Introduction to Q-curves

Transcription

1 Introduction to Q-curves Imin Chen Simon Fraser University November 24, 2008

2 Introduction Some Galois cohomology Abelian varieties of GL 2 -type Explicit splitting maps for example References

3 Let K be a number field and let C be an elliptic curve defined over K such that there is an isogeny µ C (σ) : σ C C defined over K for each σ G Q. Such an elliptic curve C is called a Q-curve defined over K. This notion was originally defined and studied for a CM-elliptic curve by [Gross-1980] [Buhler-Gross-1985] but was extended by Ribet [Ribet-1992] to the non-cm case using different methods. Further explicit considerations were developed in [Quer-2000] which we will use in the sequel. The exposition below of the theory follows closely the citations above as well as [Ellenberg-Skinner-2001].

4 Q-curves are interesting classes of elliptic curves over number fields because their isogeny class is defined over Q. A priori, if one has an elliptic curve C defined over a number field K, one can attach Galois representations φ C,p : G K GL 2 (Q p ) of G K from its Tate modules. If C is a Q-curve, we will explain how one can attach a Galois representation ρ C,β,π : G Q Q p GL 2 (Q p ). This allows us to make sense of the modularity of C using classical modular forms f S 2 (Γ 0 (N), ɛ), rather than going into the realm of more general automorphic forms. Q-curves are thus one form of generalization of the notion of a modular elliptic curve over Q.

5 Consider the elliptic curve C defined over Q( 5) given by where u, v Q. ( C : Y 2 = X ) 5 ( 3 ( )v 2 3u 2) X 2 ( + 4v (17 4 5)v 2 ( )u 2). Its discriminant is C = η 3 ( s (5 + 2 ) 2 ( 5)t s (5 2 ) 5)t. where s = v 2, t = u 2, η = κ 3, κ = , which we require to be non-zero. The elliptic curve C has a 2-torsion point defined over Q( 5) given by (X, Y ) = ((2 5 2)v, 0).

6 Replacing X by X + (2 5 2)v gives a model where the 2-torsion point defined over Q( 5 is now (X, Y ) = (0, 0). The equation of C is now ( Y 2 = X 3 + ( + (6 5 6)v ) X 2 ) u v 2 X. 2 Recall if E is an elliptic curve given by Y 2 = X 3 + ax 2 + bx and φ : E E is the 2-isogeny with kernel equal to the subgroup (0, 0), then E is given by Y 2 = X 3 + AX 2 + BX where A = 2a, B = a 2 4b, and φ(x, Y ) = (Y 2 /X 2, Y (b X 2 )/X 2 ).

7 Let σ be an automorphism such that σ( 5) = 5. Consider the conjugate σ C given by ( Y 2 = X 3 + ( 6 ) 5 6)v X 2 ( 27 ) u v 2 X. 2 2 Taking the quotient by the subgroup (0, 0), we see that σ C is 2-isogenous over Q( 5) to the elliptic curve C given by ( Y 2 = X 3 + (12 ) )v X 2 ( + ( )u 2 + ( )v 2) X.

8 Let α = Then ψ(x, Y ) = (X α 2, Y α 3 ) gives an isomorphism ψ : C C defined over Q( 5, 2). Thus, there is a 2-isogeny σ C C defined over Q( 5, 2). C is thus a Q-curve defined over Q( 5, 2), but not over Q( 5), even though C is defined over Q( 5). From here on, we choose the isogenies so that µ C (σ) factors through G K/Q and µ C (σ) is the identity on G K. In our example, we take K = Q( 5, 2). If σ( 5) = 5, then σ C = C and we take µ(σ) = 1. If σ( 5) = 5, then we take µ(σ) to be the 2-isogeny σ C C described earlier.

9 Q-curves have applications to resolving diophantine equations, for instance, special classes of the generalized Fermat equation Ax a + By b = Cz c. This was first used in [Ellenberg-2004] to resolve the case x 2 + y 4 = z p. The idea is to apply a version of the modular method which uses Q-curves. For example, suppose we wish to resolve an equation such as x 2 + y 2p = z 5.

