Functions and differentials on the non-split Cartan modular curve of level 11

TL;DR

This paper studies the genus 4 modular curve X_{ns}(11) related to a non-split Cartan group, providing explicit generators for its function field and revealing its Jacobian's isomorphism to a component of X_0(121).

Contribution

It explicitly computes generators for the function field of X_{ns}(11) and establishes an isomorphism of its Jacobian with a part of the Jacobian of X_0(121).

Findings

Generators for the function field are expressed in terms of Siegel modular functions.

The Jacobian of X_{ns}(11) is isomorphic over a0 to the new part of the Jacobian of X_0(121).

The curve admits a model defined over a0.

Abstract

The genus $4$ modular curve $X_{ns}(11)$ attached to a non-split Cartan group of level $11$ admits a model defined over $\mathbb{Q}$. We compute generators for its function field in terms of Siegel modular functions. We also show that its Jacobian is isomorphic over $\mathbb{Q}$ to the new part of the Jacobian of the classical modular curve $X_0(121)$.