On a result of Imin Chen

TL;DR

This paper provides a new proof of Imin Chen's result on the isogeny of Jacobians of certain modular curves, utilizing representation theory of GL_2(Z/pZ), and generalizes it to broader algebraic objects.

Contribution

It introduces a novel proof method based solely on representation theory and extends Chen's result to objects with GL_2(Z/pZ) actions in general categories.

Findings

Proof of Chen's Jacobian isogeny using representation theory

Generalization to objects with GL_2(Z/pZ) action in pseudo-abelian categories

Applicable to a broad class of algebraic objects with group actions

Abstract

We give another proof of Imin Chen's result that the jacobian of the modular curve X(p)_{non-split}, for p a prime number, is isogeneous to the new part of the jacobian of X_0(p^2), using only the representation theory of the group GL_2(Z/pZ). In fact, we prove a generalization of Chen's result for objects with an action by GL_2(Z/pZ) in any pseudo-abelian Q-linear category.