Kohnen-Waldspurger formula

introduction

chapter 15 of Unearthing the visions of a master: harmonic Maass forms and number theory
study central values and derivatives of weight 2 modular L-functions
let $g(z)$ be a Kohnen newform
There is a unique newform, say $f(z)\in S^{\text{new}}_{2k}(N)$, associated to $g(z)$ under Shimura's correspondence.
The coefficients of $g(z)$ determine the central critical values of many of the quadratic twists $L(f, \chi_D, s)$

\[L(E(\Delta),1)=\alpha(\cdots)b_{E}(\Delta)^{2}\]

In the 1980s, Waldspurger [205], and Kohnen and Zagier [135, 136, 137] established that certain half-integral weight modular forms serve as generating functions of a new type.
Using the Shimura correspondence [192], they proved that certain coefficients of half-integral weight cusp forms essentially are square-roots of central values of quadratic twists of modular L-functions.
Ono and Bruinier [67] have generalized this theorem of Waldspurger and Kohnen to prove that the Fourier coefficients of weight 1/2 harmonic Maass forms encode the vanishing and nonvanishing of both the central values and derivatives of quadratic twists of weight 2 modular L-functions.
- Mock modular periods and L-functions

[67] Bruinier, Jan H., and Ken Ono. 2007. “Heegner Divisors, $L$-Functions and Harmonic Weak Maass Forms.” arXiv:0710.0283 [math] (October 1). http://arxiv.org/abs/0710.0283., Annals of Mathematics
[135] W. Kohnen, Newforms of half-integral weight, J. Reine Angew. Math. 333 (1982) 32-72.
[136] W. Kohnen, Fourier coefficients of modular forms of half-integral weight, Math. Ann. 271 (1985), 237-268. http://www.springerlink.com/content/p52527460724p36m/
[137] W. Kohnen and D. Zagier, Values of L-series of modular forms at the center of the critical strip, Invent. Math. 64 (1981), no. 2, pages 175–198.
[205] J.-L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, J. Math. Pures Appl. (9) 60 (1981), no. 4, pages 375–484.