Computing systems of Hecke eigenvalues associated to Hilbert modular forms

We utilize effective algorithms for computing in the cohomology of a Shimura curve together with the Jacquet-Langlands correspondence to compute systems of Hecke eigenvalues associated to Hilbert modular forms over a totally real field F . The design of algorithms for the enumeration of automorphic forms has emerged as a major theme in computational arithmetic geometry. Extensive computations have been carried out for elliptic modular forms and now large databases exist of such forms [5, 35]. As a consequence of the modularity theorem of Wiles and others, these tables enumerate all isogeny classes of elliptic curves over Q up to a very large conductor. The algorithms employed to list such forms rely heavily on the formalism of modular symbols, introduced by Manin [26] and extensively developed by Cremona [4], Stein [34], and others. For a positive integer N , the space of modular symbols on Γ0(N) ⊂ SL2(Z) is defined to be the group H c (Y0(N)(C),C) of compactly supported cohomology classes on the open modular curve Y0(N)(C) = Γ0(N)\H, where H denotes the upper half-plane. Let S2(Γ0(N)) denote the space of cuspidal modular forms for Γ0(N). By the Eichler-Shimura isomorphism, the space S2(Γ0(N)) embeds into H 1 c (Y0(N)(C),C) and the image can be characterized by the action of the Hecke operators. In sum, to compute with the space of modular forms S2(Γ0(N)), one can equivalently compute with the space of modular symbols H c (Y0(N)(C),C) together with its Hecke action. This latter space is characterized by a natural isomorphism of Hecke-modules H c (Y0(N)(C),C) ∼= HomΓ0(N)(Div P(Q),C), where a cohomology class ω is mapped to the linear functional which sends the divisor s− r ∈ Div P(Q) to the integral of ω over the image on Y0(N) of a path in H between the cusps r and s. Modular symbols have proved to be crucial in both computational and theoretical roles. They arise in the study of special values of L-functions of classical modular forms, in the formulation of p-adic measures and p-adic L-functions, as well as in the conjectural constructions of Gross-Stark units [8] and Stark-Heegner points [7]. It is therefore quite inconvenient that a satisfactory formalism of modular symbols is absent in the context of automorphic forms on other Shimura varieties. Consequently, the corresponding theory is not as well understood. From this point of view, alternative methods for the explicit study of Hilbert modular forms are of particular interest. Let F be a totally real field of degree n = [F : Q] and let ZF denote its ring of integers. Let S2(N) denote the Hecke module of (classical) Hilbert modular cusp forms over F of parallel weight 2 and level N ⊂ ZF . Dembélé [12] and Dembélé and Donnelly [13] have presented methods for computing with the space S2(N) under the assumption that n is even. Their Date: February 19, 2010.