Eisenstein series modulo prime powers

2025/09/02 by Ahlgren, Scott, Castillo, Cruz, Williams, Clayton
#11F33 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.02427

Abstract

If p≥ 5 is prime and k≥ 4 is an even integer with (p-1)\nmid k we consider the Eisenstein series G_k on SL₂(ℤ) modulo powers of p. It is classically known that for such k we have G_k≡ G_k'\pmod p if k≡ k'\pmodp-1. Here we obtain a generalization modulo prime powers p^m by giving an expression for G_k\pmodp^m in terms of modular forms of weight at most mp. As an application we extend a recent result of the first author with Hanson, Raum and Richter by showing that, modulo powers of E_p-1, every such Eisenstein series is congruent modulo p^m to a modular form of weight at most mp. We prove a similar result for the normalized Eisenstein series E_k in the case that (p-1)| k and m

Citations

• Eisenstein series modulo p²

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