Congruences for traces of singular moduli and Hurwitz - Kronecker class numbers

TL;DR

This paper offers a new perspective on congruences for traces of singular moduli by analyzing modular forms of weight 3/2 under the U-operator, leading to general results and specific applications, especially for small primes.

Contribution

It introduces an alternative approach to congruences of singular moduli using modular forms and the U-operator, unifying and extending previous results.

Findings

Derived a general congruence result encompassing earlier findings

Explained numerical observations for small primes

Established congruences between singular moduli traces and class numbers modulo 11

Abstract

Traces of singular moduli were introduced and studied by Zagier in 1998. Being simultaneously the (traces of) values of a modular function ($j$-invariant) and Fourier coefficients of modular forms - which constitutes Zagier's duality - these integers are quite interesting. Since then, a substantial amount of research was devoted to various properties of these numbers, congruences in particular. We present an alternative point of view on these congruences, specifically, we view them as congruences between certain weight $3/2$ modular forms under repeated action of $U$-operator. That allows us to obtain a general result which includes some previously known results as special cases. Our approach is especially effective when the prime modulus is relatively small. In these cases, we obtain explanations for certain numerical observations and quantification of some previously known qualitative results. As an application, we obtain modulo $11$ congruences between the traces of singular moduli and class numbers of quadratic fields in the case when the twisted central special value of the $L$-function associated with the elliptic curve of conductor $11$ vanishes.