An Algorithm for Checking Injectivity of Specialization Maps from Elliptic Surfaces
Tyler Raven Billingsley

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 be an elliptic curve and let be a rational number for which the specialization is an elliptic curve. Given a subgroup of with mild conditions and coming from a relatively large subset , we provide an algorithm that can show that the specialization map is injective when restricted to . The set is effectively computable in certain cases, and we carry out this computation for some explicit examples where is given by a Weierstrass equation.
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Taxonomy
TopicsPolynomial and algebraic computation · Algebraic Geometry and Number Theory · Advanced Numerical Analysis Techniques
