33 research outputs found
A local-global principle for rational isogenies of prime degree
Let K be a number field. We consider a local-global principle for elliptic
curves E/K that admit (or do not admit) a rational isogeny of prime degree n.
For suitable K (including K=Q), we prove that this principle holds when n = 1
mod 4, and for n < 7, but find a counterexample when n = 7 for an elliptic
curve with j-invariant 2268945/128. For K = Q we show that, up to isomorphism,
this is the only counterexample.Comment: 11 pages, minor edits, to appear in Journal de Th\'eorie des Nombres
de Bordeau
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
A database of genus 2 curves over the rational numbers
We describe the construction of a database of genus 2 curves of small
discriminant that includes geometric and arithmetic invariants of each curve,
its Jacobian, and the associated L-function. This data has been incorporated
into the L-Functions and Modular Forms Database (LMFDB).Comment: 15 pages, 7 tables; bibliography formatting and typos fixe