Defining newforms in characteristic $p$

TL;DR

This paper explores algebraic approaches to defining newforms in characteristic p, extending the classical theory of modular forms and addressing challenges in generalization.

Contribution

It details Deo and Medvedovsky's algebraic methods for newforms and generalizes their results to broader contexts.

Findings

Detailed exposition of Deo and Medvedovsky's methods

Generalization of algebraic newform definitions in characteristic p

Potential applications to modular form decomposition

Abstract

The theory of newforms, due to Atkin and Lehner, provides a powerful method for decomposing spaces of modular forms. However, many problems occur when trying to generalise this theory to characteristic $p$. Recently, Deo and Medvedovsky have suggested a way around these problems by using purely algebraic notions to define newforms. In this thesis, we describe the methods of Deo and Medvedovsky in detail and generalise their results where possible.