Congruences for coefficients of level 2 modular functions with poles at 0

TL;DR

This paper establishes congruences modulo powers of 2 for Fourier coefficients of specific level 2 modular functions with poles at 0, linking the congruences to the binary expansion of the form's order of vanishing.

Contribution

It provides new congruences for Fourier coefficients of level 2 modular functions, addressing a question by Andersen and the first author.

Findings

Congruences involve powers of 2 and binary expansion of vanishing order.

Answers a previously posed question about modular function coefficients.

Enhances understanding of the arithmetic properties of level 2 modular functions.

Abstract

We give congruences modulo powers of 2 for the Fourier coefficients of certain level 2 modular functions with poles only at 0, answering a question posed by Andersen and the first author. The congruences involve a modulus that depends on the binary expansion of the modular form's order of vanishing at $\infty$.