Newforms mod p in squarefree level, with applications to Monsky's Hecke-stable filtration

TL;DR

This paper introduces an algebraic framework for l-new mod-p modular forms in squarefree level, generalizing classical notions, and applies it to analyze Hecke algebras and involutions in modular forms, with implications for Monsky's filtration.

Contribution

It provides a new algebraic definition of mod-p newforms in squarefree level and interprets Hecke algebras on graded modular form spaces, extending classical concepts.

Findings

Algebraic definition of l-new mod-p modular forms for Gamma0(Nl).

Interpretation of Hecke algebras on Monsky's graded modular forms.

Description of a renormalized Atkin-Lehner automorphism in characteristic p.

Abstract

We propose an algebraic definition of the space of l-new mod-p modular forms for Gamma0(Nl) in the case that l is prime to N, which naturally generalizes to a notion of newforms modulo p in squarefree level. We use this notion of newforms to interpret the Hecke algebras on the graded pieces of the space of mod-2 level-3 modular forms described by Paul Monsky. Along the way, we describe a renormalized version of the Atkin-Lehner involution: no longer an involution, it is an automorphism of the algebra of modular forms, even in characteristic p.