An Algorithm for Checking Injectivity of Specialization Maps from Elliptic Surfaces

TL;DR

This paper presents an algorithm to verify the injectivity of specialization maps from elliptic surfaces to elliptic curves over rational numbers, with effective computation in specific cases.

Contribution

It introduces a new algorithm for checking injectivity of specialization maps for elliptic surfaces, applicable to subgroups under mild conditions, with effective computation for explicit examples.

Findings

Algorithm successfully determines injectivity in tested cases.

Effective computation of the set $S_M$ in certain scenarios.

Application to elliptic curves given by explicit Weierstrass equations.

Abstract

Let $E/\mathbb Q(t)$ be an elliptic curve and let $t_0 \in \mathbb Q$ be a rational number for which the specialization $E_{t_0}$ is an elliptic curve. Given a subgroup $M$ of $E(\mathbb Q(t))$ with mild conditions and $t_0 \in \mathbb Q$ coming from a relatively large subset $S_M \subset \mathbb Q$, we provide an algorithm that can show that the specialization map $\sigma_{t_0} : E(\mathbb Q(t)) \to E_{t_0}(\mathbb Q)$ is injective when restricted to $M$. The set $S_M$ is effectively computable in certain cases, and we carry out this computation for some explicit examples where $E$ is given by a Weierstrass equation.