10 One first tries to construct Frey curves for this class of equations, which means we try to attach to each non-trivial proper solution a curve which gives rise to a Galois representation which has bounded ramification independent of the solution and exponents in the class. Assuming we can show their modularity, the problem is then reduced to one of rational points on modular curves and/or congruences between modular forms. It turns out for the equation x 2 + y 2p = z 5, one choice of Frey curve is the Q-curve example given previously [Chen-2007]. Because it is a Q-curve, we can attach a Galois representation ρ C,β,π : Gal(Q/Q) GL 2 (Q p ) to it, prove its modularity, and then apply the modular method.

11 Using classical descent, one can show non-trivial proper solutions to x 2 + y 2p = z 5 for an odd prime p give rise to solutions to s 2 10st + 5t 2 = 5 j γ p where 5 γ v = β p and j = 0 or v = 5 kp 1 β p and j = 1, where 5 β and k 1 s = v 2, t = u 2 (s, t) = 1. If we look at the discriminant of C, it has the form ( C = η 3 s (5 + 2 ) 2 ( 5)t s (5 2 ) 5)t. This shows C has the properties to be a Frey curve for the equation x 2 + y 2p = z 5.

12 The example we considered was such that C was defined over a quadratic extension K and the non-trivial conjugate σ C was 2-isogenous to C over Q. Such examples are classified by the Q-rational points on the modular curve X 0 (2)/ w 2, where w 2 is the Fricke involution on X 0 (2). If we want to specify the quadratic extension, such examples are classified by the Q-rational points on the twist X 0 (2) χ of X 0 (2) by the cocycle in H 1 (G Q, Aut(X 0 (2))) given by χ : G Q {±1} = w 2 Aut(X 0 (2)), where χ is the character associated to the quadratic extension K.

13 Both X 0 (2) and X 0 (2)/ w 2 have genus 0 and are isomorphic to P 1. If t is a uniformizer for X 0 (2), then the map X 0 (2) X (1) is given by j = (t+256)3 t 2. If s is a uniformizer for X 0 (2)/ w 2, then the correspondence between X 0 (2)/ w 2 and X (1) is given by j 2 +( s 2 49s +6656)j +s t s = 0, and s = t /t.

14 When we speak of a Q-curve, we will assume that it does not have complex multiplication. In our example, C will have complex multiplication if and only if 1. v/u = 0, j C = 1728, d(o) = 4 2. v/u = ±1, j C = 8000, d(o) = 8 3. v/u = ±3, j C = , d(o) = v/u =, j C = , d(o) = 20

15 This follows from the formula from the j-invariant of C. j C = η (( )s 3t ) 3 ( s ( )t ) 2 ( s (5 2 5)t ) (1) where s = v 2, t = u 2, η = κ 3, κ = Assume s/t Q. Then j C does not lie in Q unless s/t = 0, j C = 1728 s/t = 1, j C = 8000.

16 The elliptic curves with complex multiplication by an imaginary quadratic order O of class number 2 are listed below (c.f. [Montgomery-Weinberger-1973], [Stark-1975]). d(o) j 15 ( ± )/ ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

17 For a left G-module M, recall H 0 (G, M) = Z 0 (G, M)/B 0 (G, M) where Z 0 (G, M) = { φ : G 0 M φ 1 σ φ = 1 }, B 0 (G, M) = {1}. So we see that H 0 (G, M) = G M. In fact, it is probably better to take the point of view that H 0 (G, M) := G M (see later slide about short exact sequences giving rise to long exact sequences).

18 For a left G-module M, recall H 1 (G, M) = Z 1 (G, M)/B 1 (G, M) where Z 1 (G, M) = { φ : G M φ(σ) σ φ(τ) φ(στ) 1 = 1 } the submodule of 1-cocycles, B 1 (G, M) = { ψ 1 σ ψ ψ M } the submodule of 1-coboundaries. If we let b(σ) = ψ 1 σ ψ, then we see that b(σ) σ b(τ) b(στ) 1 = ψ 1 σ ψ σ ( ψ 1 τ ψ ) ( ψ 1 στ ψ ) 1 = ψ 1 σ ψ σ ψ 1 στ ψ στ ψ 1 ψ = 0. The other thing to note is that we can generalize the construction to allow for more general objects than ψ M in the formula b(σ) = ψ 1 σ ψ. More precisely, let ψ X where X has a left action of G and is a right M-torsor. Then b(σ) = ψ 1 σ ψ will formally satisfy the conditions to be in Z 1 (G, M) and the different choices for ψ will modify b(σ) by a 1-coboundary. Thus, there is a well-defined class b(σ) = ψ 1 σ ψ H 1 (G, M) attached to ψ X.

