On families of 13-congruent elliptic curves
Tom Fisher

TL;DR
This paper investigates families of elliptic curves that are 13-congruent, computes related modular curve twists, and demonstrates the existence of infinitely many such non-isogenous pairs over rationals.
Contribution
It explicitly constructs and analyzes the surfaces parametrizing 13-congruent elliptic curves, revealing infinite families of non-trivial pairs over ${f Q}$.
Findings
Identifies equations for the surfaces parametrising 13-congruent elliptic curves.
Shows there are infinitely many non-trivial 13-congruent pairs over ${f Q}$.
Distinguishes between 13-congruences that do and do not respect the Weil pairing.
Abstract
We compute twists of the modular curve that parametrise the elliptic curves 13-congruent to a given elliptic curve. Searching for rational points on these twists enables us to find non-trivial pairs of 13-congruent elliptic curves over , i.e. pairs of non-isogenous elliptic curves over whose 13-torsion subgroups are isomorphic as Galois modules. We also find equations for the surfaces parametrising pairs of 13-congruent elliptic curves. There are two such surfaces, corresponding to 13-congruences that do, or do not, respect the Weil pairing. We write each as a double cover of the projective plane ramified over a highly singular model for Baran's modular curve of level 13. By finding suitable rational curves on these surfaces, we show that there are infinitely many non-trivial pairs of 13-congruent elliptic curves over .
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Advanced Algebra and Geometry · Algebraic structures and combinatorial models
