphic, then χ 1, χ 2 = 1. Proof. For (i), one considers a mapping, as in the proof of Theorem 2.1, defined as f 0 = 1 G (ρ 2,g ) 1 fρ 1,g g G where f is a linear map from V 1 to V 2 as before. One may note that ρ 2,g f 0 = f 0 ρ 1,g for all g G. By Schur s lemma, f 0 0. In this case, one considers f to be the linear map which, as a matrix, has an element 1 in a particular entry, and zeroes elsewhere. The inner product of χ 1 and χ 2, with ρ 1 and ρ 2 written in matrix form as ρ 1,g = ( ρ 1 ij(g) ) i,j and ρ 2,g = ( ρ 2 ij(g) ) i,j, satisfies χ 1, χ 2 = i,j ρ 1 ii, ρ 2 jj. With f as the linear map mapping the jth basis element for V 1 onto the ith basis element for V 2 and all else to zero, one must have < ρ 1 ii, ρ 1 jj >= 0 for all i and j. Therefore the inner product of χ 1 and χ 2 is equal to zero. For (ii), if ρ 1 and ρ 2 are irreducible and isomorphic, then one immediately identifies from the definitions that χ 1 χ 2. Thus to consider the inner product of these two characters, one may assume that V = V 1 = V 2, as has been done in the proof of Schur s lemma. With ρ = ρ 1 ρ 2, one has T r(f 0 ) = 1 G g G T r(ρ 1 g fρ g ) = T r(f) as matrices, and as, by the Schur lemma, f 0 λ for some λ C, it follows that λ = 1 T r(f). With the above matrix notations, by considering f to be the linear dim(ρ) map sending the jth basis element in V 1 to the ith in V 2, and all else to zero, one 7

14 must have ρ ii, ρ jj = 1 dim(ρ) χ 1, χ 2 = χ 1, χ 1 = if i = j and zero otherwise. Therefore ρ ii, ρ ii = dim(ρ) i=1 dim(ρ) i=1 1 dim(ρ) = 1. This yields immediately the uniqueness, up to isomorphism, of the decomposition of a representation into a direct sum of irreducible representations. In the sequel, it will also be useful to note the following fact. Lemma 2.1 χ(g) = dim(ρ) or χ(g) = dim(ρ) if and only if it holds respectively that ρ g = I or that ρ g = I, where I denotes the identity matrix. Proof. One may represent χ(g) as the trace of the diagonal matrix with entries along the diagonal consisting of the eigenvalues of ρ g. The matrix ρ g is conjugate to this diagonal matrix of eigenvalues. Also, because ρ g has finite order, each of its eigenvalues is a root of unity, and thus has modulus one as a complex number. Therefore if χ(g) = dim(ρ), then ρ g is conjugate, and hence equal, to the identity matrix, as each eigenvalue of ρ g is equal to 1. If χ(g) = dim(ρ), then ρ g is conjugate, and hence equal, to the diagonal matrix I, as each eigenvalue of ρ g is equal to 1. The converse holds trivially. The following result will be necessary to the proof of Brauer s theorem on induction of characters. First, one must define the regular representation of G to be the C- algebra C[G] which has elements of G as its basis, where the action of g on C[G] is defined as 8

15 g c g g = c g gg. g G g G Lemma 2.2 The irreducible characters of a finite group G form an orthonormal basis of the space of class functions of G. Proof. Consider a class function f, an irreducible representation (ρ, G, V ), and the linear mapping ρ(f) : V V defined by ρ(f) = g G f(g)ρ g. One may notice that ρ 1 g 1 ρ(f)ρ g1 = f(g)ρ 1 g 1 ρ g ρ g1 g G = g G f(g)ρ g 1 1 gg 1 = g G f(g 1 gg 1 1 )ρ g = g G f(g)ρ g = ρ(f). Therefore, with χ ρ denoting the character associated with ρ, ρ(f) is equivalent to a scalar λ = = 1 dim(v ) 1 dim(v ) f(g)t r(ρ g ) g G f(g)χ ρ (g) g G = G dim(v ) f, χ ρ. Suppose then that f is a class function orthogonal to χ, for every character χ that corresponds to an irreducible representation of G. Considering an arbitrary representation (ρ, G, V ) of G, one then has, by Schur s lemma and componentwise additivity of the inner product, in each component, that ρ(f) is identically zero. 9

16 Supposing now that this arbitrary representation is specially chosen to be the regular representation of G, one has ρ(f) 1 = g G f(g)g in C[G]. As ρ(f) is identically zero, one has ρ(f) 1 = 0. Thus f(g) = 0 for all g G, and f is identically zero. Therefore the characters χ from characters χ corresponding to irreducible representations of G form an orthonormal basis for the set of class functions on G. If χ is such a character, then χ is also such a character, a fact which is immediate from the definition of the inner product,. Therefore the characters of the irreducible representations of G form an orthonormal basis for G. 2.2 Induction In this section, the notion of an induced representation is introduced. As before, suppose that G is a group. Where necessary a representation ρ : G V for a group G will be denoted by (ρ, V, G). Consider then H a subgroup of G, with a representation (θ, W, H). The induced representation (ρ, Ind G H(W ), G) of (θ, W, H) is defined as Ind G H(W ) = C[G] C[H] W, where elements of H act on W via the representation θ. The following well-known result is of importance for the proof of Brauer s theorem on induced characters. It is due to Frobenius. In what follows, R will denote a set of left coset representatives for H in G, and G will denote a finite group. A character χ of H is viewed in the following theorem as a function on G with χ(g) = 0 for all g / H. Theorem 2.5 If (θ, W, H) has character χ θ, and (ρ, V, G) is the induced representa- 10

