Modular forms invariant under non-split Cartan subgroups

TL;DR

This paper develops a method to compute bases of weight 2 cusp forms invariant under non-split Cartan subgroups and applies it to explicitly determine equations for related modular curves over Q.

Contribution

It introduces a novel computational approach for invariant cusp forms under non-split Cartan subgroups and derives explicit equations for the associated modular curves.

Findings

Computed bases for invariant cusp forms at small primes p

Derived explicit equations for modular curves over Q

Enhanced understanding of modular forms under non-split Cartan subgroups

Abstract

In this paper we describe a method for computing a basis for the space of weight $2$ cusp forms invariant under a non-split Cartan subgroup of prime level $p$. As an application we compute, for certain small values of $p$, explicit equations over $\bf Q$ for the canonical embeddings of the associated modular curves.