Heegner divisors, $L$-functions and harmonic weak Maass forms

TL;DR

This paper demonstrates that harmonic weak Maass forms can generate central values and derivatives of quadratic twists of modular L-functions, extending classical results and connecting to various aspects of number theory.

Contribution

It generalizes the Borcherds lift to harmonic weak Maass forms and links these forms to the values and derivatives of quadratic twist L-functions.

Findings

Harmonic weak Maass forms serve as generating functions for L-function values.

Construction of differentials of the third kind with twisted Heegner divisors.

Connections established between Maass forms, periods, Fourier coefficients, and Jacobian points.

Abstract

Recent works, mostly related to Ramanujan's mock theta functions, make use of the fact that harmonic weak Maass forms can be combinatorial generating functions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove that such forms also serve as "generating functions" for central values and derivatives of quadratic twists of weight 2 modular $L$-functions. To obtain these results, we construct differentials of the third kind with twisted Heegner divisor by suitably generalizing the Borcherds lift to harmonic weak Maass forms. The connection with periods, Fourier coefficients, derivatives of $L$-functions, and points in the Jacobian of modular curves is obtained by analyzing the properties of these differentials using works of Scholl, Waldschmidt, and Gross and Zagier.