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… (More)
We enumerate all Shimura curves X D 0 (N) of genus at most two: there are exactly 858 such curves, up to equivalence. The elliptic modular curve X 0 (N) is the quotient of the completed upper half-plane H * by the congruence subgroup Γ 0 (N) of matrices in SL 2 (Z) that are upper triangular modulo N ∈ Z >0. The curve X 0 (N) forms a coarse moduli space for… (More)
We utilize effective algorithms for computing in the cohomology of a Shimura curve together with the Jacquet-Langlands correspondence to compute systems of Hecke eigenvalues associated to Hilbert modular forms over a totally real field F. The design of algorithms for the enumeration of automorphic forms has emerged as a major theme in computational… (More)
We enumerate all totally real number fields F with root discriminant δF ≤ 14. There are 1229 such fields, each with degree [F : Q] ≤ 9.
Let & v2(R) be a cocompact arithmetic triangle group, i.e. a triangle Fuchsian group that arises from the unit group of a quater-nion algebra over a totally real number eld. We introduce CM points dened on the Shimura curve quotient C = nH, and we algorithmi-cally apply the Shimura reciprocity law to compute these points and their Galois conjugates so… (More)
We present a heuristic that suggests that ranks of elliptic curves E over Q are bounded. In fact, it suggests that there are only finitely many E of rank greater than 21. Our heuristic is based on modeling the ranks and Shafarevich–Tate groups of elliptic curves simultaneously, and relies on a theorem counting alternating integer matrices of specified rank.… (More)
Jagy and Kaplansky exhibited a table of 68 pairs of positive definite binary quadratic forms that represent the same odd primes and conjectured that their list is complete outside of " trivial " pairs. In this article, we confirm their conjecture, and in fact find all pairs of such forms that represent the same primes outside of a finite set.
Let ∆ = ∆(a, b, c) be a hyperbolic triangle group, a Fuchsian group obtained from reflections in the sides of a triangle with angles π/a, π/b, π/c drawn on the hyperbolic plane. We define the arithmetic dimension of ∆ to be the number of split real places of the quaternion algebra generated by ∆ over its (totally real) invariant trace field. Takeuchi has… (More)
This note corrects a mathematical error in the article " Algorithmic enumeration of ideal classes for quaternion orders " [SIAM J. There is an error in the article  by the authors: The proof of Theorem B uses a diameter bound (7.1) of Chung , D(G) ≤ log(H − 1) log(k/λ) , which holds for a k-regular directed graph G of size H with adjacency matrix T… (More)