161 research outputs found
Algorithmic enumeration of ideal classes for quaternion orders
We provide algorithms to count and enumerate representatives of the (right)
ideal classes of an Eichler order in a quaternion algebra defined over a number
field. We analyze the run time of these algorithms and consider several related
problems, including the computation of two-sided ideal classes, isomorphism
classes of orders, connecting ideals for orders, and ideal principalization. We
conclude by giving the complete list of definite Eichler orders with class
number at most 2.Comment: 39 pages, includes 2 tables; corrections made to Table 8.
Quaternary quadratic lattices over number fields
We relate proper isometry classes of maximal lattices in a totally definite
quaternary quadratic space (V,q) with trivial discriminant to certain
equivalence classes of ideals in the quaternion algebra representing the
Clifford invariant of (V,q). This yields a good algorithm to enumerate a system
of representatives of proper isometry classes of lattices in genera of maximal
lattices in (V,q)
Nonvanishing of twists of -functions attached to Hilbert modular forms
We describe algorithms for computing central values of twists of
-functions associated to Hilbert modular forms, carry out such computations
for a number of examples, and compare the results of these computations to some
heuristics and predictions from random matrix theory.Comment: 19 page
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
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