12 research outputs found
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
Overconvergent cohomology and quaternionic Darmon points
We develop the (co)homological tools that make effective the construction of
the quaternionic Darmon points introduced by Matthew Greenberg. In addition, we
use the overconvergent cohomology techniques of Pollack--Pollack to allow for
the efficient calculation of such points. Finally, we provide the first
numerical evidence supporting the conjectures on their rationality.Comment: Fixed some minor typos, added authors' affiliatio
On rigid analytic uniformizations of Jacobians of Shimura curves
The main goal of this article is to give an explicit rigid analytic uniformization of the maximal toric quotient of the Jacobian of a Shimura curve over Q at a prime dividing exactly the level. This result can be viewed as complementary to the classical theorem of Cerednik and Drinfeld which provides rigid analytic uniformizations at primes dividing the discriminant. As a corollary, we offer a proof of a conjecture formulated by M. Greenberg in hispaper on Stark-Heegner points and quaternionic Shimura curves, thus making Greenberg's construction of local points on elliptic curves over Q unconditional