MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26

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1 MA 162B LECTURE NOTES: THURSDAY, FEBRUARY Abelian Varieties of GL 2 -Type 1.1. Modularity Criteria. Here s what we ve shown so far: Fix a continuous residual representation : G Q GLV, where V is a 2-dimensional vector space over a finite field k of odd characteristic l. Assume that MC1: ramifies at finitely many primes. MC2: We have a G l -stable short exact sequence V I l V V Il where both V I l and V Il are 1-dimensional, I l acts via det on V I l, and I l acts trivially on V Il. MC3: The restriction : Gal Q/Q 1 l 1/2 l GLV is absolutely irreducible. MC4: We have the dimension formula dim k Hf 1 Gl, ad = dim k H G l, ad + 1. MC5: is modular. Then any continuous lift : G Q GL 2 Q l is l-adically modular. We discuss when these criteria are satisfied for a rational abelian variety. Say that A is a simple abelian variety of GL 2 -type. Recall that this means there exists some number field K/Q such that K End Q A Z Q and the degree [K : Q] = dima. These conditions imply that the 2d-dimensional l-adic Tate module V l A = T l A Zl Q l is the direct sum of the d Galois conjugates of 2-dimensional λ-adic Tate modules V l A over K λ. Hence we have a continuous 2-dimensional representation A,λ : Gal Q/Q GL 2 Ql. It is clear that when A = E is an elliptic curve then E is an abelian variety of GL 2 -type; indeed we may choose K = Q since dime = 1. The 2-dimensional representation E,l is just the l-adic representation on the Tate module. We recall some notation. Denote O as the ring of integers in K λ, and k = O/λ as a finite field extension of F l. Then we have the composition A,λ : Gal Q/Q A,λ GL2 O mod λ GL 2 k coming from the Galois action on the module A[λ]. We discuss hypothesis sufficient to impose on A such that this residual representation satisfies the five modularity criteria above. We begin by discussing some of the properties of the Galois representations associated to such varieties. 1

2 2 MA 162B LECTURE NOTES: THURSDAY, FEBRUARY Galois Representations of Q-Curves. Let F be a number field that is Galois over Q, and E be an elliptic curve defined over F such that each Galois conjugate E σ is isogenous to E for σ GalF/Q. We say that E is a Q-curve in this case. Denote A = Res F/Q E as the Weil restriction of scalars for E. The abelian variety A is simple if E cannot be defined over any field smaller than F. We saw a few weeks back that A is an abelian variety of GL 2 -type, so what is the 2-dimensional λ-adic Galois representation associated to A? How is this representation related to the 2-dimensional l-adic Galois representation associated to E? We show how to exhibit a 2-dimensional Galois representation following ideas of Jordan Ellenberg, Ken Ribet, and Chris Skinner. The idea is to consider the projective representation E,l : Gal Q/F E,l GL2 Z l mod Z l P GL 2 Z l. As E is a Q-curve, for each σ Gal F/Q we can find isogenies α σ : E σ E. Assume that F is large enough so that each of the isogenies is defined over F. Denote the element ξσ, τ = α σ α σ τ 1 deg α στ α στ EndE Z Q. Recall that α σ : E E σ is the dual isogeny. Assuming that E does not have complex multiplication then the composition α σ α σ : E E is the multiplicationby-m map where m = ± deg α σ. One shows that ξ H 2 G Q, Q = {} = ξσ, τ = βσ βτ βστ 1 for β : G Q Q. Define the representation ϱ E,l : G Q GL 2 K λ by its Galois action on V l E = T l E Zl Q l : ϱ E,l σ P = βσ 1 α σ P σ = ϱ E,l GF E,l. We conclude that ϱ E,l GF χ E E,l in terms of the character χ E : Gal Q/F β 1 Q Q l. In fact, if we denote K = Q βσ σ GalF/Q, and λ a prime of K lying above l, then we can choose β and hence χ E such that ϱ E,l A,λ. This shows in particular that the weight of such a representation is κ = 2. Let s denote χ as that character such that det A,λ = χ ɛ l /ω l. Recall that we say an abelian variety A is modular if there is a nonconstant morphism X 1 N A. In particular, if E is a Q-curve, then we can set A = Res F/Q and there should be a nonconstant map X 1 N E. This would imply the existence of a cusp form fτ of level N and a character χ E such that f GF χ E E,l. Ken Ribet conjectured that if A is a simple abelian variety of of GL 2 -type then A is indeed modular. As a consequence, he conjectured that all Q-curves are modular.

