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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 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)

Quadratic forms and quaternion algebras: algorithms and arithmetic y tohn wihel oight hotor of hilosophy in wthemtis
niversity of gliforni t ferkeley rofessor rendrik venstrD ghir his thesis omes in two prts whih n e red independently of one notherF sn the rst prtD we prove result onerning representtion of primes y qudrti… (More)

- JOHN VOIGHT
- 2010

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)

- PETE L. CLARK, JOHN VOIGHT
- 2008

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)

- John Voight
- 2010

Contents Acknowledgements vii Chapter 1. Introduction 1 Chapter 2. Quaternion algebras over fields 3 1. Quaternion algebras 3 Exercises 5 2. Involutions 5 Involutions 5 Reduced trace and norm 6 Uniqueness 7 Quaternion algebras 7 Exercises 9 3. Quadratic forms 10 Definitions 10 Nonsingular standard involution 12 Quaternion algebras 13 Splitting 14 Hilbert… (More)

- John Voight
- 2009

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 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 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 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)