John Voight
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View article: Computing class groups and unit groups in Magma
Computing class groups and unit groups in Magma Open
We describe the computation of class groups and unit groups of number fields as implemented in Magma (V2.29). After quickly reviewing the main algorithms based on factor bases, relation collection, and analytic class number evaluation, we …
View article: Computing Hilbert modular forms as orthogonal modular forms
Computing Hilbert modular forms as orthogonal modular forms Open
We show how to efficiently compute Hilbert modular forms as orthogonal modular forms, generalizing and expanding upon the method of Birch.
View article: A framework for Tate modules of abelian varieties under isogeny
A framework for Tate modules of abelian varieties under isogeny Open
We explain the linear algebraic framework provided by Tate modules of isogenous abelian varieties in a category-theoretic way.
View article: Hypergeometric Motives from Euler Integral Representations
Hypergeometric Motives from Euler Integral Representations Open
We revisit certain one-parameter families of affine covers arising naturally from Euler's integral representation of hypergeometric functions. We introduce a partial compactification of this family. We show that the zeta function of the fi…
View article: 17T7 is a Galois group over the rationals
17T7 is a Galois group over the rationals Open
We prove that the transitive permutation group 17T7, isomorphic to a split extension of $C_2$ by $\mathrm{PSL}_2(\mathbb{F}_{16})$, is a Galois group over the rationals. The group arises from the field of definition of the $2$-torsion on a…
View article: Book review: “Hurwitz’s Lectures on the Number Theory of Quaternions” by Nicola Oswald and Jörn Steuding
Book review: “Hurwitz’s Lectures on the Number Theory of Quaternions” by Nicola Oswald and Jörn Steuding Open
published his pioneering work Vorlesungen über die Zahlentheorie der Quaternionen in 1919, just before his death.In this remarkable booklet, he sought to make the number-theoretic aspects of quaternions widely accessible and to encourage h…
View article: Ideal classes of orders in quaternion algebras
Ideal classes of orders in quaternion algebras Open
We provide an algorithm that, given any order O in a quaternion algebra over a global field, computes representatives of all right equivalence classes of right O-ideals, including the non-invertible ones. The theory is developed for a more…
View article: LuCaNT: LMFDB, Computation, and Number Theory
LuCaNT: LMFDB, Computation, and Number Theory Open
LuCaNT there was a robust panel discussion on this particular review process, as well as on best practices in computational mathematics.Eighty-five registered participants enjoyed the LuCaNT conference, comprising a diverse group from over…
View article: Rational torsion points on abelian surfaces with quaternionic multiplication
Rational torsion points on abelian surfaces with quaternionic multiplication Open
Let A be an abelian surface over ${\mathbb {Q}}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur’s theorem for elliptic curves, we show that the torsion subgroup of $A({\mathbb {Q}}…
View article: A database of basic numerical invariants of Hilbert modular surfaces
A database of basic numerical invariants of Hilbert modular surfaces Open
We describe algorithms for computing geometric invariants for Hilbert modular surfaces, and we report on their implementation.
View article: A database of paramodular forms from quinary orthogonal modular forms
A database of paramodular forms from quinary orthogonal modular forms Open
We compute tables of paramodular forms of degree two and cohomological weight via a correspondence with orthogonal modular forms on quinary lattices.
View article: Computing Euclidean Belyi maps
Computing Euclidean Belyi maps Open
We exhibit an explicit algorithm to compute three-point branched covers of the complex projective line when the uniformizing triangle group is Euclidean.
