An explicit correspondence of modular curves

TL;DR

This paper provides an explicit, representation-theory-free proof of Merel's conjecture relating Jacobians of modular curves associated with split and non-split Cartan subgroups, and generalizes it to more complex cases.

Contribution

It offers a new explicit correspondence for isogenies between Jacobians, avoiding extensive representation theory and using finite geometries, with detailed coefficients.

Findings

Proof of Merel's conjecture without extensive representation theory

Explicit correspondence expressed as linear combination of double coset operators

Generalization to more complex cases with explicit coefficients

Abstract

In this paper, we recall an alternative proof of Merel's conjecture which asserts that a certain explicit correspondence gives the isogeny relation between the Jacobians associated to the normalizer of split and non-split Cartan subgroups. This alternative proof does not require extensive representation theory and can be formulated in terms of certain finite geometries modulo $\ell$. Secondly, we generalize these arguments to exhibit an explicit correspondence which gives the isogeny relation between the Jacobians associated to split and non-split Cartan subgroups. An interesting feature is that the required explicit correspondence is considerably more complicated but can expressed as a certain linear combination of double coset operators whose coefficients we are able to make explicit.