CM values of regularized theta lifts and harmonic weak Maa{\ss} forms of weight one

TL;DR

This paper investigates the values of regularized theta lifts at CM points, expressing them through Fourier coefficients of harmonic weak Maa{ ext} forms, and connects these to algebraic integers and special cycles, confirming conjectures and enhancing previous results.

Contribution

It provides a new explicit formula for CM values of theta lifts in terms of harmonic weak Maa{ ext} forms and proves related conjectures, improving upon prior partial results.

Findings

CM values expressed via Fourier coefficients of harmonic Maa{ ext} forms

Fourier coefficients are logarithms of algebraic integers

Confirmed a conjecture of Duke and Li

Abstract

We study special values of regularized theta lifts at complex multiplication (CM) points. In particular, we show that CM values of Borcherds products can be expressed in terms of finitely many Fourier coefficients of certain harmonic weak Maa{\ss} forms of weight one. As it turns out, these coefficients are logarithms of algebraic integers whose prime ideal factorization is determined by special cycles on an arithmetic curve. Our results imply a conjecture of Duke and Li and give a new proof of the modularity of a certain arithmetic generating series of weight one studied by Kudla, Rapoport and Yang. The results of the paper are much improved in comparison to the 2012 preprint arxiv:1208.2386 which contained partial results in the same direction. Moreover, they are also an improvement of the main result of the authors thesis (CM values of regularized theta lifts, TU Darmstadt, 2013).