The Eisenstein ideal and Jacquet-Langlands isogeny over function fields

Abstract

Let $\mathfrak{p}$ and $\mathfrak{q}$ be two distinct prime ideals of ${\mathbb{F}}_q[T]$. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve $X_0(\mathfrak{pq})$ to compare the rational torsion subgroup of the Jacobian $J_0(\mathfrak{pq})$ 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 $\mathbb{Q}$.