Convergence of the Zagier type series for the Cauchy kernel

TL;DR

This paper extends Zagier's series for the Cauchy kernel associated with Hecke operators through analytic continuation and explores the differential form of the logarithm of the difference of two j-invariant values.

Contribution

It constructs an analytic continuation of a formally divergent series related to the Cauchy kernel for Hecke operators and derives an expression for the differential form of the logarithm of the j-invariant difference.

Findings

Extended the convergence of Zagier's series via analytic continuation.

Derived an explicit expression for the differential form of |j(z_1)-j(z_2)|.

Provided insights into the kernel functions for Hecke operators.

Abstract

In 1975 prof. Don Zagier derived a preliminary formula for the trace of the Hecke operators acting on the space of cusp forms (\cite{5}, \cite{6}). Actually, it is an expression in terms of an integral over a fundamental domain of $SL_2(\mathbb{Z}).$ His theorem tells us that if $f$ is a cusp form of weight $k$, then we can identify the Peterson scalar product of $f$ and a certain series $\omega_m(z_1,\bar{z_2}, k)$ with the action of the Hecke operator $T(m)$ on the function $f$, up to a constant that depends only on $k$ and $m$. It follows that $\omega_m(z_1,\bar{z_2}, k)$ is kind of "kernel function" for the operator $T(m)$. Don Zagier proved this theorem using the Rankin-Selberg method. Other evidence was proposed by prof. A. Levin. He suggested to construct a Cauchy kernel. Formally, the Cauchy kernel expressed by the series, which doesn't converge absolutely. The main purpose of this paper is to extend this series to the edge of convergence by analytic continuation. The second part of the paper is devoted to getting an expression for differential form of logarithm of difference of two $j$-invariant values $|j(z_1)-j(z_2)|$.