Eisenstein series modulo $p^2$

TL;DR

This paper investigates congruences of Eisenstein series modulo p^2, revealing new relationships between their weights and modular forms, including a detailed analysis of E_2 modulo p^2.

Contribution

It establishes that Eisenstein series of higher weights are determined modulo p^2 by forms of lower weights, refining classical results and providing explicit descriptions for E_2.

Findings

Eisenstein series of weight ≥ 4 are determined by those of weight ≤ p^2 - p + 2.

Up to powers of E_{p-1}, each Eisenstein series is determined by forms of weight ≤ 2p - 4.

E_2 modulo p^2 is expressed in terms of a modular form of weight p + 1.

Abstract

We study congruences for Eisenstein series on $\mathrm{SL}_2(\mathbb{Z})$ modulo $p^2$, where $p \geq 5$ is prime. It is classically known that all Eisenstein series of weight at least $4$ are determined modulo $p^2$ by those of weight at most $p^2-p+2$. We prove that up to powers of $E_{p-1}$, each such Eisenstein series is in fact determined modulo $p^2$ by a modular form of weight at most $2p-4$. We also determine $E_2$ modulo $p^2$ in terms of a modular form of weight $p+1$.