17 tion with character χ ρ, then one has. χ ρ (g) = r R r 1 gr H χ θ (r 1 gr) Proof. Consider the direct sum V = r R ρ r W, and some g G. One may note that χ ρ (g) consists of the sums of traces of those indices r R for which ρ gr W = ρ r W. Denoting by R g the set of indices in R so fixed by ρ g, one has χ ρ (g) = r R g T r ρrw (ρ g,r ) with here ρ g,r denoting the restriction of ρ g to the subspace ρ r W. In fact, the set R g consists exactly of those elements r R for which r 1 gr H, and thus, one may note that ρ r is an isomorphism from W onto ρ r W for every r R g, and therefore, as ρ g θ r 1 gr = ρ g,r ρ r if r 1 gr H, one has χ θ (r 1 gr) = T r W (θ r 1 gr) = T r ρrw (ρ g,r ), and the result holds. This may be used to define for any class function f : H C the induced function Ind G H(f) = r R r 1 gr H χ θ (r 1 gr), where as with characters of H, f is viewed as a function extended to G by zero. With this machinery, one is now prepared to prove Brauer s theorem, whose final steps are put forth in the last section of this chapter. 11

18 2.3 Brauer s theorem on induction of characters This section establishes Brauer s theorem on induction of characters. Lemma 2.3 Let p be a prime number, G a finite group, and g G. Two elements g 1 and g 2 exist so that g = g 1 g 2, the order of g 1 is a power of p, the order of g 2 is relatively prime to p, and g 1 and g 2 commute. An element of G commutes with g if and only if it commutes with both g 1 and g 2. Also, g 1 and g 2 are uniquely determined by the element g Proof. All of these facts follow easily from the following construction. Suppose that g = p α m, where m and p are relatively prime. Then there exist integers a and b satisfying ap α + bm = 1. In this case, one may set g 1 = g bm and g 2 = g apα, and one will evidently have g = g 1 g 2. Evidently, also g 1 commutes with g 2. The order of g 1 is a power of p by its construction, and the order of g 2 is relatively prime to p by its construction. That an element of G commutes with g if and only if it commutes with both g 1 and g 2 follows from the construction of g 1 and g 2 as powers of g. One must now establish uniqueness of g 1 and g 2. As p α and m are relatively prime, one must have that g 1 has order p α and g 2 has order m. Thus, one must have g m 1 = g m, and g pα 2 = g pα. Therefore, g 1 = g bm 1 = g bm, and likewise g 2 = g apα 2 = g apα. This proves the claim. Define the function 1 G (g) = 1, for all g G. 12

19 Let the symbol p stand for a prime number dividing the order of G, let S denote the ring Z[ε], where ε is a primitive mth root of unity, and m is a natural number where g m = 1 for all g G. For a G, denote the centralizer of a by C G (a) = {g G ga = ag}. Then, suppose that R is a subring of C containing the integers. Denote by E the set of elementary subgroups of G, i.e., the subgroups of G of the form a B where a G is an element of order relatively prime to p, and B is a p-subgroup of C G (a). Denote by Ch(G) the set of characters of representations of G. Define the following three sets, ascending in the inclusion ordering: V R (G) = {ρ ρ = r i Ind G E i (χ i ), E i E, r i R, for all i}, finite Ch R (G) = {ρ ρ = r i χ i, χ i Ch(G), r i R, for all i}, finite U R (G) = {ρ ρ E Ch R (E) for each E E}. One may note that V R (G) is naturally an ideal of U R (G), for with ρ V R (G), µ U R (G), one has, with ρ = r i Ind G E i (χ i ), finite that µρ = r i µind G E i (χ i ) finite = r i Ind G E i ((µ χ i ) Ei ), finite with the denoting multiplication. Of course, (µ χ i ) Ei = µ Ei χ i Ch(E i ), for both µ Ei and χ i are characters of representations, and thus their product is the character of the tensor product of the representations for which µ Ei and χ i are characters. The following lemmas are necessary precursors to Brauer s theorem on induced characters. 13

20 Lemma 2.4 Let E be a supersolvable group. Then E is monomial, i.e., an irreducible representation ρ of E is induced from a representation of degree one for a subgroup of E. Proof. This proof is by induction on the order of E. If E = 1, then the lemma is obvious. Also, if E is abelian, then an irreducible representation ρ of E is automatically of dimension one. Suppose then that E > 1, and that E is nonabelian. As E is supersolvable, E admits a tower 1 = E 0 E 1 E 2 E n = E where each E i is a normal subgroup of E, and E i+1 /E i is a cyclic group of prime order. One may then take the quotient of E with its center Z(E), and may thus obtain such a cyclic tower Z(E) = F 0 F 1 F 2 F n = E where, again, each F i is a normal subgroup of E, and quotient groups of successive groups in this chain are cyclic of prime order. One may note that F 1 must be an abelian, normal subgroup of E, and may suppose that F 1 is not equal to Z(E). Suppose that the character χ ρ corresponds to the irreducible representation ρ of E. By induction, one may assume that ρ is injective. Then, as F 1 is not contained in the center of E, there must be some a F 1 so that ρ(a) is not identical to a scalar. If the decomposition of the restriction of ρ to F 1 into irreducible representations consisted of a collection of pairwise isomorphic representations, then, as F 1 is abelian, one would have a decomposition of ρ as (ρ, V, F 1 ) = r (ρ i, V i, F 1 ) i=1 where V i has dimension one, for each i {1, 2,..., r}. 14

