Eisenstein series modulo $p^2$
Scott Ahlgren, Michael Hanson, Martin Raum, Olav K. Richter
TL;DR
This paper analyzes congruences for Eisenstein series on ${\rm SL}_2(\mathbb{Z})$ modulo $p^2$ for primes $p\ge 5$, refining the classical description that such series are determined by weights up to $2+p(p-1)$. It shows that, up to powers of $E_{p-1}$, every Eisenstein series with $(p-1) \nmid k$ is determined by a $p$-integral modular form of weight at most $2p-4$, and it gives a description of $E_2$ modulo $p^2$ in terms of a weight $p+1$ form via the relation $E_2 E_{p-1} \equiv f_{p+1} + p E_{p+1}^p$ with $f_{p+1} = \partial E_{p-1}$. The proof combines standard $p$-adic modular-form techniques (Bernoulli numbers, Kummer congruences) with Popa’s derivative relations arising from Eichler–Shimura duality, arranged in an inductive framework that carefully tracks $p$-adic valuations. Consequently, for $2\le k_0 \le p-3$, one has $G_{k_0+(n+1)(p-1)} \equiv E_{p-1}^n f_{k_0+(p-1)} \pmod{p^2}$ with $f_{k_0+(p-1)} \in \mathrm{M}_{k_0+(p-1)}$, and the weights divisible by $p-1$ admit a simple description. These results illuminate the structure of Eisenstein congruences modulo higher powers of $p$ and furnish a framework for understanding related $p$-adic phenomena in modular forms.
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 $p+1$.