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 H1 c (Y0(N)(C), C) of compactly supported cohomolo...

• Type: Journal
• Year: 2011
• Where: MOC
• Authors: Matthew Greenberg, John Voight
• Keywords: Intelligent Agents, MOC 2011, Modular Form, Modular Symbols, Space