Derivatives of Eisenstein series of weight 2 and intersections of modular correspondences

TL;DR

This paper derives formulas for derivatives of Siegel Eisenstein series and explores their connections to modular correspondences and intersection theory, providing new insights into Fourier coefficients and arithmetic applications.

Contribution

It introduces explicit formulas for derivatives of Siegel Eisenstein series and links them to intersection numbers of modular correspondences, advancing understanding of their arithmetic significance.

Findings

Formulas for derivatives of Siegel series and Eisenstein series.

Connection between Fourier coefficients and intersections of modular correspondences.

Applications to representation numbers of symmetric matrices.

Abstract

We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight g/2 and genus g. When g=4, the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices.