On Jacquet–Langlands isogeny over function fields

Abstract

We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of D-elliptic sheaves. The kernel of the isogeny is a subgroup of the cuspidal divisor group constructed by examining the canonical maps from the cuspidal divisor group into the component groups.

On the Eisenstein ideal over function fields

“…Both [32] and [21] assume that the level is prime. In [27], in connection with the problem of Jacquet-Langlands isogenies over function fields, we examined the Eisenstein ideal on Drinfeld modular curves whose level is a product of two distinct primes. We discovered that some of the properties of the Eisenstein ideal in that case are quite different from its prime level counterpart.…”

Rational torsion of generalized Jacobians of modular and Drinfeld modular curves

“…On the other hand, the Eichler-Shimura congruence relation shows that the Hecke correspondence T ℓ acts on J /F ℓ (F ℓ ) by the multiplication by ℓ + 1, whence the statement. (2) is shown in[13, Lemma 7.1]. (3)is well-known, and it also follows from Proposition 5.2.5 (3) below.5.2.Generalized Jacobian.…”

D-elliptic sheaves and odd Jacobians

“…Let r := x∈R p x . The Jacquet-Langlands correspondence over F in combination with some other deep results implies that there is a surjective homomorphism J 0 (r) → J D defined over F ; see [18,Theorem 7.1]. Since by construction X (y) is a quotient of X D , there is also a surjective homomorphism J D → J (y) defined over F .…”