9 Artin representations
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1 9 Artin representations Let K be a global field. We have enough for G ab K. Now we fix a separable closure Ksep and G K := Gal(K sep /K), which can have many nonabelian simple quotients. An Artin representation (ρ, V ) is a continuous homomorphism ρ : G K GL(V ) for some finite dimensional C-vector space V. Here GL(V ) is the space of automorphisms of V equipped with the natural topology from that of C. We say (ρ, V ) and (ρ, V ) are equivalent if there is an isomorphisms f : V V such that ρ = f ρ f 1. We identify equivalent Artin representations. It is common to abbreviate (ρ, V ) to ρ when there is no ambiguity. For any decomposition of vector space V = V 1 V 2, we have an inclusion of groups ι : GL(V 1 ) GL(V 2 ) GL(V ) where GL(V 1 ) acts trivially on V 2 and vise versa. For ρ i : G K GL(V i ), i = 1, 2, we say ρ = ρ 1 ρ 2 if ρ = ι (ρ 1, ρ 2 ) and say ρ is the direct sum of the subrepresentations ρ 1 and ρ 2. If this happens for some V 1, V 2 non-trivial we say ρ is reducible, and irreducible otherwise. Representation theory of the finite group G K / ker(ρ) gives Lemma 9.1. Every Artin representation is a unique direct sum (up to permutation) of irreducible subrepresentations. We have also seen Lemma 9.2. For any Artin representation ρ, Im(ρ) is finite. The kernel of ρ is a finite index open normal subgroup, and thus ker ρ = for some finite Galois extension L/K. Recall also that for any place v for K, we can fixed an embedding of separable closures K sep Kv sep, which induces ι v : G Kv G K. Different choices simply conjugate ι v by elements in G K. Definition 9.3. ρ is said to be unramified (resp. tamely ramified) at a non-archimedean place v iff the following equivalent condition holds: (i) L/K is unramified (resp. tamely ramified) at v. (ii) The image of the inertia I v (resp. wild inertia P v ) of G Kv is contained in ker ρ. It is obvious that any Artin representation is unramified almost everywhere. When ρ is unramified at v, the composition G Kv G K ρ GL(V ) factors through GKv /I v = Frob v. In general, let V Iv V be the subspace fixed by ρ(i v ), then Frob v acts on V Iv. Definition 9.4. Let (ρ, V ) be an Artin representation and v a non-archimedean place of K. We define the Artin local L-factor Recall Γ R (s) = π s/2 Γ(s/2). L v (s, ρ) := det(id V Iv q s v ρ(frob v ) V Iv ) 1. 1
2 Definition 9.5. Let (ρ, V ) be an Artin representation and v an archimedean place of K. We define the Artin local L-factor at v as follows: either K v = R and ρ = (triv) n 1 + (sgn) n 2, or K v = C and ρ = (triv) dim V is trivial. In the former case, we put L v (s, ρ) = Γ R (s) n 1 Γ R (s + 1) n 2. In the latter case put L v (s, ρ) = Γ R (s) dim V Γ R (s + 1) dim V. Remark 9.6. Despite that we chose a global language, one sees that all these definitions are purely local. For example, for F a local field and ρ : G F GL(V ) a continuous representation (i.e. homomorphism) we can define L F (s, ρ) likewise, so that L v (s, ρ) = L Kv (s, ρ GKv ). Definition 9.7. Let (ρ, V ) be an Artin representation. The Artin L-function is the Euler product L(s, ρ) = L v (s, ρ) v The Euler product converges again for Re(s) > 1. This is because as Im(ρ) is finite, Frob v is diagonalizable with eigenvalues being roots of unity. One sees directly that when dim V = 1, our Artin representations become Dirichlet characters, and our Artin L-functions agrees with the Dirichlet L-functions we had last time. In fact, they are compatible in even greater generality. Suppose G is any group and H G a (say finite index) subgroup, and ρ H : H GL(V ) a representation of H. One may think of V as a left C[H]-module, and form C[G] C[H] V as a left C[G]-module. This gives a representation ρ G : G GL(C[G] C[H] V ) and we write ind G H ρ H := ρ G. For any finite separable extension L/K and an Artin representation ρ L : GL(V ) for L, one checks that ind G K ρ L is an Artin representation for K. Proposition 9.8. Let L/K be a finite separable extension and ρ L be an Artin representation for L. Then L v (s, ind G K ρ L ) = w v L w (s, ρ L ) where w runs over places of L above v. Corollary 9.9. Let L/K be a finite separable extension and ρ L be an Artin representation for L. Then L(s, ind G K ρ L ) = L(s, ρ L ) where the first Artin L-function is of the Artin representation ind G K ρ L for K. To prove the proposition, we first establish its purely local part: 2
