Eisenstein series modulo prime powers

If $p\geq 5$ is prime and $k\geq 4$ is an even integer with $(p-1)\nmid k$ we consider the Eisenstein series $G_k$ on $\operatorname{SL}2(\mathbb{Z})$ modulo powers of $p$. It is classically known that for such $k$ we have $G_k\equiv G{k’}\pmod p$ if $k\equiv k’\pmod{p-1}$. Here we obtain a generalization modulo prime powers $p^m$ by giving an expression for $G_k\pmod{p^m}$ 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 $(p-1)\mid k$ and $m<p$.

💡 Research Summary

The paper investigates congruences for Eisenstein series on the full modular group when reduced modulo powers of a prime (p\ge5). For an even integer (k\ge4) with ((p-1)\nmid k) the classical result states that the Eisenstein series (G_k) satisfies (G_k\equiv G_{k’}\pmod p) whenever (k\equiv k’\pmod{p-1}). The authors extend this to arbitrary prime powers (p^m) and also treat the normalized Eisenstein series (E_k) when ((p-1)\mid k).

The main technical tool is a family of coefficients
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