John Voight

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.

Explicit Methods for Hilbert Modular Forms

The study of modular forms remains a dominant theme in modern number theory, a consequence of their intrinsic appeal as well as their applications to a wide variety of mathematical problems. This subject has seen dramatic progress during the past half-century in an environment where both abstract theory and explicit computation have developed in parallel. Experiments will remain an essential tool in the years ahead, especially as we turn from classical contexts to less familiar terrain.

Enumeration of totally real number fields of bounded root discriminant

We enumerate all totally real number fields F with rootdiscriminant δF ≤ 14. There are 1229 such fields, each with degree[F : Q] ≤ 9.

COMPUTING SYSTEMS OF HECKE EIGENVALUES ASSOCIATED TO HILBERT MODULAR FORMS

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.

Characterizing quaternion rings over an arbitrary base

Abstract We consider the class of algebras of rank 4 equipped with a standard involution over an arbitrary base ring. In particular, we characterize quaternion rings, those algebras defined by the construction of the even Clifford algebra.

Identifying the Matrix Ring: Algorithms for Quaternion Algebras and Quadratic Forms

We discuss the relationship between quaternion algebras and quadratic forms with a focus on computational aspects. Our basic motivating problem is to determine if a given algebra of rank 4 over a commutative ring R embeds in the 2 ×2-matrix ring M2(R) and, if so, to compute such an embedding. We discuss many variants of this problem, including algorithmic recognition of quaternion algebras among algebras of rank 4, computation of the Hilbert symbol, and computation of maximal orders.

Shimura curves of genus at most two

We enumerate all Shimura curves X D 0 (R) of genus at most two: there are exactly 858 such curves, up to equivalence.

ON COMPUTING BELYI MAPS

We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples.

Lattice Methods for Algebraic Modular Forms on Classical Groups

We use Kneser’s neighbor method and isometry testing for lattices due to Plesken and Souveigner to compute systems of Hecke eigenvalues associated to definite forms of classical reductive algebraic groups.

Computing CM points on shimura curves arising from cocompact arithmetic triangle groups

Let