Barcelona Fall Workshop on Number Theory II

Implementing Algorithms to Compute Elliptic Curves Over $\mathbb{Q}$

Presenters

A. Gherga (UBC Vancouver)

Abstract

Let $S$ be a set of rational primes and consider the set of all elliptic curves over $\mathbb{Q}$ having good reduction outside $S$ and bounded conductor $N$. Currently, using modular forms, all such curves have been determined for $N \leq 500000$, the bulk of this work being attributed to Cremona. Early attempts to tabulate all such curves often relied on reducing the problem to one of solving a number of certain integral binary forms called Thue-Mahler equations. These are Diophantine equations of the form $F(x,y) = u$, where $F$ is a given binary form of degree at least $3$ and $u$ is an $S$-unit. A theorem of Bennett-Rechnitzer shows that the problem of computing all elliptic curves over $\mathbb{Q}$ of conductor $N$ reduces to solving a number of Thue-Mahler equations. To resolve all such equations, there exists a practical method of Tzanakis-de Weger using bounds for linear forms in $p$-adic logarithms and various reduction techniques. In this talk, we describe our refined implementation of this method and discuss the key steps used in our algorithm.

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