10 l-adic representations
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1 0 l-adic representations We fix a prime l. Artin representations are not enough; l-adic representations with infinite images naturally appear in geometry. Definition 0.. Let K be any field. An l-adic Galois representation (for K) is a continuous homomorphism ρ : G K GL(V ) where V is a finite-dimensional vector space over Q l. Example. There is a natural surjection G Q G ab Q = Ẑ. Explicitly, Q ab = Q(ζ ) is the extension of Q with all roots of unity. The Galois group G Q acts on the m-th roots of unity, giving a map G Q (Z/mZ). Taking the (inverse) limit of all such maps gives G Q Ẑ. One may also concentrate at a prime l by taking Ẑ Z l. This amounts to looking at the action of G Q on all l n -th roots of unity. We call the Galois representation χ cyc : G Q Z l the (l-adic) cyclotomic character. We have a geometric interpretation of the cyclotomic character: Let G m be the variety for which G m (k) = {(a, b) k 2 ab = } for any field k. Write µ l n the l n -th roots of unity (as a G Q -module), one has lim µ l n = H ét (G m, Z l ). We may restrict the Galois representation to G Qp for any prime p. The resulting map χ cyc : G Qp Z l is always still surjective. When p l, it s easy to see that the cyclotomic character is unramified, i.e. trivial on the inertia in G Qp. When l = p, this is no longer the case, but the geometry suggests no difference for l. This amounts to the phenomenon that when l = p, there is a notion in p-adic Hodge theory that corresponds to the usual unramified property, that χ cyc : G Qp Z p is crystalline. Lemma 0.2. Any compact subgroup H GL n ( Q l ) is contained in GL n (E) for some E/Q l finite. Proof. The set H is compact Hausdorff, and thus a Baire space; it is never a countable union of nowhere-dense closed subset. Let E/Q l be any finite extension. Either the intersection H GL n (E) has finite index in H, so that by adding finitely many elements into E we have H GL n (E), or H GL n (E) has infinite index in H. We now assume the latter is the case for any E/Q l finite. There are countably many (why?) finite extensions of Q l. Thus H = E H GL n(e) as a countable union. However each H GL n (E) cannot contain an open subgroup and thus are nowhere-dense, a contradiction. Corollary 0.3. Any l-adic Galois representation is realized on a E-vector space for some E/Q l finite. Example 2. Another difference that (the category of) l-adic Galois representations is different from (that of) Artin representations or representations of compact groups is that an l-adic Galois representation need not be semisimple. For example, let F be a non-archimedean local field with residue field k, char(k) l, and consider the composition
2 χ : G F Ẑ Z l where the first map is the natural projection from G F to Gal( k/k) = Ẑ and the second map is the natural projection from Ẑ to Z l. Consider the representation χ χ(σ) ρ =, i.e. ρ(σ) =, σ G F. It s obvious that ρ is non-trivial, yet it fits into an exact sequence of l-adic representations and thus ρ is not semisimple. triv ρ triv. In general, for a (ρ, V ) be an l-adic Galois representation we will find a filtration 0 = V 0 V... V m = V of subrepresentations, so that each V i /V i is an irreducible Galois m representation. The semi-simplification of V is V ss := V i /V i. Theorem 0.4. (Jordan-Hölder) The semi-simplification V ss is unique up to isomorphism. Theorem 0.5. (Brauer-Nesbitt) Two representations ρ and ρ of G K (in fact, any group) on a finite-dimensional vector space over any field have isomorphic semi-simplification if and only if ρ(σ) and ρ (σ) have the same characteristic polynomial for all σ G K. Let (ρ, V ) be any l-adic Galois representation of G K, where by Lemma 0.2 we take V to be an E-vector space, E/Q l finite. Let Λ V be any lattice, i.e. a O E -submodule of rank equal to dim E V. Let GL(Λ ) := {g GL(V ) gλ Λ }. This is an open subgroup of GL(V ), and thus H := ρ (GL(Λ )) is open. Let σ,..., σ s be a set of representative for G K /H. Then Λ = s i= ρ(σ i)λ is a ρ(g K )-stable lattice. Now