3 Lemma Let E/F be a finite separable extension of local fields and ρ E : GL(V ) a continuous representation. Then for L F (s, ind G F ρ E ) = L E (s, ρ E ). Proof. The case for F archimedean is obvious; we assume F non-archimedean. Let E be the maximal unramified subextension of E/F. Then we have from definition ind G F ρ E = ind G F G ind G E G E E ρ E, and the desired equality will follow from the two equalities L F (s, ind G F ρ E ) = L E (s, ind G E ρ E ) = L E (s, ρ E ). In other words, it suffices to deal with the case E = E and F = E. Let us write ρ F = ind G F ρ E. Also write I F the inertia in G F and likewise I E = I F the inertia in. First we consider the case E = F, so that E/F is totally ramified and equivalently G F = I F. In particular G F /I F = GE /I E and they share the same Frobenius σ. Let {θ 0 = 1, θ 1..., θ e 1 } be a set of representatives of G F / = I F / = IF /I E. Then e 1 V := C[G F ] C[GE ] V = θ i V and one has V I E V I F by v (v, θ 1 (v),...,, θ e 1 (v)). This is actually an isomorphism; every element in V fixed by I F, i.e. by I E and by θ 1,..., θ e 1 has to be of this form. Hence L F (s, ind G F ρ E ) = det(id V I F ρ F (σ) V I F ) 1 = det(id V I E ρ E (σ) V I E ) 1 = L E (s, ρ E ). Now suppose E = E, so that E/F is unramified and equivalently I F = I E. Let σ G F be any Frobenius. We have {1,..., σ f 1 } as a set of representatives of G F /, and σ f is a Frobenius in. We have f 1 f 1 V := C[G F ] C[GE ] V = σ i V, and V I F = σ i V I E. i=0 i=0 i=0 By the elementary properties of determinants this gives L F (s, ind G F ρ E ) = det(id V I F ρ F (σ) V I F ) 1 = det(id V I E ρ E (σ n ) V I E ) 1 = L E (s, ρ E ). Proof of Proposition 9.8. Now we are ready to prove the global result. The idea is that if {α i } is a set of representatives of G K /. Then we have V := C[G K ] C[GL ]V = i α iv. Now Frob v acts on V. The action is related to the action of G Kv on G K / We have a G Kv -equivariant (left action) bijection G K / = G Kv /w. w v Thus one has V = w v C[G K v ] C[GLw ] V as representations of G Kv, which is enough for defining L-factors. The desired identity thus follows from Lemma
4 Example Let L/K be any quadratic (separable) extension of global field. Let triv K (resp. triv L ) be the trivial representation of G K (resp. ) and χ L/K be the non-trivial character of Gal(L/K) pulled-back to G K. One has ind G K ind L = ind K χ L/K, and thus by Corollary 9.9 we have L(s, ind L ) = L(s, ind G K ind K ) = L(s, ind K )L(s, χ L/K ) which is well-known at least when K = Q. By the standard character theory of finite (or compact Hausdorff) groups, we know we can identify representations of Gal(L/K) (or G K ) as characters, i.e. certain functions on Gal(L/K) (or G K ), and one can talk about possibly negative integral linear combination of representations as virtual representations. Suppose now ρ as a representation of G = Gal(L/K) (or G = G K ) is an integral combination of certain ind G H χ for finite index open subgroups H G and 1-dimensional representations χ : H C. Then by Corollary 9.9 and our results for L-functions of Dirichlet characters, we see that L(s, ρ) is meromorphic for s C. This is indeed the case by a theorem of Brauer Theorem (Brauer) Let G be a finite group. Then any representation of G is an integral linear combination of inductions of 1-dimensional characters of subgroups H G. In fact, all H can be chosen among subgroups for which each is a direct product of a cyclic group and a p-group. Proof. We refer to the 7-page paper of