Limits of traces of singular moduli

TL;DR

This paper investigates the limits of traces of singular moduli as points approach rational numbers at the cusps, linking these limits to special values of regularized twisted L-functions and establishing conditions for trace equality.

Contribution

It establishes a connection between boundary limits of modular trace sums and special values of regularized twisted L-functions, including a trace equality criterion for square-free levels.

Findings

Limits of trace sums relate to special values of L-functions.

Regularized L-functions are meromorphic and satisfy functional equations.

Trace equality implies equality of all trace values for square-free levels.

Abstract

Let $f$ and $g$ be weakly holomorphic modular functions on $\Gamma_0(N)$ with the trivial character. For an integer $d$, let $\Tr_d(f)$ denote the modular trace of $f$ of index $d$. Let $r$ be a rational number equivalent to $i\infty$ under the action of $\Gamma_0(4N)$. In this paper, we prove that, when $z$ goes radially to $r$, the limit $Q_{\hat{H}(f)}(r)$ of the sum $H(f)(z) = \sum_{d>0}\Tr_d(f)e^{2\pi idz}$ is a special value of a regularized twisted $L$-function defined by $\Tr_d(f)$ for $d\leq0$. It is proved that the regularized $L$-function is meromorphic on $\mathbb{C}$ and satisfies a certain functional equation. Finally, under the assumption that $N$ is square free, we prove that if $Q_{\hat{H}(f)}(r)=Q_{\hat{H}(g)}(r)$ for all $r$ equivalent to $i \infty$ under the action of $\Gamma_0(4N)$, then $\Tr_d(f)=\Tr_d(g)$ for all integers $d$.