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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 exhibit an algorithm to compute a Dirichlet domain for a Fuchsian group Γ with cofinite area. As a consequence, we compute the invariants of Γ, including an explicit finite presentation for Γ. Let Γ ⊂ PSL 2 (R) be a Fuchsian group, a discrete group of orientation-preserving isometries of the upper half-plane H with hyperbolic metric d. A fundamental… (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 construct certain subgroups of hyperbolic triangle groups which we call " congruence " subgroups. These groups include the classical congruence subgroups of SL 2 (Z), Hecke triangle groups, and 19 families of arithmetic triangle groups associated to Shimura curves. We determine the field of moduli of the curves associated to these groups and thereby… (More)
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 Clif-ford algebra. A quaternion algebra is a central simple algebra of dimension 4 over a field F. Generalizations of the notion of quaternion… (More)
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).
We exhibit algorithms to compute systems of Hecke eigenvalues for spaces of Hilbert modular forms over a totally real field. We provide many explicit examples as well as applications to modularity and Galois representations.
We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. We prove that every curve of genus g ≤ 4 over an algebraically closed field is nondegenerate in the above sense. More generally, let M nd g be the locus of nondegenerate curves inside the moduli space of… (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 enumerate all totally real number fields F with root discriminant δF ≤ 14. There are 1229 such fields, each with degree [F : Q] ≤ 9.