Petersson inner products of weight one modular forms

TL;DR

This paper investigates regularized Petersson products between theta series and weight 1 modular forms, revealing their connection to algebraic numbers in ring class fields and providing explicit factorization formulas.

Contribution

It provides an explicit factorization formula for algebraic numbers arising from Petersson products of theta series and weight 1 modular forms, extending previous conjectures.

Findings

Petersson products relate to algebraic numbers in ring class fields.

Explicit factorization formulas are derived for these algebraic numbers.

The work supports conjectures linking modular forms and algebraic number theory.

Abstract

In this paper we study regularized Petersson products between a holomorphic theta series associated to a positive definite binary quadratic form and a weakly holomorphic weight 1 modular form with integral Fourier coefficients. In our recent work \cite{Via CM Green} motivated by the conjecture of B. Gross and D. Zagier on the CM values of higher Green's functions we have discovered that such a Petersson product is equal to the logarithm of a certain algebraic number lying in a ring class field associated to the binary quadratic form. The main result of the present paper is the explicit factorization formula for the obtained algebraic number.