The Eisenstein Ideal and Jacquet-Langlands Isogeny over Function Fields

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

2010 Mathematics Subject Classification: 11G09, 11G18, 11F12

Keywords and Phrases: Drinfeld modular curves; Cuspidal divisor group; Shimura subgroup; Eisenstein ideal; Jacquet-Langlands isogeny

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