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

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.

42 pages, tables added in Section 8.3, references updated

References in corpus (1)

• Mock Theta Functions

Cited by in corpus (7)

• Twisted Traces of CM values of Harmonic Weak Maass Forms
• Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products
• On a theta lift related to the Shintani lift
• Borcherds lifts of harmonic Maass forms and modular integrals
• Congruences for Ramanujan's f and ω functions via generalized Borcherds products
• Period Relations, Jacobi Forms and Eichler Integral
• Derivatives of L-functions