21 Of course, as each representation in the direct sum is isomorphic to every other, one would have in particular for i, j {1, 2,..., r} some F ij : V j V i satisfying F ij ρ j,f = ρ i,f F ij, for each f F 1. As each representation in the direct sum is also of one dimension, it must be that ρ j,f = ρ i,f for all i, j {1, 2,..., r}, for each f F 1 and thus ρ f would be a scalar for every f F 1. This is a contradiction, and so the restriction of ρ to F 1 may not be decomposed in this way. Nonetheless, in restricting to F 1, one may write (ρ, V, F 1 ) = s j=1 r j i=1 (ρ i,j, V i,j, F 1 ) where each collection B j = {(ρ i,j, V i,j, F 1 )} r j i=1 is a maximal collection of irreducible representations of F 1 in its decomposition into irreducible representations. The linear transformation ρ permutes the subspaces V j = r j i=1 V i,j. Considering some V j, and the subgroup H j E where ρ h V j = V j, one notes that F 1 H j, and that ρ is induced by the representation ρ Hj when viewed as acting on V j. Also, one must have H j E. This establishes the result. Theorem 2.6 A character χ of a representation of G is a Z-linear combination of characters of representations of G as χ = finite Ind G H i (χ i ) where, as before, H i is a subgroup of G, and χ i is a character of a representation of dimension one for H i. 15

22 Proof. One needs only to show that 1 G V Z (G). For, once this has been established, one may note that if 1 G V Z (G), then as Ch(G) U Z (G) and V Z (G) is an ideal of U Z (G), then so must Ch(G) V Z (G). Consider a character χ of a representation of G as a finite Z-linear combination of characters of representations G where each character in the linear combination is induced from a character of a representation of a particular element, say, E, of E. Writing χ = n n j Ind G E j (χ j ) j=1 with E j E and n j Z for each j {1, 2,..., n}, and with χ j = l j χ i,j i=1 the expression of χ j as a sum of characters from irreducible representations of E j, for each j {1, 2,..., n}, it follows that χ = n n j Ind G E j (χ j ) j=1 = n n j Ind G E j ( l j j=1 i=1 χ i,j ) n = l j j=1 i=1 n j Ind G E j (χ i,j ), where the last equality holds because induction is additive on characters of representations, from Frobenius theorem in Section 2.2. And each χ i,j is induced by a character of a representation of dimension one on a subgroup of E j, for each i {1, 2,..., l j }, for each j {1, 2,..., n}, because each E j is nilpotent, and thus supersolvable. Therefore χ will be the desired Z-linear combination. Noting this, let us now must only establish that 1 G V Z (G). Consider then an element E E. Suppose, to begin, that E = a, 16

23 and let n = a. Suppose then that ω S is a primitive nth root of unity. The class function defined as η(a) = a and η(a i ) = 0 for all a i a may, by the above, be written as η = a i ω i finite where each ω i denotes a character of an irreducible representation of E and each a i denotes a complex number. Here, one may, in particular, take ω i to be the character uniquely defined by ω i (a) = ω i. In this case, the set of these characters ω i constitute all of the characters for irreducible representations of E. So written, one has explicitly that a i = η, ω i = ω i S. Therefore one must have that η Ch S (E), noting here that this definition does apply because S = Z[ε] is a subring of C containing Z. Then, considering a general element of E, one may see that this argument extends to these groups of the form a B where B need not be trivial, by defining η to be equal to the function η constructed immediately above on a, and extended as a function on E that is constant in its second coordinate. Of course, one may do the same with the characters ω i, and this yields a set of irreducible characters of E, denoted by ω i for each i, with, by the above, η = a i ω i. finite In this fashion one has for an arbitrary element of E a function η Ch S (E) with η(a) = a and η(a i ) = 0 for all a i a that is constant in its second coordinate as a function of the product a B. 17

24 One may write, for a representation (ρ, G, V ) induced from (θ, H, W ), that χ ρ (g) = 1 H r G r 1 gr H χ θ (r 1 gr) by the theorem of Frobenius of Section 2.2 noticing now that the sum is over all relevant a set of left coset representatives for H in G. Suppose now that B is a Sylow p-subgroup of C G (a). Define N CG (a)(b) = {r C G (a) rbr 1 = B} and Note that the induced function n = card({p a p-sylow subgroup of C G (a) b / P }). Ind G E(η ) = finite a i Ind G E(ω i) satisfies Ind G E(η )(ab) = 1 B {r G r 1 (ab) = ab for some b B} = 1 B {r C G(a) r 1 br B} = 1 B {r C G(a) b rbr 1 } = n N C G (a)(b) B [N CG (a)(b) : B] mod p by the first theorem in this section, and the fact that n = 1 mod p, which holds trivially when b = 1, and in any other case because b 1 permutes by the action of conjugation the Sylow p-subgroups of C G (a) which do not contain b in orbits of cardinality equal to some power of p. Of course, p does not divide the index of B in its normalizer in C G (a), as it is a Sylow p-subgroup of C G (a), and therefore, there is some integer α so that αind G E(η )(ab) 1 mod p. 18

