Cusp forms as p-adic limits circumventing p-adic version of the Legendre period relation

2025/06/08 by Pavel Guerzhoy, Guerzhoy, Pavel
Mathematics · #11F11 #11F33 #14H52 #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2506.07107

openalex publication_date 2025/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Several authors have recently proved results which express a cusp form as a p-adic limit of weakly holomorphic modular forms under repeated application of Atkin's U-operator. Initially, these results had a deficiency: one could not rule out the possibility when a certain quantity vanishes and the final result fails to be true. Later on, Ahlgren and Samart \citeAS found a method to prove that no exceptions happen in the specific case considered by El-Guindy and Ono, Hanson and Jameson, and (independently) Dicks. generalized this method to finitely many other cases. In this paper, we present a different approach which allows us to prove a similar non-vanishing result for an infinite family of similar cases. Our approach also allows us to return back to the original example considered by El-Guindy and Ono, where we calculate the (manifestly non-zero) quantity explicitly in terms of Morita's p-adic Γ-function.

Citations

• On the p-adic values of the weight two Eisenstein series for supersingular primes
• The p-adic constant for mock modular forms associated to CM forms
• Cusp forms as p-adic limits
• Weight 2 CM newforms as p-adic limits
• A note on cusp forms as p-adic limits
• Mock modular forms as p-adic modular forms
• Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues

Related

• A note on cusp forms as p-adic limits
• Cusp forms without complex multiplication as p-adic limits
• Cusp forms as p-adic limits
• A construction of \mathfrak v-adic modular forms
• Weight 2 CM newforms as p-adic limits