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

Deo, Shaunak; Medvedovsky, Anna

doi:10.1090/btran/35

Newforms mod $p$ in squarefree level with applications to Monsky’s Hecke-stable filtration
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by Shaunak V. Deo and Anna Medvedovsky; with an appendix by Alexandru Ghitza HTML | PDF

Trans. Amer. Math. Soc. Ser. B 6 (2019), 245-273

Abstract:

We propose an algebraic definition of the space of $\ell$-new mod-$p$ modular forms for $\Gamma _0(N\ell )$ in the case that $\ell$ 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 $p$.

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