Brauer and Tate, On the characters of finite groups, Ann. of Maths., 62, As taking dual of a representation is a conjugate-linear action on the space of characters, we have a functional equation relating L(s, ρ) and L(1 s, ρ ) where ρ is the dual representation. By Brauer s theorem we thus have Corollary Let ρ be any Artin representation. Then L(s, ρ) is meromorphic for s C, and there exists constants C(ρ), N(ρ) Q >0 as well as ɛ(ρ) C, ɛ(ρ) = 1 such that L(1 s, ρ ) = (C(ρ)N(ρ)) 1/2 ɛ(ρ)l(s, ρ). We would like to say a few words about C(ρ), N(ρ) and ɛ(ρ). Recall that when dim(ρ) = 1, we had N = v q nv v where the product runs over non-archimedean places v and n v is such that our chosen 1 additive character ψ K is trivial on ϖv nv O Kv but non-trivial on ϖv nv 1 O Kv. When K is a number field, N is the norm of the different to Q. When K contain F q as its constant field, we have N = q 2g 2. 1 When one scales the choice by some x K, the product formula ensures that N is scaled by x = 1, thus invariant. 4
5 Now we would like to put simply N(ρ) = N dim(ρ) ; in particular N(ρ) Z when K is a number field. It is then not an easy task to find a general formula for C(ρ). This had also been carried out by E. Artin and involves substantial interplay with ramification groups and class field theory. Recall that for a finite Galois extension of local fields E/F with G = Gal(E/F ), the lower ramification groups are defined by G i = {σ G σ(x) x ϖ i+1 E O E, x O E }, i 0 so that G 0 is the inertia and G 1 is the wild inertia. We define c E/F : G C as 0, g G 0 c E/F (g) = (i + 1) G G 0, g G i\g i+1 g G\{1} c E/F (g), g = 1 One directly computes that for any representation ρ of G with character χ ρ : G C we have (c E/F, χ ρ ) = 1 c E/F (g)χ ρ (g) = 1 χ ρ (1) χ ρ (σ). G G 0 g G i 0 σ G i Theorem (Artin) (c E/F, χ ρ ) Z 0. In other words, c is the character of some representation. Now let ρ be an Artin representation for our global field K which factor through Gal(L/K) for some L/K finite Galois. For each non-archimedean place v, take any w v of L, let G v = Gal(L w /K v ) Gal(L/K) and let χ ρv be the character of ρ Gv. Let the Artin conductor c(ρ) be c(ρ) := p (c Lw/Kv,χρv ) v. v and we also put C(ρ) := N(c(ρ)) = [O K : c(ρ)]. (When K is a global field, we should think of c(ρ) as a divisor and C(ρ) = q deg(c(ρ)).) In particular we see C(ρ) Z. Theorem (Artin) Our C(ρ) and N(ρ) are what is needed in Corollary There remains the question of expressing ɛ(ρ). It will be desirable to write ɛ(ρ) = v ɛ v(ρ, ψ). This turns out to be difficult: Theorem (Deligne) There exist unique constants ɛ F (ρ, ψ F ) C of absolute value 1 for any local field F, non-trivial additive character ψ F and virtual C-representation ρ of G F, such that ɛ F (ρ, ψ F ) is the same as what we had when ρ is a 1-dimensional (actual) represen- (i) tation. (ii) ɛ F (ρ 1 ρ 2, ψ F ) = ɛ F (ρ 1, ψ F )ɛ F (ρ 2, ψ F ). 5
6 (iii) Suppose E/F is a finite extension, ψ E = ψ F Tr E/F, and ρ E is any virtual representation of degree 0 of, we have ɛ E (ρ E, ψ E ) = ɛ F (ind G F ρ E, ψ F ). (iv) Let K be our global field with chosen ψ K : A K C. Write ψ Kv ɛ(ρ) = ɛ Kv (ρ GKv, ψ Kv ). v := ψ K Kv. Then Lastly, we mention the Artin Conjecture, which is one of the many motivations for Langlands program: Conjecture Let ρ be an irreducible non-trivial Artin representation. Then L(s, ρ) is entire. In the case when K is a global function field (say the function field of the algebraic curve X), the conjecture was proved by Weil in the case when K is a global function field. Nowadays this case is a standard application of Grothendieck-Lefschetz fixed point theorem, which allows us to express L(s, ρ) in terms of the l-adic cohomology of the local system on X associated to ρ. (In particular, the poles of L(s, ρ) comes from even degree l-adic cohomology in Hét even (X, ρ) = Hét 0 (X, ρ) H2 ét (X, ρ) = Hét 0 (X, ρ) H0 ét (X, ρ ) = 0, thus there are no poles.) 6
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