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2021 | Book

Arithmetic Geometry, Number Theory, and Computation

Editors: Jennifer S. Balakrishnan, Noam Elkies, Brendan Hassett, Prof. Bjorn Poonen, Andrew V. Sutherland, John Voight

Publisher: Springer International Publishing

Book Series: Simons Symposia

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About this book

This volume contains articles related to the work of the Simons Collaboration “Arithmetic Geometry, Number Theory, and Computation.” The papers present mathematical results and algorithms necessary for the development of large-scale databases like the L-functions and Modular Forms Database (LMFDB). The authors aim to develop systematic tools for analyzing Diophantine properties of curves, surfaces, and abelian varieties over number fields and finite fields. The articles also explore examples important for future research.
Specific topics include● algebraic varieties over finite fields● the Chabauty-Coleman method● modular forms● rational points on curves of small genus● S-unit equations and integral points.

Table of Contents

Frontmatter
A Robust Implementation for Solving the S-Unit Equation and Several Applications
Abstract
Let K be a number field, and S a finite set of places in K containing all infinite places. We present an implementation for solving the S-unit equation x + y = 1, \(x,y \in \mathcal {O}_{K,S}^\times \) in the computer algebra package SageMath. This paper outlines the mathematical basis for the implementation. We discuss and reference the results of extensive computations, including exponent bounds for solutions in many fields of small degree for small sets S. As an application, we prove an asymptotic version of Fermat’s Last Theorem for totally real cubic number fields with bounded discriminant where 2 is totally ramified. In addition, we use the implementation to find all solutions to some cubic Ramanujan-Nagell equations.
Alejandra Alvarado, Angelos Koutsianas, Beth Malmskog, Christopher Rasmussen, Christelle Vincent, Mckenzie West
Computing Classical Modular Forms for Arbitrary Congruence Subgroups
Abstract
In this paper, we prove the existence of an efficient algorithm for the computation of systems of Hecke eigenvalues of modular forms of weight k and level Γ, where \(\Gamma \subseteq SL_{2}({\mathbb {Z}})\) is an arbitrary congruence subgroup. We also discuss some practical aspects and provide the necessary theoretical background.
Eran Assaf
Square Root Time Coleman Integration on Superelliptic Curves
Abstract
Since Kedlaya first introduced a p-adic algorithm for computing zeta functions of hyperelliptic curves, many related algorithms for computing both zeta functions and Coleman integrals on various classes of algebraic curves have been studied. These algorithms compute in the Monsky-Washnitzer cohomology or the rigid cohomology of the curve to determine the action of Frobenius on this cohomology.
We give a new algorithm for explicitly computing Coleman integrals on superelliptic curves over unramified extensions of p-adic fields. The runtime is softly linear with respect to the square root of the size of the residue field, bringing the runtime in line with that of the corresponding zeta function algorithms. We also describe the implementation of this algorithm in Nemo, a new package for the Julia programming language, which adds functionality for computational number theory. We compare Nemo with other systems in use in this area.
Alex J. Best
Computing Classical Modular Forms
Abstract
We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and modular forms database (LMFDB).
Alex J. Best, Jonathan Bober, Andrew R. Booker, Edgar Costa, John E. Cremona, Maarten Derickx, Min Lee, David Lowry-Duda, David Roe, Andrew V. Sutherland, John Voight
Elliptic Curves with Good Reduction Outside of the First Six Primes
Abstract
We present a database of rational elliptic curves, up to https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-80914-0_5/510803_1_En_5_IEq1_HTML.gif -isomorphism, with good reduction outside {2, 3, 5, 7, 11, 13}. We provide a heuristic involving the abc and BSD conjectures that the database is likely to be the complete set of such curves. Moreover, proving completeness likely needs only more computation time to conclude. We present data on the distribution of various quantities associated to curves in the set. We also discuss the connection to S-unit equations and the existence of rational elliptic curves with maximal conductor.
