On the space of theta functions for a prime level

TL;DR

This paper investigates the subspace of modular forms spanned by theta series associated with quaternion algebras, providing a new proof of existing results using arithmetic and geometric methods.

Contribution

It offers an alternative proof to Boecherer and Schulze-Pillot's results on the dimension of theta series subspaces using modular curve properties.

Findings

Confirmed the dimension of theta series subspace matches previous results.

Demonstrated the effectiveness of arithmetic and geometric methods in modular form analysis.

Provided insights into Hecke's basis problem for prime level N.

Abstract

Hecke expected that an explicit set of theta series obtained from maximal orders of the definite quaternion algebra over Q which is ramified at a prime N will be a basis of the space of holomorphic modular forms of weight 2 and level N. However, later Eichler noticed that Hecke's conjecture does not hold in general. It is natural to ask for the dimension of the subspace of the modular forms spanned by the theta series. This question is called Hecke's basis problem. Boecherer and Schulze-Pillot have given an answer using the theory of theta liftings. In this paper we will give another proof of their results using arithmetic and geometric properties of the modular curve.