John Voight

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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)
Let & € ƒv2(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 ˆ 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)
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 [2] by the authors: The proof of Theorem B uses a diameter bound (7.1) of Chung [1], 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)