25 Then one will have αind G E(η )(ab) 1 mod p for all b B. One may note that this induced character is zero for every g which is not conjugate to any ab a B. Also, if g is conjugate to ab for some b B, then one must have αind G E(η )(g) = αind G E(η )(ab). Note then that g is conjugate to such an element if and only if the decomposition according to Lemma 2.3 gives that the element g 2, of order relatively prime to p and uniquely determined by g, is conjugate to a. One may then consider the conjugacy classes C 1,..., C k of the elements g 2, over all g G. As above, one has for each such conjugacy class a function θ j in V S (G) where θ j (g) = 0 if the element g 2 in its decomposition does not lie in the conjugacy class C j, and θ j (g) 1 mod p if g 2 lies in C j. By taking k θ j, one then obtains an element, denoted by θ p, in V S (g) satisfying j=1 θ(g) 1 mod p for all g G. Suppose then that the order of G is written as p β n p, where (n p, p) = 1. The following analysis will hold for each prime p dividing the order of G. One has θ pβ 1 p 1 mod p β. 19

26 Of course, taking an arbitrary element a of G, and considering its conjugacy class, one may notice that the function η constructed to lie in V S ( a ) induces to a function Ind G a (η) in V S (G) satisfying Ind G a (η)(g) = 0 if g is not conjugate to a, and Ind G a (η)(g) = C G (a) if g lies in the conjugacy class of a. Therefore, a class function whose values are divisible by G must lie in V S (G). Thus, in particular, the class function n p (θ pβ 1 p 1 G ) must be contained in V S (G). As in the proof that V R (G) is an ideal of U R (G), one must have θ pβ 1 p V S (G), and thus n p 1 G V S (G). Noting then that the set of natural numbers n p as constructed above, with p ranging over all primes dividing the order of G, has greatest common divisor one, one may find integers c p satisfying c p n p = 1. Therefore p 1 G = p c p n p 1 G V S (G). Writing 1 G = finite c i Ind G E i (χ i ) where χ i is a character of a representation of E i and c i S for each i, one may notice that, with c i = j c i,jε j for integers c i,j, one has 1 G = φ(m) 1 j=0 ε j ( i c i,j Ind G E i (χ i ) ) 20

27 where φ( ) denotes Euler s phi-function, as {1, ε,...ε φ(m) 1 } is a Z-basis for S. Now, one may notice that each of the characters appearing in this sum may be written as a sum of characters of irreducible representations of G, and so one has 1 G = i c i,0 Ind G E i (χ i ) V Z (G) as soon as the elements {1, ε,..., ε φ(m) 1 } are proven to be linearly independent over Ch Z (G). For that, a relation φ(m) 1 j=0 ε j ( i c i,j χ i ) = 0 with the sum over the index i sweeps across characters χ i of irreducible representations of G requires i φ(m) 1 j=1 c i,j ε j χ i = 0. But as these characters χ i are orthonormal with respect to the inner product,, so must they be linearly independent, and thus φ(m) 1 j=1 c i,j ε j = 0 for each i. Therefore c i,j = 0 for each i and each j {1, 2,..., φ(m) 1}, and the elements {1, ε,..., ε φ(m) 1 } are linearly independent over Ch Z (G). This establishes Brauer s theorem. This concludes the chapter on representation theory. The next chapter introduces Artin s L-function and addresses its basic properties. 21

28 CHAPTER 3 Artin s L-function 3.1 Artin s L-function This chapter introduces Artin s L-function. One considers now a finite extension K of the rational numbers Q, called an algebraic number field, and a finite extension F of K. In this work, G(F K) will denote the Galois group of F over K. Artin s L-function hinges upon a representation ρ : G(F K) GL(V ) where, as before, V denotes a vector space over the complex numbers C. An element x K will be called integral over Z if it satisfies an equation x n + a n 1 x n a 0 = 0 where a i Z for i {1, 2,..., n 1}, where here n is understood to be at least one. The set of such elements in K forms a ring, called the integers of K, and is denoted by o K. One may note the following properties of o K : (i) o K is an integral domain; (ii) o K is noetherian as a ring; (iii) an element in K that is integral over o K must be contained in o K ; (iv) every nonzero prime ideal in o K is maximal. A ring possessing these four properties is called a Dedekind domain. One then defines a fractional ideal of K to be an o K -submodule a of K where ca o K for some c o K. The following result is necessary for the construction of Artin s L-function. Theorem 3.1 The nonzero fractional ideals form a group under multiplication of ideals, equal to the free abelian group on the set of prime ideals of o K. 22

29 Proof. Consider a nonzero ideal a of o K. Since o K is noetherian, some ideal a 0 is maximal with respect to the property that there is not product of prime ideals p 1 p 2 p r a. In particular, this ideal a cannot be a prime ideal, so that there will be b 1, b 2 o K where b 1 b 2 a, but neither of b 1 and b 2 lies in a. Then, if a 1 = (a, b 1 ), and a 2 = (a, b 2 ), one must have a 1 a 2 a, and each of a 1 and a 2 is not equal to a. And a is maximal for the property as above, so that there is a product of prime ideals contained in each of a 1 and a 2. The product of these products of prime ideals is contained in the product of a 1 and a 2, hence in a, which is a contradiction. Hence every nonzero ideal of a contains a product of prime ideals. Consider then a maximal ideal p, and define p 1 = {x K xp o K }. This will play the role of the multiplicative inverse of p in the group of fractional ideals. One evidently has that p 1 o K. Take a nonzero element p p and consider the smallest natural number r for which there exists a product of prime ideals satisfying p 1 p 2 p r (p) p. Such a product exists as above. In this case, p is, of course, prime, and so one of the ideals p i is contained in p, and because p i is maximal, it must equal p. But also, the product of primes contained in (p) as above, with p i removed, is no longer contained in (p), by the minimality of r as selected. Thus there is an element b j i p j where b / (p). However, one must have bp (a), and thus ba 1 p o K, whence ba 1 p 1. Also, ba 1 / o K. Therefore in this case p 1 o K, and p pp 1 o K. The ideal p is maximal, so one must have that one of these inclusions must be equality. If cannot, however, be the first, because then p 1 would contain only elements integral 23

