Lacunary recurrences and 2-adic properties of Eisenstein series

TL;DR

This paper investigates the 2-adic properties of Eisenstein series coefficients, proving a conjecture relating minimal 2-adic valuation to binary expansion using lacunary recurrences.

Contribution

It provides a precise formula for the minimal 2-adic valuation of Eisenstein series coefficients, confirming a recent conjecture.

Findings

Exact formula for minimal 2-adic valuation in terms of binary expansion

Proof of a recent conjecture on Eisenstein series coefficients

Use of lacunary recurrences in the proof

Abstract

We study the rational coefficients that arise when the Eisenstein series $G_k$ is expressed as a polynomial in $G_4$ and $G_6$. We prove a recent conjecture giving an exact formula for the minimal 2-adic valuation of these coefficients in terms of the binary expansion of the weight. The proof uses lacunary recurrences for Eisenstein series.