Alex J. Best, Benjamin Matschke
Efficient Computation of BSD Invariants in Genus 2
Abstract
Recently, all Birch and Swinnerton-Dyer invariants, except for the order of https://static-content.springer.com/image/chp%3A10.1007%2F978-3-030-80914-0_6/510803_1_En_6_IEq1_HTML.gif , have been computed for all curves of genus 2 contained in the L-functions and Modular Forms Database [LMFDB]. This report explains the improvements made to the implementation of the algorithm described in [vBom19] that were needed to do the computation of the Tamagawa numbers and the real period in reasonable time. We also explain some of the more technical details of the algorithm, and give a brief overview of the methods used to compute the special value of the L-function and the regulator.
Raymond van Bommel
Restrictions on Weil Polynomials of Jacobians of Hyperelliptic Curves
Abstract
Inspired by experimental data, we investigate which isogeny classes of abelian varieties defined over a finite field of odd characteristic contain the Jacobian of a hyperelliptic curve. We provide a necessary condition by demonstrating that the Weil polynomial of a hyperelliptic Jacobian must have a particular form modulo 2. For fixed g ≥ 1, the proportion of isogeny classes of g-dimensional abelian varieties defined over \(\mathbb {F}_q\) which fail this condition is 1 − Q(2g + 2)∕2g as q → ranges over odd prime powers, where Q(n) denotes the number of partitions of n into odd parts.
Edgar Costa, Ravi Donepudi, Ravi Fernando, Valentijn Karemaker, Caleb Springer, Mckenzie West
Zen and the Art of Database Maintenance
Abstract
The last decade has seen a proliferation of online mathematical resources. We discuss some of the technical challenges involved in creating and maintaining a mathematical database. In particular, we report on the transition of the L-functions and Modular Forms Database (LMFDB) between two database systems. We also highlight some of the improvements to the LMFDB that we have made as part of this transition.
Edgar Costa, David Roe
Effective Obstruction to Lifting Tate Classes from Positive Characteristic
Abstract
We give an algorithm that takes a smooth hypersurface over a number field and computes a p-adic approximation of the obstruction map on the Tate classes of a finite reduction. This gives an upper bound on the “middle Picard number” of the hypersurface. The improvement over existing methods is that our method relies only on a single prime reduction and gives the possibility of cutting down on the dimension of Tate classes by two or more. The obstruction map comes from p-adic variational Hodge conjecture and we rely on the recent advancement by Bloch–Esnault–Kerz to interpret our bounds.
Edgar Costa, Emre Can Sertöz
Conjecture: 100% of Elliptic Surfaces Over have Rank Zero
Abstract
Based on an equation for the rank of an elliptic surface over \(\mathbb {Q}\) which appears in the work of Nagao, Rosen, and Silverman, we conjecture that 100% of elliptic surfaces have rank 0 when ordered by the size of the coefficients of their Weierstrass equations, and present a probabilistic heuristic to justify this conjecture. We then discuss how it would follow from either understanding of certain L-functions, or from understanding of the local behaviour of the surfaces. Finally, we make a conjecture about ranks of elliptic surfaces over finite fields, and highlight some experimental evidence supporting it.
Alex Cowan
On Rational Bianchi Newforms and Abelian Surfaces with Quaternionic Multiplication
Abstract
We study the rational Bianchi newforms (weight 2, trivial character, with rational Hecke eigenvalues) in the LMFDB that are not associated to elliptic curves, but instead to abelian surfaces with quaternionic multiplication. Two of these examples exhibit a rather special kind of behaviour: we show they arise from twisted base change of a classical newform with nebentypus character of order 4 and eight inner twists.
J. E. Cremona, Lassina Dembélé, Ariel Pacetti, Ciaran Schembri, John Voight
A Database of Hilbert Modular Forms
Abstract
We describe the computation of tables of Hilbert modular forms of parallel weight 2 over totally real fields.
Steve Donnelly, John Voight
Isogeny Classes of Abelian Varieties over Finite Fields in the LMFDB
Abstract
