MA 162B LECTURE NOTES: FRIDAY, JANUARY 30 1. Examples of Cohomology Groups (cont d) 1.1. H 2 and Projective Galois Representations. Say we have a projective Galois representation ρ : G P GL(V ) where either G G Q or G G p i.e. either ρ is a local or global representation. I claim that there exists a continuous Galois representation ρ : G GL(V ) such that ρ : G ρ GL(V ) mod Z(V ) P GL(V ) i.e. the proejctive representation ρ has a lift to a bona fide representation ρ. To see why the claim is true, for each σ G choose matrices α(σ) GL(V ) such that α(σ) ρ(σ) mod Z(V ). Of course, these matrices do not form a group in general i.e. α(σ 1 σ 2 ) α(σ 1 ) α(σ 2 ). However, we do have α(σ 1 σ 2 ) ρ(σ 1 σ 2 ) ρ(σ 1 ) ρ(σ 2 ) α(σ 1 ) α(σ 2 ) mod Z(V ) so the element α(σ 1 ) α(σ 2 ) α(σ 1 σ 2 ) 1 is a scalar, say ξ(σ 1, σ 2 ) K. I will show that this continuous map ξ : G G K is actually a 2-cocycle i.e. d 2 (ξ) (σ 1, σ 2, σ 3 ) ξ(σ 2, σ 3 ) ξ(σ 1 σ 2, σ 3 ) 1 ξ(σ 1, σ 2 σ 3 ) ξ(σ 1, σ 2 ) 1 1 where the action of G on X K is trivial. Indeed, ξ(σ 2, σ 3 ) ξ(σ 1, σ 2 σ 3 ) 1 d ξ(σ 1, σ 2 σ 3 ) ξ(σ 2, σ 3 ) 1 d [α(σ 1 ) α(σ 2 σ 3 ) α(σ 1 σ 2 σ 3 ) 1] [ α(σ 2 ) α(σ 3 ) α(σ 2 σ 3 ) 1] [ ] [ ] α(σ 1 σ 2 σ 3 ) 1 α(σ 1 ) α(σ 2 σ 3 ) α(σ 2 σ 3 ) 1 α(σ 2 ) α(σ 3 ) α(σ 1 ) α(σ 2 ) α(σ 3 ) α(σ 1 σ 2 σ 3 ) 1 [α(σ 1 ) α(σ 2 ) α(σ 1 σ 2 ) 1] [ α(σ 1 σ 2 ) α(σ 3 ) α(σ 1 σ 2 σ 3 ) 1] ξ(σ 1, σ 2 ) ξ(σ 1 σ 2, σ 3 ) 1 d Hence, ξ Z 2 (G, K ). Note that ξ corresponds to a homothetic transformation on V, so it makes sense that the action of G on K is trivial. By a classical theorem of Tate, H 2 (G, Q ) {1} is trivial where the action by either G G Q or G G p is trivial, so H 2 ( G, K ) H 2 ( G, Q ) K {1} is trivial as well. Hence Z 2 (G, K ) B 2 (G, K ) so ξ d 1 (β) is a 2-coboundary i.e. there is a map β : G K which may not be a homomorphism! such that ξ(σ 1, σ 2 ) β(σ 1 ) β(σ 2 ) β(σ 1 σ 2 ) 1. Define the map ρ : G K GL(V ) as the product ρ(σ) β(σ) 1 α(σ) by construction ρ ρ mod Z(V K K). This is a 1
2 MA 162B LECTURE NOTES: FRIDAY, JANUARY 30 continuous homomorphism: for σ 1, σ 2 G we have ρ(σ 1 ) ρ(σ 2 ) ρ(σ 1 ) ρ(σ 2 ) ρ(σ 1 σ 2 ) 1 ρ(σ 1 σ 2 ) [β(σ 1 ) β(σ 2 ) β(σ 1 σ 2 ) 1] 1 [ α(σ 1 ) α(σ 2 ) α(σ 1 σ 2 ) 1] ρ(σ 1 σ 2 ) ρ(σ 1 σ 2 ) thereby proving the claim. In particular, say that we have a projective residual representation Gal ( Q/Q ) P GL 2 (F l ), such as a tetrahedral, octahedral, or icosahedral representation. Then it lifts to a bonafide residual representation ρ : Gal ( Q/Q ) F l GL 2 (F l ). In general, given a projective representation ρ : Gal ( Q/Q ) P GL d (K) we have the exact diagram 1 Gal (L/F ) Gal (L/Q) Gal (F/Q) 1 χ ρ eρ 1 K K GL d (K) P GL d (K) 1 2. Galois Cohomology of the Adjoint Representation In this section we fix a residual Galois representation ρ : G Q GL 2 (k), where we assume that k is a finite extension of F l. Ultimately we wish to discuss the properties of lifts ρ to characteristic zero i.e. representations such that there is a composition We assume that l 2. ρ : Gal ( Q/Q ) ρ GL d (O) mod λ GL 2 (k). 