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

Abstract

We propose an algebraic definition of the space of ℓ \ell -new mod- p p modular forms for Γ 0 ( N ℓ ) \Gamma _0(N\ell ) in the case that ℓ \ell is prime to N N , which naturally generalizes to a notion of newforms modulo p 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 2 level- 3 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 p .