This document is intended to summarize the theory and methods behind fq_isog collection inside the ab_var database in the LMFDB as well as some observations gleaned from these databases. This collection consists of tables of Weil q-polynomials, which by the Honda-Tate theorem are in bijection with isogeny classes of abelian varieties over finite fields.
Taylor Dupuy, Kiran Kedlaya, David Roe, Christelle Vincent
Computing Rational Points on Rank 0 Genus 3 Hyperelliptic Curves
Abstract
We compute rational points on genus 3 odd degree hyperelliptic curves C over \(\mathbb {Q}\) that have Jacobians of Mordell–Weil rank 0. The computation applies the Chabauty–Coleman method to find the zero set of a certain system of p-adic integrals, which is known to be finite and include the set of rational points \(C(\mathbb {Q})\). We implemented an algorithm in Sage to carry out the Chabauty–Coleman method on a database of 5870 curves.
María Inés de Frutos-Fernández, Sachi Hashimoto
Curves with Sharp Chabauty-Coleman Bound
Abstract
We construct curves of each genus g ≥ 2 for which Coleman’s effective Chabauty bound is sharp and Coleman’s theorem can be applied to determine rational points if the rank condition is satisfied. We give numerous examples of genus two and rank one curves for which Coleman’s bound is sharp. Based on one of those curves, we construct an example of a curve of genus five whose rational points are determined using the descent method together with Coleman’s theorem.
Stevan Gajović
Chabauty–Coleman Computations on Rank 1 Picard Curves
Abstract
We provably compute the full set of rational points on 1403 Picard curves defined over \(\mathbb {Q}\) with Jacobians of Mordell–Weil rank 1 using the Chabauty–Coleman method. To carry out this computation, we extend Magma code of Balakrishnan and Tuitman for Coleman integration. The new code computes p-adic (Coleman) integrals on curves to points defined over number fields where the prime p splits completely and implements effective Chabauty for curves whose Jacobians have infinite order points that are not the image of a rational point under the Abel–Jacobi map. We discuss several interesting examples of curves where the Chabauty–Coleman set contains points defined over number fields.
Sachi Hashimoto, Travis Morrison
Linear Dependence Among Hecke Eigenvalues
Abstract
We prove an explicit upper bound on the absolute value of the coefficients of a non-trivial integral linear relation among Hecke eigenvalues of a given cuspidal eigenform. Our motivation lies in its algorithmic application. For any fixed positive integer n, the bound established here yields an algorithm that computes cuspidal Hecke eigenforms with a given weight k whose Hecke eigenvalues generate a number field of degree n. The resulting algorithm reduces to Cremona’s when n = 1 and k = 2.
Dohyeong Kim
Congruent Number Triangles with the Same Hypotenuse
Abstract
In this article, we discuss whether a single congruent number t can have two (or more) distinct corresponding triangles with the same hypotenuse. We describe and carry out computational experimentation providing evidence that this does not occur.
David Lowry-Duda appendix by Brendan Hassett
Visualizing Modular Forms
Abstract
We examine several currently used techniques for visualizing complex-valued functions applied to modular forms. We plot several examples and study the benefits and limitations of each technique. We then introduce a method of visualization that can take advantage of colormaps in Python’s matplotlib library, describe an implementation, and give more examples. Much of this discussion applies to general visualizations of complex-valued functions in the plane.
David Lowry-Duda
A Prym Variety with Everywhere Good Reduction over
Abstract
We compute an equation for a modular abelian surface A that has everywhere good reduction over the quadratic field \(K = \mathbb {Q}(\sqrt {61})\) and that does not admit a principal polarization over K.
Nicolas Mascot, Jeroen Sijsling, John Voight
The S-Integral Points on the Projective Line Minus Three Points via Finite Covers and Skolem’s Method
Abstract
We describe a p-adic proof of the finiteness of \((\mathbb {P}^1-\{0,1,\infty \})(\mathbb {Z}[S^{-1}])\) using only Skolem’s method applied to finite covers.
Bjorn Poonen
Metadata
Title
Arithmetic Geometry, Number Theory, and Computation
Editors
Jennifer S. Balakrishnan
Noam Elkies
Brendan Hassett
Prof. Bjorn Poonen
Andrew V. Sutherland
John Voight
Copyright Year
2021
Electronic ISBN
978-3-030-80914-0
Print ISBN
978-3-030-80913-3
DOI
https://doi.org/10.1007/978-3-030-80914-0

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