let Λ be an arbitrary ρ(g K )-stable lattice of V. We may consider the quotient Λ := Λ/ϖ E Λ. We have a natural induced action of G K on Λ. Corollary 0.6. The semi-simplification of the G K -representation Λ does not depends on the choice of Λ. Proof. The characteristic polynomial of any element in G K on Λ comes from reducing the characteristic polynomial on V modulo ϖ E, and thus independent of Λ. The result then follows from Theorem 0.5. Example 3. Let p be a prime with l p +. We have a natural surjection Z p 2 F l. By taking any isomorphism F l {x Q l x l = } and Q p = Z 2 p 2 Z, we have a character χ : Q p Q 2 l, which extends to a -dimensional l-adic Galois representation χ : G Qp 2 Q l. Let ρ = indg Qp G Qp χ : G Qp GL 2 (Q l ). This is an irreducible representation, 2 while ρ ss = triv sgn Qp 2 /Q p. 2 i=
3 We now fix some E/Q l finite and a non-archimedean local field F with residual characteristic p l. Let G F I F P F be the absolute Galois group, the inertia, and the wild inertia, respectively. For an l-adic Galois representation ρ : G F GL(V ), we first look at how ρ IF looks like. Recall that we have I F /P F = ( Ẑ p ) := p p Z p. (Sketch of proof: By Hensel lemma style argument, any degree m extension of F ur with p m is of the form F ur ( m ϖ F ) for a fixed uniformizer ϖ F.) Write t l : I F Z l the composition I F I F /P F Z l. Definition 0.7. Say a compact topological group G is prime to l if x x l is a homeomorphism. We have ker(t l ) is prime to l. One easily proves the following (see also exercise) Lemma 0.8. If G is a prime-to-l group and ρ : G GL n (E) an l-adic representation, then the image of ρ is finite. Corollary 0.9. (Grothendieck s l-adic monodromy theorem) Let ρ : G F GL n (E) be an l-adic Galois representation. Then there exists a relatively open subgroup U I F and a necessarily unique n n nilpotent matrix N M n (E) such that ρ(τ) = exp(t l (τ)n), τ U. Proof. By Lemma 0.8, ker(ρ) contains an open subgroup of ker(t l ). By shrinking U we may assume ρ U ker(tl ) is trivial. We have seen that the image of ρ stabilize a lattice, and thus by conjugation we may assume Im(ρ) GL n (O E ). By shrinking U again we may assume ρ(u) K (2) := {g GL n (O E ) g = Id mod l 2 }, so that it makes sense to talk about log ρ(τ) for τ U. We have t l (U) log = U/(U ker(t l )) K (2) l 2 M n (O E ). Since t l (U) = l m Z l for some m, the above composition has to be given by x xn for some N M n (E). This is the N we seek. It remains to prove that N is nilpotent. Let Φ G F be any (lift of) arithmetic Frobenius. Then for any τ I, we have t l (ΦτΦ ) = q F t l (τ) where q F is the order of the residue field of F. Applying this to the definition of N we see ρ(φ)nρ(φ) = q F N. The eigenvalues of N thus have to be all zero and N has to be nilpotent. Example 4. We have the following l-adic Galois representation ρ : G F GL 2 (Q l ) given by χcyc t ρ = l. 0 This is somewhat an example of Galois representation that is not semisimple, but the ρ(φ) is semisimple, i.e. diagonalizable over Q l. It is a general conjecture that Frobenius should always acts semisimply. In fact, this is probably the most important non-trivial Galois representation; it s the Tate module of an elliptic curve that has semistable but not good reduction. 3
4 We have understood the difference between local Artin representations and l-adic Galois representations of G F up to the inertia I F. To state the final result in a better way and to connect with Langlands program, we d like to switch from the absolute Galois group G F to the so-called Weil group. Recall that we have an exact sequence I F G F Ẑ We also suppose that our local class field theory is normalized so that an arithmetic Frobenius Φ is mapped to the inverse of a uniformizer. We then define the Weil group W F of F to be the preimage of Z Ẑ, but equipped with a new topology similar to that of Z Ẑ. That is, the topology on W F is such that I F has the same subspace topology as before but is open in W F. Note also that by local class field theory we have WF ab = F. We also denote by : W F C the normalized norm, so