Torsion points of elliptic curves over multi-quadratic number fields

TL;DR

This paper computes Mordell-Weil groups of certain modular Jacobians over multi-quadratic fields and establishes criteria for the existence of elliptic curves with specific torsion points over these fields.

Contribution

It extends the understanding of torsion points and Mordell-Weil groups of modular Jacobians to multi-quadratic fields, generalizing previous quadratic field results.

Findings

Mordell-Weil groups computed for hyperelliptic modular curves over multi-quadratic fields

Criteria established for the existence of elliptic curves with prescribed torsion over these fields

Generalization of Kamienny and Najman's results to multi-quadratic number fields

Abstract

We compute the Mordell-Weil groups of the modular Jacobian varieties of hyperelliptic modular curves $X_1(M, MN)$ over every number field which is the composition of quadratic fields. Also we prove criteria for the existence of elliptic curves over such number fields with prescribed torsion points generalizing the results for quadratic number fields of Kamienny and Najman.