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

Pavel Guerzhoy

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.

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

Abstract

Traces of singular moduli were introduced and studied by Zagier in 1998. Being simultaneously the (traces of) values of a modular function (-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 modular forms under repeated action of -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 congruences between the traces of singular moduli and class numbers of quadratic fields in the case when the twisted central special value of the -function associated with the elliptic curve of conductor vanishes.
Paper Structure (3 sections, 5 theorems, 61 equations)

This paper contains 3 sections, 5 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Modified Zagier - Eisentstein series $\tilde{{\mathcal{H}}}$
  3. Proofs of the Theorems

Key Result

Theorem 1

Assume that $-d$ is fundamental and $\left( \frac{-d}{11} \right) = -1$. We have that $L(E^{-d},1) = 0$ if and only if where the limit is taken $p$-adically with $p=11$. In particular (note that $B(11^{2n}d) \equiv 2B(d) \pmod {11}$ for $n>0$), we have the following analog of Mazur_5.

Theorems & Definitions (12)

  • Theorem 1
  • Remark
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • proof
  • Remark
  • Proposition 2
  • proof
  • proof : Proof of Theorem \ref{['thm_general']}
  • ...and 2 more