19 For a left G-module M, recall H 2 (G, M) = Z 2 (G, M)/B 2 (G, M) where Z 2 (G, M) = {φ : G G M φ(σ, τ)φ(στ, ρ) = σ φ(τ, ρ) φ(σ, τρ)} is the submodule of 2-cocycles, B 2 (G, M) = { ψ(σ) σ ψ(τ) ψ(στ) 1 ψ : G M } is the submodule of 2-coboundaries. If we let c(σ, τ) = ψ(σ) σ ψ(τ) ψ(στ) 1, then we see that c(σ, τ)c(στ, ρ)c(σ, τρ) 1 σ c(τ, ρ) 1 = ψ(σ) σ ψ(τ) ψ(στ) 1 ψ(στ) στ ψ(ρ) ψ(στρ) 1 ψ(στρ) σ ψ(τρ) 1 ψ(σ) 1 σ ψ(τρ) στ ψ(ρ) 1 σ ψ(τ) 1 = ψ(σ) σ ψ(τ) στ ψ(ρ) σ ψ(τρ) 1 ψ(σ) 1 σ ψ(τρ) στ ψ(ρ) 1 σ ψ(τ) 1 = ψ(σ) σ ψ(τ) στ ψ(ρ) σ ψ(τρ) 1 σ ψ(τρ) στ ψ(ρ) 1 σ ψ(τ) 1 ψ(σ) 1 = 1

20 The last line uses the fact that ψ(σ) 1 σ c(τ, ρ) 1 = σ c(τ, ρ) 1 ψ(σ) 1 as M is abelian. As before, we can allow for more general objects than ψ : G M in the formula c(σ, τ) = ψ(σ) σ ψ(τ) ψ(στ) 1 to produce classes in H 2 (G, M), as we will in our application.

21 Let 1 A B C 1 be a short exact sequence of G-modules. This gives rise to an exact sequence we can view as 1 G A G B G C, 1 H 0 (G, A) H 0 (G, B) H 0 (G, C). (2) The defining property of the cohomology groups H q (G, M) is that this extends to a long exact sequence 1 H 0 (G, A) H 0 (G, B) H 0 (G, C) H 1 (G, A) H 1 (G, B) H 1 (G, C) H 2 (G, A) H 2 (G, B) H 2 (G, C).... In fact, it can be shown there exists one and only one (up to canonical equivalence) cohomological extension of (2) (see Atiyah-Wall in Brighton Notes).

22 The maps H q (G, A) H q (G, B) and H q (G, B) H q (G, C) are the evident ones induced from the maps A B and B C. The map δ : H q (G, C) H q+1 (G, A) is called a connecting homomorphism. The connecting homomorphism δ : H 0 (G, C) H 1 (G, A) is given by δ(φ)(σ) = φ 1 σ φ where φ Z 0 (G, C). We know φ 1 σ φ = 1 as elements of C so φ 1 σ φ A. The connecting homomorphism δ : H 1 (G, C) H 2 (G, A) is given by δ(φ)(σ, τ) = φ(σ) σ φ(τ) φ(στ) 1 where φ Z 1 (G, C). We know φ(σ) σ φ(τ) φ(στ) 1 = 1 as elements of C so φ(σ) σ φ(τ) φ(στ) 1 A.

23 In the case of G = G K = Gal(K/K), where K is a number field, G = lim Gal(L/K) is profinite where the inverse system run through all finite Galois extensions L/K. The topology on G is given by a basis of open subsets containing the identity which consists of normal subgroups of finite index. We then require that the functions φ and ψ are continuous where we put the discrete topology on M.

24 Let c C (σ, τ) = µ C (σ) σ µ C (τ) µ C (στ) 1 (Hom K (C, C) Z Q) = Q, where µ 1 C := (1/ deg µ C )µ C and µ C is the dual of µ C. Then c C (σ, τ) determines a class in H 2 (G Q, Q ). More precisely, c C (σ, τ) Z 2 (G Q, Q ) and making different choices for µ C (σ) modifies this 2-cocycle by a 2-coboundary in B 2 (G Q, Q ). This is because for any two µ C (σ), µ C (σ) Hom( σ C, C) Q we have that µ C (σ) = µ C (σ)b(σ) = b(σ)µ C (σ) for b(σ) Q.

