Computing Automorphic Forms on Shimura Curves over Fields with Arbitrary Class Number

13 years 7 months ago

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We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with two examples which illustrate the effectiveness of our algorithms. The development and implementation of algorithms to compute with automorphic forms has emerged as a major topic in explicit arithmetic geometry. The first such computations were carried out for elliptic modular forms, and now very large and useful databases of such forms exist [2, 13, 14]. Recently, effective algorithms to compute with Hilbert modular forms over a totally real field F have been advanced. The first such method is due to Demb

John Voight

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