The Eisenstein ideal and Jacquet-Langlands isogeny over function fields

TL;DR

This paper investigates the Eisenstein ideal in the context of Drinfeld modular curves over function fields to compare torsion subgroups of Jacobians and construct explicit Jacquet-Langlands isogenies, surpassing known results over rational numbers.

Contribution

It introduces new methods using the Eisenstein ideal to analyze torsion subgroups and explicitly construct Jacquet-Langlands isogenies over function fields.

Findings

Comparison of rational torsion and cuspidal divisors in Jacobians.

Explicit examples of Jacquet-Langlands isogenies.

Results stronger than existing analogues over $\

Abstract

Let $\frak{p}$ and $\frak{q}$ be two distinct prime ideals of $\mathbb{F}_q[T]$. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve $X_0(\frak{p}\frak{q})$ to compare the rational torsion subgroup of the Jacobian $J_0(\frak{p}\frak{q})$ with its subgroup generated by the cuspidal divisors, and to produce explicit examples of Jacquet-Langlands isogenies. Our results are stronger than what is currently known about the analogues of these problems over $\mathbb{Q}$.