25 The class of c C (σ, τ) H 2 (G Q, Q ) only depends on the Q-isogeny class of C. This follows from the isomorphism between Hom( σ C, C) Q Hom( σ C, C ) Q given by f λ 1 f σ λ where λ : C C is an isogeny.

26 If C is a Q-curve defined over K, then the class c C (σ, τ) factors through H 2 (G K/Q, Q ) and only depends on the K-isogeny class of C. The class c C (σ, τ) in fact lies in H 2 (G Q, Q )[2]. This follows by taking degrees of the formula we get the identity c C (σ, τ) = µ C (σ) σ µ C (τ) µ C (στ) 1 d(σ)d(τ)d(στ) 1 = c C (σ, τ) 2, where d(σ) is the degree of µ C (σ).

27 Tate (cf. [Serre-1977-weight-one, Theorem 4]) showed that H 2 (G Q, Q ) is trivial where the action of G Q on Q is trivial. Thus, there is a continuous map β : G Q Q such that c C (σ, τ) = β(σ)β(τ)β(στ) 1 as elements of Z 2 (G Q, Q ), not just as classes in H 2 (G Q, Q ). In such a case, we say that β is a splitting map for C (or more precisely, for the cocycle c C (σ, τ)).

28 Let A be an abelian variety defined over Q. The endomorphism algebra End Q A of A is defined as the ring of endomorphisms of A defined over Q tensored over Z with Q. Let R C be the Q-algebra generated over Q by λ σ for σ G K/Q with multiplication given by where we recall that λ στ c C (σ, τ) = λ σ λ τ, c C (σ, τ) = µ C (σ) σ µ C (τ) µ C (στ) 1 depends on the function µ C.

29 The only thing to check is that the multiplication given on R C is associative. This follows from the computation below. (λ σ λ τ ) λ ρ = λ στ c C (σ, τ) 1 λ ρ = c C (σ, τ) 1 c C (στ, ρ) 1 λ στρ λ σ (λ τ λ ρ ) = λ σ c C (τ, ρ) 1 λ τρ c C (τ, ρ) 1 c C (σ, τρ) 1 λ στρ Because we have c C (σ, τ)c C (στ, ρ) = σ c C (τ, ρ) c C (σ, τρ) = c C (τ, ρ)c C (σ, τρ), both sides are equal.

30 Consider the restriction of scalars Res K Q C, for which we recall its defining functorial property that Hom(S, Res K Q C) Hom(S K, C). There is a natural isomorphism R C End Q Res K Q C which sends λ σ to the endomorphism of Res K Q C defined by on στ C. P τ µ C (σ)(p) A simple example is C = Spec(Q( 5)[X ]). Write X = X 1 + X 2 5. Then Res Q( 5) Q C = Spec(Q[X 1, X 2 ]). Another simple example is C = Spec(Q[X ]/(X 2 5)). Write X = X 1 + X 2 5. Then Res Q( 5) Q C = Spec(Q[X 1, X 2 ]/(X X 2 2 5, 2X 1X 2 ).

31 Given a splitting map β for C, we now enlarge K if necessary so that β factors through G K/Q. The map given by θ β : λ σ β(σ) gives a surjective homomorphism θ β : R C M β = Q(β(σ)). This is because c C (σ, τ) = β(σ)β(τ)β(στ) 1 so from the identity λ σ λ τ = c C (σ, τ)λ στ, we obtain after applying θ β, the identity holds, which shows β(σ)β(τ) = c C (σ, τ)β(στ) θ β (λ σ )θ β (λ τ ) = θ β (λ στ ). As R C is a semi-simple Q-algebra, there is a projection from R C onto the isomorphic copy of M β in R C. Let A β be the image of this projection in the category of abelian varieties defined over Q up to isogeny over Q.

32 We note the following twist on the construction of A β above which is useful in practice to minimize the degree of the extension K required Recall we need to find a β(σ) so that c C (σ, τ) = β(σ)β(τ)β(στ) 1 as cocycles in Z 2 (G Q, Q ) and K needs to be large enough so β(σ) factors through G K/Q. Suppose that as 2-cocycles we have have found a β(σ) such that c C (σ, τ)ɛ(σ, τ) = β(σ)β(τ)β(στ) 1 where ɛ(σ, τ) is the 2-coboundary obtained from a 1-cocycle σ γ γ where γ Q.

