Kohnen-Waldspurger formula

introduction

• 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 \(L(f, \chi_D, s)\)

formula

• There is a modular form \(g(z)=\sum b_{E}(n)q^n\) such that if \(\epsilon()=1\),

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

history

• 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

related items

• Shimura correspondence
• Gross-Zagier formula

expositions

• RAMAKRISHNAN, B. "MODULAR FORMS OF HALF-INTEGRAL WEIGHT." THE MATHEMATICS STUDENT: 101. http://indianmathsociety.org.in/mathstudent2012.pdf#page=104
• Unearthing the visions of a master: harmonic Maass forms and number theory
  • chapter 15

articles

• Nicolás Sirolli, Gonzalo Tornaría, An explicit Waldspurger formula for Hilbert modular forms, http://arxiv.org/abs/1603.03753v1
• Sergey Lysenko, Geometric Waldspurger periods II, http://arxiv.org/abs/1308.6531v2
• Wen, Jun. “Orthogonal Periods and Central Values of Rankin-Selberg L-Functions of \(GL_3 {\times} GL_2\).” arXiv:1512.09222 [math], December 31, 2015. http://arxiv.org/abs/1512.09222.
• Liu, Yifeng, Shouwu Zhang, and Wei Zhang. “On \(p\)-Adic Waldspurger Formula.” arXiv:1511.08172 [math], November 25, 2015. http://arxiv.org/abs/1511.08172.
• Cooper, Ian A., Patrick W. Morris, and Nina C. Snaith. “Beyond the Excised Ensemble: Modelling Elliptic Curve L-Functions with Random Matrices.” arXiv:1511.05805 [math-Ph], November 18, 2015. http://arxiv.org/abs/1511.05805.
• Wen, Jun. “Bhargava’s Composition Law and Waldspurger’s Central Value Theorem.” arXiv:1510.07334 [math], October 25, 2015. http://arxiv.org/abs/1510.07334.
• Schwagenscheidt, Markus. ‘Nonvanishing and Central Critical Values of Twisted \(L\)-Functions of Cusp Forms on Average’. arXiv:1502.02492 [math], 9 February 2015. http://arxiv.org/abs/1502.02492.
• [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.