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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 wi™h—el †oight ho™tor of €hilosophy in w—them—ti™s …niversity of g—liforni— —t ferkeley €rofessor rendrik venstr—D gh—ir „his thesis ™omes in two p—rts whi™h ™—n ˜e re—d independently of one —notherF sn the rst p—rtD we prove — result ™on™erning represent—tion of primes ˜y qu—dr—ti™(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 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)
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)
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 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)