33 By the way twisting affects the cocycles c C (σ, τ) [Quer-2000, p. 291] we see that the twist C γ of C is such that c Cγ (σ, τ) = c C (σ, τ)ɛ(σ, τ) = β(σ)β(τ)β(στ) 1. By the twist C γ we mean if C is given by Y 2 = X 3 + AX + B, then C γ is given by Y 2 = X 3 + Aγ 2 X + Bγ 3. Thus, replacing C by C γ allows us to use the β(σ) given and we take K be large enough so that β(σ) factors through G K/Q. This will be useful because the splitting map β(σ) for c Cγ (σ, τ) might factor through a smaller field K/Q than a splitting map for c C (σ, τ).

34 Recall an abelian variety defined over Q of GL 2 -type is one whose endomorphism algebra is isomorphic to a number field M of degree equal to the dimension of the abelian variety. An abelian variety defined over Q of GL 2 -type attached to a Q-curve C is one which has C as a quotient over Q. Theorem The abelian variety A β is an abelian variety defined over Q of GL 2 -type attached to C, with endomorphism algebra isomorphic to M β. Proof. cf. [Ribet-1992, Theorem 6.1].

35 Proposition If A is an abelian variety defined over Q of GL 2 -type attached to a Q-curve C, then A is isogenous over Q to some A β where β is a splitting map for C. Proof. If C is a quotient of A defined over K, then there is a non-zero homomorphism A Res K Q C defined over Q. Since A is simple up to isogeny over Q, it follows that A is a quotient defined over Q of Res K Q C. As R C is a semi-simple Q-algebra, there is a projection R C End Q A given by λ σ β(σ) say. We now see that β is a splitting map for C, and that A β is isogenous over Q to A.

36 Proposition Suppose that R C is a product of fields. Then Res K Q C is isogenous over Q to a product of pairwise non-isogenous abelian varieties defined over Q of GL 2 -type, each of the form A β where β is a splitting map for C. Furthermore, A β1 is isogenous over Q to A β2 if and only if β 2 = σ β 1 for some σ G Q. Proof. cf. [Quer-2000, Proposition 5.1, Lemma 5.3].

37 For an abelian variety A defined over Q, let ˆV p (C) denote the Q p [G Q ]-module which is the p-adic Tate module of C tensored over Q p. Proposition ˆV p (Res K Q C) = R C ˆV p (C) as R C Q p [G Q ]-modules. Proof. The proof is a modification of [Ribet-1992, Corollary 6.6]. Recall that it is given that C is a Q-curve defined over K and let A = Res K Q C. There is an isomorphism A = B K = σ G σ K/Q C defined over K by the defining property of restriction of scalars. Let T K = σ G K/Q C σ where C σ = C for all σ G K/Q. There is an action of R C on T K with λ g taking the factor C σ to the factor C gσ via multiplication by c C (g, σ). Let ι : T K B K be the map which takes the factor C σ to the factor σ 1 C via the map σ 1 µ C (σ). Then ι is a R C [G K ]-equivariant isomorphism.

38 Proof con t. By the defined action of R C on T K, we have that ˆV p (T K ) = R C ˆV p (C) as R C Q p [G K ]-modules. Hence, ˆV p (A) = ˆV p (B K ) = R C ˆV p (C) as R C Q p [G K ]-modules. The action of G Q on A can be transferred to an action of G Q on T K via the isomorphisms A = B K = TK. From this, it can be shown that the explicit action of G Q on the R C Q p -module ˆV p (A) = R C ˆV p (C) is given by x y x λ σ 1 ( σµc (σ 1 )) 1 (σ(y)).

39 From Proposition 4, it follows that ˆV p (A β ) = M β ˆV p (C) as M β Q p [G Q ]-modules. Picking a prime π of M β above p, we get a representation ˆρ C,β,π : G Q GL 2 (M β,π ). The explicit action of G Q on the M β Q p module ˆV p (A β ) is then given by x y x β(σ 1 ) ( σµc (σ 1 )) 1 (σ(y)), which can be simplified to the expression x y x β(σ) 1 µ C (σ)(σ(y)).

