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.

SO(3)–Donaldson invariants of ℂP2and mock theta functions

“…Zwegers's thesis has sparked a flurry of recent activity involving such Maass forms. Indeed, harmonic Maass forms are now known to play a central role in the study of Ramanujan's mock theta functions, as well as other important mathematical topics: Borcherds products, derivatives of modular L-functions, GrossZagier formulas and Faltings heights of CM cycles, partitions,and traces of singular moduli (see Bringmann and Ono [5;6], Bringmann, Ono and Rhoades [7], Bruinier [8], Bruinier and Funke [9], Bruinier and Ono [10], Bruinier and Yang [12], Ono [40], Zagier [55] and Zwegers [56]). …”

Arithmetic Properties of Certain Level One Mock Modular Forms

“…We will also find that the function M f k (z), the holomorphic projection of R f k (z), has coefficients c + f k (n), and is given by The theory of harmonic weak Maass forms and mock theta functions is of great current interest. See, for example, the works [3], [4], [5], [6], [7], [8], [9], [13]. Let λ ∈ {1/2, 3/2}.…”

Theta lifts and local Maass forms

“…Recent interest in harmonic weak Maass forms began with their systematic treatment by Bruinier and Funke [13]. Following their appearance in the theory of mock theta functions due to Zwegers [40], it has been shown that harmonic weak Maass forms have applications ranging from partition theory (for example [2,4,6,9,11]) and Zagier's duality [39] relating "modular objects" of different weights (for example [10]) to derivatives of L-functions (for example [14,15]). They also arise in mathematical physics, as recently evidenced in Eguchi et al [16] investigation of moonshine for the largest Mathieu group M 24 .…”