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


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.


Jan Bruinier

Fachbereich Mathematik
Technische Universität Darmstadt
Schlossgartenstrasse 7
D–64289 Darmstadt

Ken Ono

Department of Mathematics
University of Wisconsin
Madison, WI 53706
United States
Department of Mathematics and Computer Science
Emory University
Atlanta, Georgia 30322
United States