40 Hence, if we regard M β,π as a subfield of Q p, then ˆρ C,β,π is a representation to Q p GL 2 (Q p ), and it satisfies Pˆρ C,β,π GK = P ˆφC,p, where ˆφ C,p : G K GL 2 (Q p ) is the representation of G K on ˆV p (C). Let ɛ β : G Q Q be defined by Then ɛ β is a character and ɛ β (σ) = β(σ) 2 / deg µ C (σ). det ˆρ C,β,π = ɛ 1 β χ p, (3) where χ p : G Q Z p is the p-th cyclotomic character.

41 Given two splitting maps β, β for C, there is a character χ : G Q Q such that β = χβ. Conversely, if β is a splitting map, then β = χβ is a splitting map for any character χ : G Q Q. When M β = M β, we see that ρ C,β,π = χ ρ C,β,π are twists of each other, as are A β and A β.

42 We say that a Q-curve C is modular if for some positive integer N it is the quotient over Q of J 1 (N). If a Q-curve C is modular, then there is a newform f S 2 (Γ 0 (N), ɛ 1 ) such that A f is an abelian variety defined over Q of GL 2 type attached to C. This follows because J 1 (N) decomposes into product of A f s up to isogeny over Q [Ribet-1980-twist]. By Proposition 2, A f is isogenous to A β for some splitting map β, and hence for some splitting map β for C we have that ρ C,β,π = ρf,π for a newform f S 2 (Γ 0 (N), ɛ 1 ).

43 Since any two splitting maps differ by a character, we see that for every splitting map β we have that ρ C,β,π = ρf,π for some f S 2 (Γ 0 (N), ɛ 1 ). Conversely, if ρ C,β,π = ρf,π for some newform f S 2 (Γ 0 (N), ɛ 1 ), then A β is isogenous over Q to A f and hence the Q-curve C is modular. In summary, we have shown that ρ C,β,π = ρf,π for some f S 2 (Γ 0 (N), ɛ 1 ) if and only if the Q-curve C is modular.

44 We have constructed representations attached to the Q-curve C. ˆρ C,β,π : G Q GL 2 (M β,π ) However, the construction depends on a choice of splitting map β : G Q Q for C, which is related to picking a Q-curve C defined over K in the Q-isomorphism class of C such that the decomposition of Res K Q E up to isogeny over Q is a product of non-isogenous abelian varieties of GL 2 -type We now explain how this is done explicitly in our example.

45 Let G Q( 5)/Q = {σ 1, σ 5 }. There is a 2-isogeny σ 5 C C defined over Q( 5, 2), whence we set µ C (σ 5 ) to be this isogeny and µ C (σ 1 ) = 1. The cocycle c C (σ, τ) = µ C (σ) σ µ C (τ)µ C (στ) 1 can also be described as arising from a cocycle α C H 1 (G Q, Q /Q ) given by µ C (σ) (ω C ) = α C (σ)ω C, with ω C, ω C being the invariant differentials on C, C = σ C, through the formula which results from the map c C (σ, τ) = α C (σ) σ α C (τ)α C (στ) 1, H 1 (G Q, Q /Q ) H 2 (G Q, Q ), which is derived from the short exact sequence 1 Q Q Q /Q 1.

46 Explicitly, α C (σ 1 ) = 1 α C (σ 5 ) = This can be computed using the discussion in [Quer-2000, p. 288]. Let G Q( 5, 2)/Q = {σ 1, σ 2, σ 5, σ 10 }. Then c C (σ, τ) factors through this group and has the representative values c C (σ 2, σ 2 ) = 1 c C (σ 10, σ 10 ) = 2 c C (σ 2, σ 10 ) = c C (σ 10, σ 2 )(= 1).

47 It follows that R C = M2 (Q) and hence Res Q( 5, 2) Q E is isogenous over Q to B B where B is an abelian surface defined over Q with End Q B = Q. This means that taking K = Q( 5, 2) and E = E is not a suitable choice for our purposes because the decomposition of Res Q( 5, 2) Q E up to isogeny over Q does not include any abelian varieties of GL 2 -type. The reason for this is the cocycle c C (σ, τ) isn t equal to a cocycle of the form β(σ)β(τ)β(στ) 1 for a splitting map β(σ) as cocycles in B 2 (G Q, Q ).

48 Proposition The map on cocycles given by induces an isomorphism c(σ, τ) (sgn c(σ, τ), c(σ, τ) ) H 2 (G Q, Q )[2] H 2 (G Q, {±1}) H 2 (G Q, P/P 2 ) where P is the group of positive rational numbers. Proof. cf. [Quer-2000, p. 294].

