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

By Jan H. Bruinier, Ken Ichi Ono. 2007

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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.

Subfields: Algebra and Number Theory, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3ad3c16a1a5335605319f294302739f8774ab941

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In Algebra and Number Theory

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Citations in the corpus

Fourier transform and the global Gan–Gross–Prasad conjecture for unitary groups (2014, background): “Zhang in 1990’s, completed by Yuan–Zhang–Zhang ([54]) recently, and Kudla–Rapoport–Yang ([37]), Bruinier–Ono ([5]) in various prospectives.”
Unearthing the visions of a master: harmonic Maass forms and number theory (2008, background): “The following lemma 15 , which is a straightforward refinement of a proposition of Bruinier and Funke (see Proposition 3.2 of [63]), shall play a central role throughout this paper.”
Cycle integrals of the j-function and mock modular forms (2011, methodology): “... of our basis hd. This is first done for jm(!,s ) with Re(s) > 1 by applying Proposition 1 to transform the sum of exponential sums in Proposition 7 into a sum of Kloosterman sums, which is then related to the coe$cients of hd(!,s ). The method of using Kloosterman sums in this way was first appl”