On Kudla's Green function for signature (2,2), part I

TL;DR

This paper explores Kudla's conjectures linking arithmetic intersection theory, Eisenstein series, and L-series derivatives, focusing on the product of two modular curves, and proves the modularity of a generating series of modified arithmetic special cycles.

Contribution

It provides a detailed proof of Kudla's conjectures for the case of two modular curves by modifying Kudla's Green function to satisfy Arakelov theory requirements.

Findings

Modified Kudla's Green function satisfies Arakelov theory.

Generated series of arithmetic special cycles is a modular form.

Supports Kudla's conjectures relating geometry and automorphic forms.

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 The aim of this text is to work out the details of an old unpublished draft on the second authors attempt to prove these conjectures for the case of the product of two modular curves. The mayor difficulties in our situation are of analytical nature, therefore this text assembles some material concerning Kudla's Green function associated to this situation. We present a mild modification of this Green function that satisfies the requirements of being a Green function in the sense of Arakelov theory on the natural compactification in addition. Only this allows us to define arithmetic special cycles. We then prove that the generating series of those modified arithmetic special cycles is as predicted by Kudla's conjectures a modular form with values in the first arithmetic Chow group.