49 We call c ± (σ, τ) = sgn c(σ, τ) the sign component of c(σ, τ). Proposition The sign component c ± C (σ, τ) H2 (G Q, {±1}) of c C (σ, τ) is given by the quaternion algebra (5, 2) H 2 (G Q, {±1}). Proof. Let d(σ) = deg µ C (σ) be the degree map. In the terminology of [Quer-2000, p. 294], we have that {a 1 } = {5} and {d 1 } = {2} are dual bases with respect to d(σ). The conclusion then follows from [Quer-2000, Theorem 3.1].

50 Let ɛ : G Q Q be a character and let θ ɛ (σ, τ) = ɛ(σ) ɛ(τ) ɛ(στ) 1. Then θ ɛ (σ, τ) H 2 (G Q, {±1}). Proposition Let β(σ) = ɛ(σ) d(σ). Then β(σ) is a splitting map for E if and only θ ɛ (σ, τ) = c ± C (σ, τ) as classes in H2 (G Q, {±1}). Proof. cf. [Quer-2000, Theorem 4.2]. Proposition We have that θ ɛ (σ, τ) = c ± C (σ, τ) as classes in H2 (G Q, {±1}) if and only if θ ɛ (σ, τ) = c ± C (σ, τ) as classes in H2 (G Qp, {±1}) for all finite primes p. Proof. cf. [Quer-2000, p. 302].

51 Proposition H 2 (G Qp, {±1}) = {±1} for all finite primes p. Proof. This follows from the fact that H 2 (G Qp, {±1}) is contained in the 2-torsion of H 2 (G Qp, Q p) which can be identified with isomorphism classes of simple algebras over Q p with center Q p and dimension 4 over Q p, namely, quaternion algebras over Q p (c.f. [Serre-1968-local-fields, Chapitre X, 5, Chapitre XIII, 4]). It is also known that over Q p, there are precisely two isomorphism classes of quaternion algebras (c.f. [Vigneras-1980, Theorem 1.1]). Proposition We have that θ ɛ (σ, τ) p = ɛ p ( 1) as classes in H 2 (G Qp, {±1}) = {±1}. Proof. cf. [Quer-2000, p. 302].

52 The above results imply that a possible choice of splitting map β for E is given by β(σ) = ɛ(σ) d(σ), (4) where d(σ) = deg µ C (σ), ɛ = ɛ 4 ɛ 5, and ɛ 4 is the non-trivial character of (Z/4Z), and ɛ 5 is a non-trivial character of (Z/5Z). For this choice of β, we have that ɛ β = ɛ and M β = Q(i). The character ɛ has kernel {±1}, regarded as a character of (Z/20Z). To fix choices, let us suppose that ɛ(±3) = i C.

53 Explicitly, the coboundary relating the cocycles c C (σ, τ) and c β (σ, τ) = β(σ)β(τ)β(στ) 1 can be described as follows. We will use this coboundary to find a Q-curve C β defined over a number field K β in the Q-isomorphism class of E such that c Cβ (σ, τ) = c β (σ, τ) as cocycles (not just as classes). σ γ1 Let α 1 (σ) = α C (σ) γ1, where γ 1 = Then we have that α 1 (σ 1 ) = 1 α 1 (σ 5 ) = 2. Recall that the cocycles α(σ), α 1 (σ) have values in Q /Q so any equality is regarded up to multiplication by an element in Q.

54 We wish to find a γ 2 such that satisfies α 2 (σ) = α 1 (σ) c β (σ, τ) = α 2 (σ) σ α 2 (τ)α 2 (στ) σ γ2 γ2 (5) Let K β = Q(z) where z = 2 is a root of X 4 5X and let G Kβ /Q = { σ 1 ±, } σ± 5. The unit group of Kβ is generated by u 1 = 1 u 2 = 2 z 2 u 3 = z 2 + z + 2 u 4 = z 3 + z 2 + 3z 3 and is isomorphic to Z/2Z Z Z Z.

