The Eisenstein ideal and Jacquet-Langlands isogeny over function fields

Abstract

Let p and q 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 (pq) to compare the rational torsion subgroup of the Jacobian J 0 (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 Q.

The rational torsion subgroup of J0(N)

“…Utilizing Thm. 6.5, we demonstrate that E n generates E(n, R)[U p ] over R. Considering the values of E n on specific edges of T , we observe that the subgroup [12,Lem. 7.2], let R := Z/ℓ k Z for a prime ℓ ∤ q(q − 1) and a sufficiently large k, then there exists an injective map:…”

“…In the early 1970s, for any prime p, Ogg [10,Conj. 2] conjectured that C(p) = T (p) and computed that C(p) is a cyclic group of order p−1 (p− 1,12) . Later in 1977, Mazur [9,Thm.…”

“…1.5] determined the prime-to-(q − 1) part of C(n) for any square-free ideal n ⊳ A. In particular, when n = pq is square-free with two prime ideals p and q of A, the structure of C(pq) was entirely determined in [12,Thm. 1.1].…”