On the non-existence of simple congruences for quotients of Eisenstein series

TL;DR

This paper proves that for most ratios of Eisenstein series, there are no simple congruences modulo prime numbers greater than or equal to 13, extending previous findings on congruences of modular forms.

Contribution

It establishes a general theorem showing the non-existence of simple congruences for a broad class of Eisenstein series quotients, except in one specific case.

Findings

Most ratios of Eisenstein series do not satisfy simple congruences modulo p for p >= 13.

A general non-existence theorem is proved for congruences in products of Eisenstein series.

One specific case of congruence remains an exception to the general non-existence result.

Abstract

A recent article of Berndt and Yee found congruences modulo 3^k for certain ratios of Eisenstein series. For all but one of these, we show there are no simple congruences a(pn+c) = 0 modulo p when p>= 13 is prime. This follows from a more general theorem on the non-existence of congruences in (E_2^r)(E_4^s)(E_6^t) where r is non-negative and r,s,t are integers.