30 over o K, as p is finitely generated, and so would equal o K. Thus the latter inclusion is equality, and p has an inverse. Then one may notice that a nonzero ideal of o K is invertible by a fractional ideal, for if this were not true for every such nonzero ideal of o K, then one would have a maximal non-invertible ideal a 0, because o K is noetherian. By the above analysis, this a could not itself be maximal, and thus one obtains a ap 1 aa 1 o K. One cannot have the first inclusion as equality, or else one would again have that p 1 = o K, this time because a is finitely generated. But then ap 1 would have an inverse by maximality of a in this ordering, and this inverse may be multiplied by p 1 to obtain an inverse for a. In this fashion every nonzero ideal of o K has an inverse, and, in fact, this inverse must be a fractional ideal. Considering then a nonzero fractional ideal a, one finds c o K so that ca o K, and so this ideal ca has an inverse b that must be a fractional ideal. Therefore cab = o K, and, as one may easily show, cb = {x K xa o K }, and so this fractional ideal forms the inverse for a in what is now the group of fractional ideals of K. To establish that this group forms a unique factorization domain, one may notice that if there is a nonzero ideal that is not equal to a product of prime ideals, then there is some maximal ideal a with respect to this property. This ideal a cannot be prime, and so with a p for some prime ideal p, one must have a ap 1 o K as before. But then by maximality of a according to this ordering, the ideal ap 1 must have a prime factorization, which may be multiplied by p to obtain a prime 24

31 factorization for a. One may then consider two fractional ideals a and b, and say that a divides b if and only if a b. By considering two prime factorizations p 1 p 2 p r = q 1 q 2 q s of an ideal in o K, one notices easily that by maximality of each of the prime ideals in each product, and the fact that each p i is contained in a particular q j, and hence equal to it, one must have r = s and uniqueness of the factorization. Then one may consider fractional ideals a in general, and with c o K satisfying ca o K, one will have with (c) = q 1 q 2 q s and ca = p 1 p 2 p r that q 1 q 2 q s a = (c)a = ca = p 1 p 2 p r, whence a = p 1p 2 p r q 1 q 2 q s. This establishes the unique factorization, and the proof is complete. With this in mind, one may consider the prime ideals of o K to be the finite primes, for every such prime p defines a non-archimedean valuation v p (x) = e p on o K corresponding to the prime factorization (x) = p p ep. Later, one will deal with infinite primes, which are created by the embeddings of K into the complex numbers. For now, use of the finite primes will suffice. One may observe that for the extension F of K, the analogous ring o F enjoys the above properties mentioned for o K, and thus is also a unique factorization domain. There is a fundamental identity that will be used later, and it is stated here, for its clarification of a relation between the prime ideals of o K and o F. Before proceeding 25

32 further, one may notice that the prime ideal p of o K extends via multiplication by o F to an ideal of o F, and therefore, po F has a prime factorization, po F = P e 1 1 P e 2 2 P er r. For any one of the primes P i appearing in this factorization, one writes P i p, and calls P i a prime of o F lying above p. One calls the field o K /p the residue class field of p, and similarly in o F for its prime ideals. For the above factorization of p in o F, one calls e i the ramification index of P i over p, and defines f i = [o F /P i : o K /p], calling it the inertia degree of P i over p. For the following theorem, the notion of localization will be used. One may consider for o K what is called its localization at p, equal to the ring { a b a o K, b o K \p}. This will be denoted by o K,p. The ring o F,p will be defined similarly as { a b a o F, b o K \p}. The ring o K,p is a principal ideal domain with a unique maximal ideal po K,p and admits a discrete valuation equal to the exponent n Z in the prime factorization x = uπ n of one of its elements, where π is a prime element and u is a unit, where each element in o K,p has such a factorization because the unique maximal ideal is principal. Theorem 3.2 (Fundamental identity) One has the relation [F : K] = r e i f i. i=1 26

33 Proof. The inclusion o K o K,p induces an isomorphism o K /po K = ok,p /po K,p, and likewise that o F /po F = of,p /po F,p. Thus to prove the result, one notes that o F,p /po F,p is a vector space of dimension [F : K] over o F,p /po F,p, and that the factorization of po F,p in o F,p yields po F,p = P e 1,p 1,p P e 2,p 2,p P er,p r,p with f i,p = [o F,p /P 1,p : o K,p /po K,p ]. One also notices that o F,p is the set of elements in F that are integral over o K,p. It is easy to see that e i,p = e i, and that f i,p = f i for each i {1, 2,..., r}. Also, o F,p is a Dedekind domain with finitely many primes, and, in particular, its primes are exactly those P 1,p, P 2,p,..., P r,p lying above po K,p. This means that o F,p is a principal ideal domain. For an ideal a p of o F,p factors uniquely as a p = r i=1 P e 1 1,pP e 2 2,p P er r,p where e 1, e 2,..., e r are integers, each at least equal to zero. By the Chinese remainder theorem, one may select an element x o F,p so that 27

