On Kudla's Green function for signature (2,2) Part II

TL;DR

This paper advances the understanding of Kudla's conjectures by proving that certain arithmetic generating series related to modular curves are modular forms, and connects their intersection numbers to Eisenstein series derivatives.

Contribution

It provides a detailed proof of Kudla's conjectures for products of modular curves, linking arithmetic intersection theory with Eisenstein series derivatives.

Findings

Generated series are modular forms with values in arithmetic Chow groups.

Arithmetic intersection numbers match predictions and relate to Eisenstein series derivatives.

Green function integrals over modular curves align with conjectured formulas.

Abstract

Around 2000 Kudla presented conjectures about deep relations between arithmetic intersection theory, Eisenstein series and their derivatives, and special values of Rankin $L-$series. The aim of this text is to work out the details of an old unpublished draft on the second author's attempt to prove these conjectures for the case of the product of two modular curves. In part one we proved that the generating series of certain modified arithmetic special cycles is as predicted by Kudla's conjectures a modular form with values in the first arithmetic Chow group. Here we pair this generating series with the square of the first arithmetic Chern class of the line bundle of modular forms. Up to previously known Faltings heights of Hecke correspondences only integrals of the Green functions $\Xi (m)$ over $X$ had to be computed. The resulting arithmetic intersection numbers turn out to be as predicted by Kudla to be strongly related to the Fourier coefficients of the derivative of the classical real analytic Eisenstein series $E_2(\tau, s)$.