that Φ = q F. An l-adic Weil representation for F is a homomorphism ρ : W F GL(V ). One easily sees that the result of Corollary 0.9 also holds for Weil representations. We now fix a Weil representation (ρ, V ) where V is an E-vector space. Let Φ again be any (lift of) Frobenius. We have Theorem 0.0. (Deligne) Let t l be as above and N as in Corollary 0.9. The formula ρ (Φ m τ) := ρ(φ) m ρ(τ) exp( t l (τ)n), m Z, τ I F defines a new representation of W F on V. Moreover, the isomorphism class of (ρ, V ) does not depend on the choice of Φ. Proof. Recall that we had t l (ΦτΦ ) = q F t l (τ) ρ(φ)nρ(φ) = q F N. A similar argument gives t l (στσ ) = σ t l (τ) and ρ(σ)nρ(σ) = σ N for any σ W F. Similarly we have ρ(σ) exp(xn)ρ(σ) = exp( σ xn). Now for the first statement of the theorem, suppose we have Φ m τ Φ m 2 τ 2 = Φ m +m 2 τ, i.e. τ = Φ m 2 τ Φ m 2 τ 2. Then ρ (Φ m +m 2 τ) = ρ(φ m +m 2 τ) exp( t l (τ)n) = ρ(φ m τ )ρ(φ m 2 τ 2 ) exp( t l (Φ m 2 τ Φ m 2 )N) exp( t l (τ 2 )N) = ρ(φ m τ )ρ(φ m 2 τ 2 ) exp( q m 2 F t l (τ )N) exp( t l (τ 2 )N) = ρ(φ m τ ) exp( t l (τ )N)ρ(Φ m 2 τ 2 ) exp( t l (τ 2 )N) = ρ (Φ m τ )ρ (Φ m 2 τ 2 ). For the second statement, suppose Φ is a different choice of Frobenius, and Φ = Φ τ, with t l (τ ) = s. Then the resulting ρ has ρ (Φ) = ρ (Φ τ ) = ρ (Φ) exp( sn). On the other hand, we have ρ(φ) exp(xn)ρ(φ) = exp(xn) q F ρ(φ) = exp(xn) q F ρ(φ) exp(xn) ρ(φ) exp(xn) q F = exp(xn) q F ρ(φ)(exp(xn) q F ). Taking x = s, we get ρ q F (Φ) = ρ(φ) is conjugate to ρ (Φ) exp( sn) = ρ (Φ) by exp(xn)q F. Since for any τ I we have ρ (τ) = ρ (τ ) commutes with N and exp(xn) by the formula, we have ρ is conjugate to ρ by exp(xn). 4
5 Remark 0.. ρ IF is trivial on the open subgroup U I F in Corollary 0.9, and thus has finite image and is a semisimple representation. On the other hand, in Corollary 0.9 we have N = 0 iff ρ IF has finite image, which is also equivalent to ρ = ρ. Definition 0.2. A Weil-Deligne representation is a pair (ρ, N) where ρ : W F GL n (C) is a continuous homomorphism, and N M n (C) is a nilpotent n n matrix in C such that ρ (σ)nρ (σ) = σ N for any σ W F. When N = 0, it s also common to call it a Weil representation. We fix an isomorphism ι : Q l C. Corollary 0.3. Upon the choice of ι, Theorem 0.0 gives a bijection between the set of isomorphism classes of l-adic Weil representations and the set of isomorphism classes of Weil-Deligne representations. We can also talk about L-functions of Weil-Deligne representations. Definition 0.4. Let ρ = (ρ, N) be an n-dimensional Weil-Deligne representation. Let V := ker(n) I F be the subspace of ker(n) on which ρ (I F ) acts trivially. Then L(s, ρ) := det(id V q s ρ (Φ) V ). Lemma 0.5. Let (ρ, N) be a Weil-Deligne representation. If ρ (Φ) is semisimple, then ρ (σ) is semisimple for any σ W F. Proof. Let U I F be as in Corollary 0.9. By shrinking U we may assume U is normalized by W F. For any σ = Φ m τ W F, we my write σ a = Φ am τ a for any a Z. The image of τ a in I F /U has to be periodic, and thus τ a U for some a > 0. This gives ρ (σ) a = ρ (Φ) am is semisimple. Since C has characteristic zero, ρ (σ) is also semisimple. (If m = 0, then σ I F and the semisimplicity of ρ (σ) is automatic as ρ IF has finite image. Definition 0.6. A Weil-Deligne representation is called Frobenius-semisimple if it satisfies the property in Lemma 0.5. A general conjecture is that Weil-Deligne representations that correspond to interesting (i.e. those that come from geometry) l-adic Weil representations should be Frobeniussemisimple. What is very exciting is then that Frobenius-semisimple Weil-Deligne representations can be related to representation theory by the local Langlands conjecture. Remark. Theorem 0.0, Definition 0.2 and all those follow also work for arbitrary reductive group. For example, let V be any quadratic space over Q. Then SO(V ) is a reductive group, and one may consider ρ : W F SO(V Q Ql ) and ρ : W F SO(V Q C). Then N should lie in the Lie algebra of SO(V Q C), namely a anti-self-adjoint nilpotent operator on V. 5
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