34 x = Π e i i mod P e i+1 i,p where Π i is an element of P i,p not contained in P 2 i,p. In this way the factorization for the principal ideal (x) is (x) = r i=1 P e 1 1,pP e 2 2,p P er r,p, and therefore (x) = a p. Thus o F,p is a principal ideal domain. One then notes that the primes P 1,p, P 2,p,..., P r,p are pairwise relatively prime, and thus the Chinese remainder theorem gives an isomorphism o F,p /po F,p r i=1 o F,p /P e i,p i,p. In view of this isomorphism, and the natural isomorphism o F,p /P i,p P j i,p /Pj+1 i,p for each i {1, 2,..., r}, in each case given by multiplication by an element of o F,p by Π i j, where Π i is a generator of the principal ideal P i,p, the identity easily follows. With this machinery in mind, one now considers a finite Galois extension F of K, and defines for a prime P of o F the decomposition group of P, G P = {σ G(F K) σp = P}, with G intended to denote G(F K). The homomorphism σ σ given by σx = σx mod P for each x o F yields for each σ G P an associated σ : o F./P o F /P, and thereby a map from G P to the Galois group of o F /P over o K /P, which has kernel called the inertia group of P, and is denoted by I G,P. The following theorem is of great importance. 28

35 Theorem 3.3 For any two prime ideals P and P of o F lying above p in o K, there exists some σ G(F K) so that σp = P, i.e., G(F K) acts transitively on the set of primes lying above p. Proof.. If the theorem were not true, then the Chinese remainder theorem would yields some x o F where x = 0 mod P and x = 1 mod σp for all σ G(F K). Then one would have N F K (x) = σx P o K = p, σ G(F K) and yet, as x / σp for any σ G(F K), one must have σx / P for any σ G(F K), whence which is a contradiction. σ G(F K) σx / P o K = p, The following lemma is of importance in constructing Artin s L-function. Denote by F P the fixed field of the decomposition group G P. In general, for fields F F K with prime ideals P P p where P is a prime in o F, P a prime in o F, and p is a prime in o K, one may define the inertia degrees f = [o F /P : o K /p], f = [o F /P : o F /P ], and f = [o F /P : o K /p]. It follows that f = f f. With ramification indices e, e, and e defined similarly, one also has e = e e. Lemma 3.1 Let P be a prime of o F lying above the prime p of o K. One has o FP /{o FP P} = o K /p. Proof. With po F = r i=1 one has e 1 = e 2 = = e r = e and f 1 = f 2 = = f r = f from the transitivity of G(F K) over the primes lying above p, by the previous theorem. In this case, the P i e i 29

36 fundamental identity [F : K] = r i=1 e if i reduces to [F : K] = efr where r = (G : G P ). Therefore [F : F P ] = ef. Considering a particular prime P lying above p, one considers f, f, and f as above, and likewise for e, e, and e, taking in this case F = F P. Thus one has f = e = 1, and the result follows. The previous result motivates the following theorem. Theorem 3.4 The map G P G(o F /P o K /p) defined by mapping σ G P to σ G(o F /P o K /p) satisfying σx = σx mod P, for every x o K, is surjective. Proof. By the previous lemma, one may assume that K = F P, so that G(F K) = G P. Consider then a primitive element of o F /P as an extension of o K /p, i.e., an element x satisfying o F /P = {o K /p}(x), which exists because the extension o F /P over o K /p is separable. Consider then the minimal polynomial g of x over o K /p, and a lift x of x in o F, and suppose that σ G(o F /P o K /p). Then σ(x) = x, where x is also a root of the polynomial g. Also, the minimal polynomial f of x over K takes all of its roots in o F because F is normal over K, and therefore, f o K [X]. Thus one may consider f, the reduction of f modulo P, and that g must divide f. In this way g must have roots corresponding to the reductions of roots of f taken modulo P, and so x also has a lift x in o F that is a root of f. In particular, there is a σ G P so that σ(x) = x, and thus σ(x) = x, as desired. 30

37 This proves the surjectivity, because an element in G(o F /P o K /p) is completely determined by how it acts on the primitive element x. In the cases of these finite primes, one will have that the Galois group of o F /P over o K /p is cyclic, and one may choose as a generator of G P /I G,P an element mapping to what is called the Frobenius element in G(o F /P o K /p), which is the automorphism given by x x q, where q = o K /p. In this setting, this generator of G P /I G,P is itself called the Frobenius for P on account of the isomorphism G P /I G,P σ G(o F /P o K /p). Let then (ρ, V, G(F K)) be a representation, and let V I G,P be the subspace of V held fixed by I G,P. This V I G,P is called the module of invariants for IG,P, and one may also notice that the Frobenius element for P, and hence all of G P /I G,P, must map V I G,P to itself. Denoting the Frobenius element for P by φg,p, one may consider the expression det(i ρ(φ G,P )N (p) s V I G,P ). This is intended to denote the determinant of the expression I ρ(φ G,P )N (p) s as a matrix acting on V I G,P. Here, N (p) = p [o K /p:z/pz] where p lies above p Z. This determinant does not depend upon the prime P chosen, because any two primes P, P lying above p have G P and G P, I G,P and I G,P, and the Frobenius elements in each quotient group G P /I G,P and G P /I G,P as simultaneous conjugates. Also, the above determinant must depend only upon the character χ ρ of the representation ρ, and thus, so does the product p prime p o K 1 det(i ρ(φ G,P )N(p) s V I G,P ). 31