3 MA 162B LECTURE NOTES: THURSDAY, FEBRUARY Quadratic Q-Curves. We make this explicit for a certain class of Q-curves. Fix d Q Q 2, and consider the elliptic curve { E = x 1 : x 2 : x P 2 x x 2 2 = x x x } d x 2 x 1. 2 It can be shown that E is a Q-curve. In particular, E is a curve defined over F = Q d where the isogeny E σ E has degree 2 for σ : d d. Explicitly, the isogeny comes from the cyclic subgroup { : : 1, : 1 : }. Moreover, one can show that EndE Z Q Q 2, so that the 2-isogeny is actually defined over Q 2, d. Conversely, if E is an elliptic curve defined over Q d where there is an isogeny E σ E of degree 2 then E is in the form above up to twist that is. The associated abelian variety is A = {x 11 : x 12 : x 21 : x 22 : x P 4 x 11 + d x 12 : x 21 + } d x 22 : x E. From this it is clear that AQ EF and AF EF E σ F. We focus on the representation A,λ. There exists a character χ E : G F Q l such that A,λ GF χ E E,l. In fact, χ σ 1 E : Gal Q/F Gal Q 2, d/f {±1}. In particular, χ/ω l GF = χ 2 E can be chosen to be a quadratic character so that χ/ω l is a character of order 4. For p N E l, denote a p = trace A,λ Frob p. We can recover information about A,λ by noting that { a p if p has degree 1, χ E p a p E = trace A,λ Frob p = a 2 p 2 χ/ω l p p if p has degree 2. In particular, the a p lie in a quadratic extension K/Q. We give a specific example. Say that l = 5 and F = Q 5. Then d = 5 d 2 for some d Q, and we can construct χ E explicitly using Dirichlet characters because O F = Z[ ] has narrow class number 1. We find that χ/ω 5 is the character of order 4 defined by the compositions χ : ω 5 : Gal Q/Q Gal Qζ 4 /Q Gal Q/Q Gal Qζ 5 /Q µ 2 ; µ 4. Then det A,λ = χ/ω 5 ɛ l. In fact, since the ratio χ/ω 5 does not depend on the prime l, this equality holds for any prime l. We mention in passing that the ratio χ/ω 5 is unramified outside of {2, 5, } Level and Conductor. It is well-known in general that A,λ ramifies at finitely many primes; hence MC1 is always satisfied. In fact, pick a level N, and consider a newform fτ of level N. Recall that we may associate a simple abelian variety A f of GL 2 -type to fτ = n a n q n satisfying the property trace Af,λ Frob p = a p for all p N l. One can prove that the conductor of A satisfies N A = N d where d = dima. If A = E is an elliptic curve, we can consider the reduction modulo l for l 2, 3. Say if the curve has equation E = { x 1 : x 2 : x P 2 x x 2 2 = x c 4 x 2 x 1 2 c 6 x 3 }

4 4 MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26 for some constants c 4 and c 6 we say that 1 E has good reduction if l c 3 4 c 2 6; 2 E has multiplicative reduction if l c 3 4 c 2 6 yet l c 4 ; and 3 E has additive reduction if l c 4 and l c 6. We say that E is semistable at l if either E has good reduction at l or multiplicative reduction at l. It is well-known that l 2 divides the conductor N E if and only if E has additive reduction at l. Hence an elliptic curve E is semistable if and only if N E is square-free Ordinary Galois Representations. We say that the residual representation A,λ is ordinary at l if the restriction χl A,λ Gl where χ χ l = χ/χ, χ Il = 1. Recall that we set up the notation so that det A,λ = χ. The representation A,λ : G Q GLV acts on the 2-dimensional vector space V = A[λ], so we may define V I l = { v V σ v = det A,λ v for all σ I l } as that subspace such that G l acts via multiplication by χ l. Then G l acts on the quotient V Il = V λ A/V I l via multiplication by χ, and so is unramified. As long as either MC2a: det A,λ Il = ɛ l such as when A = E; or MC2b: Il i.e. A,λ is wildly ramified at l; then dim k V I l = dim k V Il = 1. It is in these cases that MC2 holds Galois Cohomology. We denote = A,λ as the mod λ reduction. Assuming that A has ordinary reduction at l, recall the definitions K 1 K 2 K 1 Hf 1 Gl, ad K 2 H 1 G l, ad 1 H 1 G l, ad K 2 H 1 K 1 K 2 I l, ad1 ad 2 H 1 I l, ad ad 2 If det Il = ɛ l then H G l, ad /ad 1 = {} so that K 2 is trivial. If is wildly ramified at l then K 1 K 2 is trivial. In these cases we have Hf 1 Gl, ad ker [H 1 G l, ad 1 ] H 1 G l, ad1 ad 2 H 1 I l, ad1 ad 2.

5 MA 162B LECTURE NOTES: THURSDAY, FEBRUARY 26 5 The kernel of the second map is easy to compute because G l acts trivially on the 1-dimensional vector space ad 1 /ad 2 : ] ker [H 1 G l, ad1 ad 2 H 1 I l, ad1 ad 2 H 1 Gl, ad1 I l ad 2 Hom k k, k k. One can verify that when either MC2a or MC2b holds then MC4 holds. It is an open question of whether MC2 always implies MC Absolute Irreducibility. First consider the case where the abelian variety A of GL 2 -type is an elliptic curve E. We claim that if E is semistable at 3 and 5 then E is modular. Fix l an odd prime. J.-P. Serre has shown that in general if E is semistable at l i.e. l 2 N E then either 1 E,l is reducible or 2 E,l is surjective. If E,l is surjective, then clearly E,l is irreducible, and this is implies condition MC4. Actually, this is equivalent to MC4 when l = 3, but stronger if l > 3. We have three cases to discuss, assuming that the elliptic curve is semistable at both l = 3 and l = 5: 1 when E,3 is irreducible; 2 when E,3 is reducible yet E,5 is irreducible; and 3 when both E,3 and E,5 are reducible. In case 3 there are only two elliptic curves up to twist having this property, and they have j-invariants je = 11 3 /2 3 and je = /2 15. One checks using various computer packages that E is modular in these cases; see for example William Stein s modular form database which is based on an algorithm due to John Cremona. As for case 2, Andrew Wiles realized he could reduce to case 1. He showed that under these assumptions there exists another curve E such that 1 E is semistable at 3 and 5, 2 E,3 is irreducible, and 3 E,5 E,5. This is known as the 3-5 Switch. We explain why this is important: Say that E,3 is irreducible. Then by a theorem of Robert Langlands and Jerrold Tunnell we know E,3 f is modular, and then by the R = T theorems we ve already verified that the modularity criteria are satisfied in this case we can deduce that E,3 is modular. But this implies there is a nonconstant morphism X N E E which is independent of l! so we deduce that E,5 is modular. Hence E,5 E,5 is modular. We deduce that if E is semistable at 3 and 5 then E is modular.