On Jacquet-Langlands isogeny over function fields

TL;DR

This paper conjectures an explicit isogeny between Jacobians of hyperelliptic Drinfeld modular curves and hyperelliptic modular curves of -elliptic sheaves, analyzing the kernel via cuspidal divisors and component groups.

Contribution

It introduces a conjectural explicit isogeny linking two classes of Jacobians in the function field setting, with a novel approach to the kernel structure.

Findings

Proposes a conjectural explicit isogeny between Jacobians.

Analyzes the kernel via cuspidal divisor groups and component maps.

Provides a framework for understanding isogenies in Drinfeld modular curves.

Abstract

We propose a conjectural explicit isogeny from the Jacobians of hyperelliptic Drinfeld modular curves to the Jacobians of hyperelliptic modular curves of $\mathcal{D}$-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.