Modular Forms and Elliptic Curves over Q(ζ 5 )

Let ζ5 be a primitive fifth root of unity, and let F = Q(ζ5). In this talk we describe recent computational work that investigates the modularity of elliptic curves over F . Here by modularity we mean that for a given elliptic curve E over F with conductor N there should exist an automorphic form f on GL2, also of conductor N , such that we have the equality of partial L-functions LS(s, f) = LS(s,E), where S is a finite set of places including those dividing N . We are also interested in checking a converse to this notion, which says that for an appropriate automorphic form f on GL2, there should exist an elliptic curve E/F again with matching of partial L-functions. Our work is in the spirit of that of Cremona and his students [7–9, 15] for complex quadratic fields, and of Socrates–Whitehouse [16] and Dembélé [10] for real quadratic fields. Instead of working with automorphic forms, we work with the cohomology of congruence subgroups of GL2(O), where O is the ring of integers of F . There are several reasons for this. First, we have the Eichler–Shimura isomorphism, which identifies the cohomology of subgroups of SL2(Z) with a space of modular forms. More precisely, if N ≥ 1 is an integer and if Γ0(N) ⊂ SL2(Z) is the usual congruence subgroup of matrices upper triangular mod N , then we have H(Γ0(N);C) ' H(X0(N);C) ' S2(N)⊕S2(N)⊕Eis2(N), where X0(N) is the open modular curve Γ0(N)\H, S2(N) is the space of weight two holomorphic cusp forms of level N , the summand Eis2(N) is the space of weight two holomorphic Eisenstein series, and the bar denotes complex conjugation. Moreover, this reason generalizes. Borel conjectured, and Franke proved [11], that all the complex cohomology of any arithmetic group can be computed in terms of certain automorphic forms, namely those with “nontrivial (g,K)-cohomology” [6, 18]. Although this is a small subset of all automorphic forms (Maass forms, for instance, can never show up in this way), all such automorphic forms are widely believed to be connected with arithmetic geometry (Galois representations, motives, . . . ). Finally, working with cohomology also has the advantage that computations can be done very explicitly using tools of combinatorial topology. In a sense the cohomology provides a concrete incarnation of exactly the automorphic forms we want. These are the automorphic forms that account for the “modular forms over Q(ζ5)” in the title. Now we explain the setting for our computations. For our field we begin with the algebraic group G = RF/Q(GL2) (R denotes restriction of scalars), which satisfies G(Q) = GL2(F ). We replace the upper halfplane H with the symmetric space X for the group G = G(R) ' GL2(C)×GL2(C). We have X ' H3×H3×R, where H3 is hyperbolic 3-space; thus X is 7-dimensional. We remark that if we were to work with G′ = RF/Q(SL2) instead, the appropriate symmetric space