38 One defines this to be the Artin L-function, and denotes it by L(F K, χ, s) for a character χ of a representation of G(F K). A prime is called unramified if I G,P = {1}. One may note that there are finitely many primes that ramify in F, so that all but finitely many p in K have V = V I G,P, so that the expression det(i ρ(φ G,P )N (p) s V I G,P ) is a polynomial of degree dim(ρ) in q s. Theorem 3.5 The Artin L-function L(F K, χ, s) converges in the half plane Re(s) > 1. Proof. One may consider the factorization det(i ρ(φ G,P )N (p) s V I G,P ) = d p i=1 (1 ε i,p N (p) s ) for a prime p of o K where each ε i,p is a root of unity, and d p dim(ρ). Taking formally the logarithm of the Artin L-function L(F K, χ, s) then yields log L(F K, χ, s) = p d p m=1 i=1 ε i,p mn (p) ms. If Re(s) > 1, one has p p = p d p m=1 i=1 d p m=1 i=1 d p dim(ρ) p m=1 ε i,p mn (p) ms ε i,p mn (p) ms 1 mn (p) mre(s) m=1 [K : Q]dim(ρ) p Q 1 mn (p) mre(s) m=1 p prime 1 mp mre(s) = [K : Q]dim(ρ) log ζ(re(s)) 32

39 where ζ denotes Riemann s zeta function, which converges in the half-plane Re(s) > 1.. The following section in this chapter outlines some basic properties of Artin L- functions. 3.2 The basic properties of this function There will be a few properties of this function that have use in this work. First, one notes the following theorem. Theorem 3.6 (i) If each of χ and χ is a character of a representation of G(F K), then one has L(F K, χ + χ, s) = L(F K, χ, s)l(f K, χ, s); (ii) if F is a Galois extension of K containing F, and χ is a character of a representation of G(F K), then with the representation yielding χ acting on G(F K) via the canonical quotient map G(F K) G(F K), one has L(F K, χ, s) = L(F K, χ, s); (iii) if F is a subfield of F containing K, and χ is a character of a representation of G(F F ), then L(F F, χ, s) = L(F K, Ind G H(χ), s), where G = G(F K), and H = G(F F ). 33

40 Proof. (i) is trivial. For (ii), one may note that the canonical map G(F K) G(F K) yields a surjection G(F K) P /I G(F K),P G P/I G,P where P P p, with P a prime of o F and P a prime of o F, which maps the Frobenius of P to the Frobenius of P. This makes clear (ii). (iii) Suppose that p is a prime ideal of K, and that q 1, q 2,..., q r are the prime ideals of o F lying above p. Choose then for each i {1, 2,..., r} a prime ideal P i of o F lying above q i. One has the equalities and One has where G Pi H = H Pi I G,Pi H = I H,Pi. N (q i ) = N (p) f i, f i = G Pi : H Pi I G,Pi. By the previous theorem, one may choose an element τ i contained in G(F K) satisfying τ i P i = P 1. Then one will have G Pi = τ 1 i G P1 τ i, and also that I G,Pi = τ 1 i I G,Pi τ i. Considering then an element φ 1 G P1 that is mapping to the Frobenius φ G,P1 G P1 /I G,P1, one will also have that φ i = τ 1 i φ 1 τ i is similarly mapped to the Frobenius φ G,Pi. Also, the image of φ f i i in H Pi /I H,Pi is the Frobenius φ H,Pi. Considering then θ : H GL(W ) a representation of H yielding χ as its character, and (ρ, G, V ) denoting the induced representation, it will suffice to show that det(i ρ(φ 1 )t V I G,P 1 ) = r i=1 34 det(i θ(φ f i i )tf i W I H,P i).

41 Henceforth in this proof, the notation ρ(σ) will be written simply as σ, and likewise for θ. For each i {1, 2,..., r}, conjugation by τ i yields det(i φ f i i tf i where also f i = G P1 : (G P1 τ i Hτ 1 i )I G,P1. W I H,P i) = det(i φ f i 1 t f i τ i W I G,P 1 τ i Hτ 1 i ), For each i {1, 2,..., r}, one then selects a system of representatives {σ i,j } of G P1 mod G P1 τ i Hτ 1 i. Then {σ i,j τ i } is a system of representatives on the left of G mod H, and one obtains a decomposition for the vector space V corresponding to the induced representation (ρ, V, G) of (θ, W, H) as V = i,j σ i,j τ i W. Then by letting V i = j σ i,j τ i W, one obtains a decomposition V = i V i of V as a G P1 -module. Therefore one must have that Now it suffices to show that det(1 φ 1 t V I G,P 1 ) = r i=1 det(i φ 1 t V I G,P 1 i ). det(1 φ 1 t V I G,P 1 i ) = det(i φ f i 1 t f i τ i W I G,P 1 τ i Hτ 1 i ) for each i {1, 2,..., r}. One has, of course, that (ρ, V i, G P1 ) is the representation induced from (θ i, τ i W, G P1 τ i Hτ 1 i ). But also, one may notice that V G,I P 1 i = Ind G P 1 /I G,P1 {G P1 τ i Hτ 1 i }/{I G,P1 τ i Hτ 1 i (τ } iw I 1 G,P 1 τ i Hτi ). Taking then a basis {w 1,..., w d } for τ i W I G,P 1 τ i Hτ 1 i and noting the decomposition f i 1 V I G,P 1 i = φ l 1τ i W I G,P 1 τ i Hτ 1 i l=0 35