2.1. Adjoint Action and the Symmetric Square. Denote V k k so that ρ : G Q GL(V ). We consider the module X End k (V ). The residual representation ρ induces an action of the absolute Galois group on the k-vector space of 2 2 matrices given by σ x ρ(σ) x ρ(σ) 1. We denote this k-vector space with such an action by ad ρ. Explicitly, denote the matrices e 1 ( 0 1 0 0 ), e 2 ( 1 0 0 1 ), and e 3 ( ) 0 0 1 0 then ad ρ is the k-vector space spanned by the identity matrix as well as the e j because l is odd. We have a filtration of submodules ad 0 a b ρ k{e 1, e 2, e 3 } a, b, c k} c a ad 1 a b ρ k{e 1, e 2 } a, b k} a ad 2 0 b ρ k{e 1 } b k}. 0 0 The submodule ad 0 ρ consists of those matrices of trace zero, while ad 1 ρ consists of those traceless matrices that are upper-triangular. The submodule ad 2 ρ consists of the strictly upper-triangular matrices, while the quotient ad 0 ρ/ad 1 ρ consists of the strictly lower-triangular matrices.
MA 162B LECTURE NOTES: FRIDAY, JANUARY 30 3 There is another way to consider the adjoint of a residual representation using the symmetric square. The Galois action on the basis {e 1, e 2, e 3 } induces a 3- dimensional representation G Q SL 3 (k). Explicitly, σ 1 a2 2 a b b 2 ( ) a c a d + b c b d a b where ρ(σ) a d b c c 2 2 c d d 2 c d and where the determinant of this 3 3 matrix is 1. On the other hand, we have a map Sym 2 : GL 2 (k) GL 3 (k), called the symmetric square, which is defined as follows. Given the vector (x, y) V we act in the canonical way: ( ) ( ) ( ) a b x a x + b y. c d y c x + d y Now consider the vector ( x 2, xy, y 2) Sym 2 (V ). We have a slightly different action: ( ) x 2 (a x + b y) 2 a 2 x 2 + 2 a b x y + y 2 a b x y (a x + b y) (c x + d y) a c x c d 2 + (a d + b c) x y + b d y 2 y 2 (c x + d y) 2 c 2 x 2 + 2 c d x y + d 2 y 2 a2 2 a b b 2 a c a d + b c b d x2 x y. c 2 2 c d d 2 y 2 We define the twisted symmetric square Ad 2 : GL 2 (k) SL 3 (k) as the adjoint: ( ) Ad 2 det 1 Sym 2 a b : 1 a 2 2 a b b 2 a c a d + b c b d. c d a d b c c 2 2 c d d 2 These matrices are the same as those above i.e. the Galois action on {e 1, e 2, e 3 } inducing the 3-dimensional representation is equivalent to the composition Gal ( Q/Q ) ρ Ad GL 2 (k) 2 SL 3 (k). 2.2. Infinitesimal Deformations. Let ρ : G Q GL 2 (k) be a continuous Galois representation, and let ɛ be an infinitesimal i.e. ɛ 2 0. I claim that the equivalence classes of infinitesimal deformations ρ ɛ : G Q GL 2 (k[ɛ]) satisfying ρ ɛ ρ mod ɛ k[ɛ] and det ρ ɛ det ρ are in one-to-one correspondence with the cohomology classes in H 1 ( G Q, ad 0 ρ ). Given σ G Q we may express an infinitesimal deformation in the form ρ ɛ (σ) (1 2 + ɛ ξ σ ) ρ(σ) for some ξ σ Mat 2 (k), where ξ σ must have trace zero since det ρ(σ) det ρ ɛ (σ) (1 + ɛ trace ξ σ ) det ρ(σ). I will show that equivalence classes of homomorphisms are in one-to-one correspondence with ξ H 1 ( G Q, ad 0 ρ ). For ρ ɛ to be a homomorphism we must have 1 2 ρ ɛ (σ τ) ρ ɛ (τ) 1 ρ ɛ (σ) 1 (1 2 + ɛ ξ στ ) ρ(σ τ) ρ(τ) 1 (1 2 ɛ ξ τ ) ρ(σ) 1 (1 2 ɛ ξ σ ) (1 2 + ɛ ξ στ ) (1 2 ɛ [ ρ(σ) ξ τ ρ(σ) 1]) (1 2 ɛ ξ σ ) 1 2 + ɛ [ξ στ σ ξ τ ξ σ ].
