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- Markus Kirschmer, John Voight
- SIAM J. Comput.
- 2010

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)

- 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 ⊂ PSL2(R) be a Fuchsian group, a discrete group of orientationpreserving isometries of the upper half-plane H with hyperbolic metric d. A fundamental domain… (More)

- JOHN VOIGHT
- 2009

For p = 3 and p = 5, we exhibit a finite nonsolvable extension of Q which is ramified only at p via explicit computations with Hilbert modular forms. The study of Galois number fields with prescribed ramification remains a central question in number theory. Class field theory, a triumph of early twentieth century algebraic number theory, provides a… (More)

- John Voight
- ANTS
- 2008

We enumerate all totally real number fields F with root discriminant δF ≤ 14. There are 1229 such fields, each with degree [F : Q] ≤ 9. In this article, we consider the following problem. Problem 1. Given B ∈ R>0, enumerate the set NF (B) of totally real number fields F with root discriminant δF ≤ B, up to isomorphism. To solve Problem 1, for each n ∈ Z>0… (More)

- JOHN VOIGHT
- 2010

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… (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 Clifford algebra. A quaternion algebra is a central simple algebra of dimension 4 over a field F . Generalizations of the notion of quaternion… (More)

In a previous paper, we proved that over a finite field k of sufficiently large cardinality, all curves of genus at most 3 over k can be modeled by a bivariate Laurent polynomial that is nondegenerate with respect to its Newton polytope. In this paper, we prove that there are exactly two curves of genus at most 3 over a finite field that are not… (More)

- John Voight
- ANTS
- 2010

We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with… (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 g be the locus of nondegenerate curves inside the moduli space of… (More)

- John Voight
- Math. Comput.
- 2009

We enumerate all Shimura curves XD 0 (N) of genus at most two: there are exactly 858 such curves, up to equivalence. The elliptic modular curve X0(N) is the quotient of the completed upper halfplane H∗ by the congruence subgroup Γ0(N) of matrices in SL2(Z) that are upper triangular modulo N ∈ Z>0. The curve X0(N) forms a coarse moduli space for… (More)