Determination of the modular Jacobian varieties $J_1(M,MN)$ with the Mordell-Weil rank zero

TL;DR

This paper classifies certain modular Jacobian varieties over cyclotomic fields that have Mordell-Weil rank zero, expanding understanding of their arithmetic properties.

Contribution

It determines all $J_1(M,MN)$ modular Jacobian varieties over $Q(zeta_M)$ with rank zero, using a specific method from recent research.

Findings

All such Jacobian varieties with rank zero are classified.

The classification is complete over the specified cyclotomic fields.

Abstract

In this paper, we determine all modular Jacobian varieties $J_1(M,MN)$ over the number field $\mathbb{Q}(\zeta_M)$ with the Mordell-Weil rank zero following the method of Derickx, Etropolski, van Hoeij, Morrow, and Zureick-Brown.