42 yields the matrix B = 0 I d d I d d A 0 0 for φ 1 acting on V G,I P 1 i, where I d d denotes the d d identity matrix and A denotes the matrix for φ f i 1 acting on τ i W I G,P 1 τ i Hτ 1 i. Thus, one must have det(i φ 1 t V I G,P 1 P i ) = det(i φ f i 1 t f i τ i W I G,P 1 τ i Hτ 1 i ), a fact which may be seen by multiplication of the first column of I fi d f i d Bt by t and subtraction from its second column, multiplication of the second column of the resulting matrix by t and subtraction from its third column, and so forth. This proves the claim. 36

43 CHAPTER 4 Class field theory 4.1 Idèles Class field theory provides an essential link between finite Galois extensions of an algebraic number field K and L-functions that ultimately leads to the functional equation for Artin s L-function, which is addressed in chapter seven of this work. In order to discuss class field theory, some definitions are in order. For this, one will deal with the finite primes of an algebraic number field K, given by p as before, but introduces now the notion of an infinite prime. In this cases, each infinite prime will be given by an embedding τ : K C, with the only restriction on this being that two such embeddings which are complex conjugates of each other are associated with the same prime. The notation p will be used to note that one is dealing with finite primes, and p will imply an infinite prime, given by an embedding as above. Together, these infinite and finite primes comprise what is called the primes of K. Each of the primes determines a valuation. In the case of a finite prime, the valuation, which is non-archimedean, is given in accordance with that assigned via the localization of K for the prime p. In the case of an infinite prime p, the valuation of K is defined to be x p = τx where denotes the modulus in C, and the embedding τ yields the infinite prime 37

44 p, where one may note that taking the complex conjugate of τ does not alter this valuation, and thus one has motivation for associating complex conjugate embeddings of K with the same infinite prime. Denote the completion of K with respect to the valuation given by the prime p as K p. One defines the idèle group I K to be the set of elements (α p ), with p ranging across all primes of K, infinite and finite, where α p K p is a unit in the ring of integers o p of K p of K with respect to the valuation given by the prime p, for almost all primes p. One equips this product space I K = p o p with a topology generated by sets of the form W p p S p S where S denotes a finite set of primes containing the infinite primes, and W p denotes a neighborhood of 1 K p in the topology corresponding to the valuation associated with the prime p. Considering then a finite Galois extension F of K, one may then define F p = F P P p a norm N Fp Kp : Fp Kp for (α P ) P p by N Fp Kp ((α P ) P p ) = det(α P ), P p with each α P here viewed as an automorphism from F P to F P over K p, and the determinant is taken according to the matrix of this automorphism of F P when viewed as a vector space over K p. In this way one obtains what is called a global norm U p N IF I K : I F I K 38

45 defined for α = (α P ) P I F by (N IF I K (α)) p = N Fp K p ((α P ) P p ), for each prime p of K. One now defines for a number field K the group C K = I K /K, known as the idèle class group, where each element x of K is viewed as diagonally embedded into I K by defining x = (x p ) p I K to have x p = x for all primes p of K. This is possible because the decomposition of the principal ideal (x) = p p ep into a product of prime ideals shows that x is a unit in o p for almost all primes p. One may then notice that the norm as defined above yields for x F that (N IF I K (x)) p = N L K (x), a fact which follows from the canonical isomorphism F K K p = F P p, P p and therefore the norm N IF I K defines a homomorphism C F C K. For a group G, Suppose now that G denotes the commutator subgroup of G. The following theorem is the main theorem of class field theory, and is known as Artin reciprocity. Theorem 4.1 There is an isomorphism A : C K /N IF I K C F G/G, 39

46 and the norm map N IF I K yields a one-to-one correspondence between finite Galois extensions F of K with abelian Galois group over K, and the subgroups of finite index in C K that are closed in the quotient topology induced by the canonical topology on I K. Denoting by N F the group N IF I K C F, one will then have the following facts for two such extensions F 1 and F 2 of K: (i) F 1 F 2 if and only if N F1 N F2 ; (ii) N F1 F 2 = N F1 N F2 ; and (iii) N F1 F 2 = N F1 N F2. In particular, this correspondence is explicitly given by the association F N F, and therefore, the field F corresponding to the closed subgroup N of C K of finite index must therefore have N = N F, and will thus satisfy C K /N = G(F K). Proof. Construction of the map A is given here to provide context; for the full proof of the theorem, the reader is referred to [Neu, VI]. First, there is a local reciprocity map, which is defined for finite primes, and also for infinite primes. For a finite prime p and an element σ G, one chooses an extension σ of σ to the maximal unramified extension of F P so that, when restricted to the maximal unramified extension K p of K p, it is a natural number power of the map φ of G( K p K p ) defined uniquely as φ(a) = a q mod p for all a in the valuation ring of Kp, where p denotes the maximal ideal of this valuation ring. Then, one considers the fixed field Σ of this extension σ, and a prime π Σ of Σ. One then constructs the map r FP K p : G(F P K p ) K p 40