55 2 = σ + 5 γ2 γ 2 Let g = α 2 (σ 5 + g 2 ). Then is a necessary constraint on g using Equation (5). As an initial guess, let us suppose that g 2 2 = u is a unit in K β. This unit u = 2 z can be obtained by noting (2) = (g 2 ) in K β. Since N Kβ /Q(u) = 1, by Hilbert 90, there is a γ 2 K β such that σ γ 2 γ 2 = u, where σ = σ 5 +. This γ 2 can be obtained from the expression γ 2 = z + uz σ + u 1+σ z σ2 + u 1+σ+σ2 z σ3 used in the proof of Hilbert 90. Then up to scaling by an element in Q, we may take γ 2 = 1 = z γ 3 + z 2 2z. 2

56 Finally, if we let α 2 (σ) = α C (σ) σ γ γ where γ = z 2 (z 3 + z 2 2z) = 3z 3 + 5z 2 5z 5 = z 3 /u 3, then the cocycle in H 2 (G Q, Q ) arising from α 2 (σ) is precisely c β (σ, τ) = β(σ)β(τ)β(στ) 1. For this fact, we list for convenience the following values σ + 5 γ γ = g 2 2, σ 5 γ γ = g 2 2 σ + 1 γ γ = 1, σ 1 γ γ = u 2 3, 1, u 2 4

57 Let C β be the Q-curve defined over K β in the Q-isomorphism class of E given by ( Y 2 = X ) 5 ( 3 (3 + 2 ) 5)s 3t γ 2 X (6) 2 + 4v ( (17 4 5)s ( )t ) γ 3 ( ) where δ = , η = κ 3, and κ = = 1/u 2. Let α Cβ (σ) be the cocycle in H 1 (G Q, Q /Q ) given by µ Cβ (σ) (ω Cβ ) = α Cβ (σ)ω C β where C β = σ C β. From consideration of how twisting affects the α C (σ) [Quer-2000, p. 291], we have that σ γ α Cβ (σ) = α C (σ) = α 2 (σ). γ

58 The values α 2 (σ) lie in K β. Hence, if we use C β instead of E, then C β is a Q-curve defined over K β and we have that c Cβ (σ, τ) = c β (σ, τ) = β(σ)β(τ)β(στ) 1 as cocycles. Now work of Quer [Quer-2000, Theorem 5.4, Case (2)] implies that Res K β Q C β Q A β A β, where A β, A β are abelian varieties defined over Q of GL 2 -type with endomorphism algebra Q(i). Here, β = χ β is a splitting map such that ɛ β = ɛ and χ = ( 5 ) is the quadratic character attached to Q( 5).

59 Proposition The elliptic curve C has the following properties. C has potentially good ordinary reduction in characteristic 3 if s 0 (mod 3) and potentially good supersingular reduction in characteristic 3 if s 0 (mod 3). The sign component c ± C (σ, τ) = sgn µ C (σ) σ µ C (τ)µ C (στ) 1 H 2 (G Q, {±1}) is trivial when restricted to G Q3. Proof. Note s 2 10st + 5t 2 0 (mod 3). The elliptic curve C has potentially good reduction because the denominator of its j-invariant is not divisible by a prime above 3 by Equation (1). Its j-invariant is zero in characteristic 3 if and only if s 0 (mod 3) so C is supersingular in characteristic 3 if and only if s 0 (mod 3). Since the sign component c ± C (σ, τ) is given by the quaternion algebra (5, 2) by Proposition 6, we see that it is trivial when restricted to G Q3.

60 Theorem The abelian varieties A β and A β are modular. Proof. In the case of potentially good ordinary reduction, C satisfies the hypotheses of [Ellenberg-2001, Theorem 5.1] because of Proposition 11 so we deduce that it is modular. In the case of potentially good supersingular reduction, we note that Pρ C,β,π is unramified at 3 so by [Ellenberg-2001, Theorem 5.2] we also deduce that C is modular. The abelian varieties A β and A β are not isogenous over Q since β σ β for any σ G Q. Let f and f be the newforms attached to A β and A β respectively.

61 Theorem A β is isogenous over Q to a twist of A β by χ 1 = χ = ( 5 ) and hence f is a twist of f by χ 1 = χ = ( 5 ). Proof. This can be seen from the Galois action on the Tate module of A β and A β which is given by Since β = χ β, we see that x y x β(σ) 1 µ C (τ)(τ(y)) x y x β (σ) 1 µ C (τ)(τ(y)). ˆρ A,β,π(σ) = ɛ 1 (σ)ˆρ A,β,π (σ), where π is a prime of M β = M β = Q(i) above p.

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