4 MA 162B LECTURE NOTES: FRIDAY, JANUARY 30 Hence ρ ɛ is a homomorphism if and only if ξ is a 1-cocycle i.e. d 1 (ξ)(σ, τ) σ ξ τ ξ στ + ξ σ 0. The equivalence class of ρ ɛ consists of conjugates A ρ ɛ A 1 where A 1 2 ɛ α for α Mat 2 (k). But A ρ ɛ (σ) A 1 (1 2 + ɛ [σ α α]) ρ ɛ (σ) so equivalence classes yield ξ σ modulo coboundaries i.e. σ α α. Hence ξ H 1 ( G Q, ad 0 ρ ) as desired. 3. Universal Deformation Ring Fix a prime l. We make the following conventions: Let G be a profinite group, such as the global Galois group G Q Gal ( Q/Q ) or the local Galois group G p Gal ( ) Q p /Q p. Let V be a d-dimensional vector space over a field K, which we assume is a subfield of either C, Q l, or F l. Let ρ : G GL(V ) be a continuous Galois representation. Recall that ρ is either complex, l-adic, or residual depending on the choice of K. 3.1. Image of Representation. I claim that we can always choose K such that either K/Q, K/Q l, or K/F l is a finite dimensional extension. To this end, let H be an open subgroup of G i.e. G/H is a finite group. Recall that we have the representation ρ : G/H GL(V H ). Since we may choose a basis so that V K d, we have a representation ρ : G GL d (K). Let K H be that subfield of K generated by the matrix entries of ρ(g/h) i.e. K H is the smallest subfield such that ρ : G/H GL d (K H ). As G/H is finite, the extension K H /Q, K H /Q l, or K H /F l is also finite. As G proj lim H G/H and V lim H V H we may choose K lim H K H. By the discrete topology, we have K is a finite extension as claimed. In particular, if we are given a residual representation ρ : Gal ( Q/Q ) ( ) GL d Fl then we know the image lies in a finite group GL d (k) for some finite extension k of F l. 3.2. Witt Vectors. Fix a prime l. We know there is an exact sequence of rings 0 l Z l Z l mod l F l 0. I claim that given any finite field extension k/f l there is a local, Noetherial ring W (k) containing Z l such that modulo the maximal ideal m we have the exact sequence of rings 0 m W (k) mod m k 0. This construction will follow the exercises at the end of chapter VI in Lang s Algebra. Write k F l [α] where α ln α 0. Then as rings, we have a surjective homomorphism Z l [T ] F l [α] given by T α mod l. Recall that Z l [T ] is a Noetherian algebra. For a countable collection of elements x 1, y 1, x 2, y 2, Z l [T ] consider the vectors x (x 1, x 2,... ) and y (y 1, y 2,... ) we let W (Z l [T ]) be the Z l -linear combination of such vectors. We explain how to turn this into a ring. We define x + y (z 1, z 2,... ) implicitly via the equation (1 z n T n ) (1 x a T a ) ( 1 y b T b). n1 a1 b1
MA 162B LECTURE NOTES: FRIDAY, JANUARY 30 5 For example, z 1 x 1 + y 1 z 2 x 2 + y 2 x 1 y 1 z 3 x 3 + y 3 x 1 x 2 y 1 y 2 x 1 y 2 x 2 y 1. We also define x y (w 1, w 2,... ) implicitly via the equation ( (1 w n T n ) 1 x c/a a y b/c b T c) ab/c, c LCM(a, b). n1 For example, w 1 x 1 y 1 a1 b1 w 2 2 x 2 y 2 + x 2 1 y 2 + x 2 y 2 1 w 3 3 x 3 y 3 2 x 1 x 2 y 1 y 2 + x 3 1 (y 3 y 1 y 2 ) + (x 3 x 1 x 2 ) y 3 1. We call the image W (k) of W (Z l [T ]) under the map T α mod l the ring of Witt vectors of k. Note that by construction W (F l ) Z l through the map T 0.