Computing automorphic forms on Shimura curves over fields with arbitrary class number

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.Comment: 15 pages; final submission to ANTS I

Modular forms and elliptic curves over the field of fifth roots of unity

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F.Comment: Added appendix by Mark Watkins, who found an elliptic curve missing from our tabl

Modular forms and elliptic curves over the cubic field of discriminant -23

Let F be the cubic field of discriminant -23 and let O be its ring of integers. By explicitly computing cohomology of congruence subgroups of GL(2,O), we computationally investigate modularity of elliptic curves over F.Comment: Incorporated referee's comment

Modular Forms and Elliptic Curves over the Field of Fifth Roots of Unity

Let F be the cyclotomic field of fifth roots of unity. We computationally investigate modularity of elliptic curves over F

Arithmetic of the Asai L-function for Hilbert Modular Forms.

Arithmetic of the Asai L-function for Hilbert modular forms Adam Kaye Chair: Kartik Prassanna We prove two results on rationality of special values of the Asai L-function attached to Hilbert modular forms at critical points. Such L-functions only admit critical values when the Hilbert modular form has non-parallel weight. Our rationality results generalize previous work of Shimura on algebraicity. The first result uses a period defined by transferring the Hilbert modular form to a Shimura curve. The second result uses a period defined using rational structures on the coherent cohomology of Hilbert modular surfaces. We also give some partial results towards integrality of such L-values. Our results are motivated by the study of a p-adic analog of the Beilinson conjecture, which is a deep conjecture relating algebraic cycles (and motivic cohomology) to values of L-functions.PhDMathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/120693/1/adamkaye_1.pd

Non-commutative Hilbert modular symbols

The main goal of this paper is to construct non-commutative Hilbert modular symbols. However, we also construct commutative Hilbert modular symbols. Both the commutative and the non-commutative Hilbert modular symbols are generalizations of Manin's classical and non-commutative modular symbols. We prove that many cases of (non-)commutative Hilbert modular symbols are periods in the sense on Kontsevich-Zagier. Hecke operators act naturally on them. Manin defines the non-commutative modilar symbol in terms of iterated path integrals. In order to define non-commutative Hilbert modular symbols, we use a generalization of iterated path integrals to higher dimensions, which we call iterated integrals on membranes. Manin examines similarities between non-commutative modular symbol and multiple zeta values both in terms of infinite series and in terms of iterated path integrals. Here we examine similarities in the formulas for non-commutative Hilbert modular symbol and multiple Dedekind zeta values, recently defined by the author, both in terms of infinite series and in terms of iterated integrals on membranes.Comment: 50 pages, 5 figures, substantial improvement of the article arXiv:math/0611955 [math.NT], the portions compared to the previous version are: Hecke operators, periods and some categorical construction

Mini-Workshop: Computations in the Cohomology of Arithmetic Groups

Explicit calculations play an important role in the theoretical development of the cohomology of groups and its applications. It is becoming more common for such calculations to be derived with the aid of a computer. This mini-workshop assembled together experts on a diverse range of computational techniques relevant to calculations in the cohomology of arithmetic groups and applications in algebraic $K$-theory and number theory with a view to extending the scope of computer aided calculations in this area