View article: On abelian varieties whose torsion is not self-dual
On abelian varieties whose torsion is not self-dual Open
We construct infinitely many abelian surfaces A defined over the rational numbers such that, for a prime ell <= 7, the ell-torsion subgroup of A is not isomorphic as a Galois module to the ell-torsion subgroup of its dual. We do this by ex…
View article: Rational torsion points on abelian surfaces with quaternionic multiplication
Rational torsion points on abelian surfaces with quaternionic multiplication Open
Let $A$ be an abelian surface over $\mathbb{Q}$ whose geometric endomorphism ring is a maximal order in a non-split quaternion algebra. Inspired by Mazur's theorem for elliptic curves, we show that the torsion subgroup of $A(\mathbb{Q})$ i…
View article: Kneser's method of neighbors
Kneser's method of neighbors Open
In a landmark paper published in 1957, Kneser introduced a method for enumerating classes in the genus of a definite, integral quadratic form. This method has been deeply influential, on account of its theoretical importance as well as its…
View article: A database of paramodular forms from quinary orthogonal modular forms
A database of paramodular forms from quinary orthogonal modular forms Open
We compute tables of paramodular forms of degree two and cohomological weight via a correspondence with orthogonal modular forms on quinary lattices.
View article: Monodromy groups of Jacobians with definite quaternionic multiplication
Monodromy groups of Jacobians with definite quaternionic multiplication Open
Let $A$ be an abelian variety over a number field. The connected monodromy field of $A$ is the minimal field over which the images of all the $\ell$-adic torsion representations have connected Zariski closure. We show that for all even $g …
View article: On unit signatures and narrow class groups of odd degree abelian number fields
On unit signatures and narrow class groups of odd degree abelian number fields Open
For an abelian number field of odd degree, we study the structure of its -Selmer group as a bilinear space and as a Galois module. We prove structural results and make predictions for the distribution of unit signature ranks and narrow cla…
View article: A database of basic numerical invariants of Hilbert modular surfaces
A database of basic numerical invariants of Hilbert modular surfaces Open
We describe algorithms for computing geometric invariants for Hilbert modular surfaces, and we report on their implementation.
View article: Counting elliptic curves over the rationals with a 7-isogeny
Counting elliptic curves over the rationals with a 7-isogeny Open
We count by height the number of elliptic curves over the rationals, both up to isomorphism over the rationals and over an algebraic closure thereof, that admit a cyclic isogeny of degree $7$.
View article: Definite orthogonal modular forms: computations, excursions, and discoveries
Definite orthogonal modular forms: computations, excursions, and discoveries Open
We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we inves…
View article: Stickelberger's discriminant theorem for algebras
Stickelberger's discriminant theorem for algebras Open
Stickelberger proved that the discriminant of a number field is congruent to 0 or 1 modulo 4. We generalize this to an arbitrary (not necessarily commutative) ring of finite rank over the integers using techniques from linear algebra. Our …
View article: Triangular modular curves of low genus
Triangular modular curves of low genus Open
Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with mono…
View article: On Galois inertial types of elliptic curves over $\mathbb{Q}_\ell$
On Galois inertial types of elliptic curves over $\mathbb{Q}_\ell$ Open
We provide a complete, explicit description of the inertial Weil--Deligne types arising from elliptic curves over $\mathbb{Q}_\ell$ for $\ell$ prime.
View article: Definite orthogonal modular forms: Computations, Excursions and Discoveries
Definite orthogonal modular forms: Computations, Excursions and Discoveries Open
We consider spaces of modular forms attached to definite orthogonal groups of low even rank and nontrivial level, equipped with Hecke operators defined by Kneser neighbours. After reviewing algorithms to compute with these spaces, we inves…
View article: Computing Euclidean Belyi maps
Computing Euclidean Belyi maps Open
We exhibit an explicit algorithm to compute three-point branched covers of the complex projective line when the uniformizing triangle group is Euclidean.
View article: 4. The $L$-Functions and Modular Forms Database by John E. Cremona, John W. Jones, Andrew V. Sutherland, and John Voight
4. The $L$-Functions and Modular Forms Database by John E. Cremona, John W. Jones, Andrew V. Sutherland, and John Voight Open
Calculation, tabulation, and experiment have always played a significant role in number theory.Here we describe the L-functions and modular forms database (LMFDB) [LMFDB],
View article: On basic and Bass quaternion orders
On basic and Bass quaternion orders Open
A quaternion order over a Dedekind domain is Bass if every -superorder is Gorenstein, and is basic if it contains an integrally closed quadratic -order. In this article, we show that these conditions are equivalent in local and global s…