Heegner divisors, $L$-functions and harmonic weak Maass forms - Mathematics > Number TheoryReport as inadecuate




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Abstract: Recent works, mostly related to Ramanujan-s mock theta functions, make use ofthe fact that harmonic weak Maass forms can be combinatorial generatingfunctions. Generalizing works of Waldspurger, Kohnen and Zagier, we prove thatsuch forms also serve as -generating functions- for central values andderivatives of quadratic twists of weight 2 modular $L$-functions. To obtainthese results, we construct differentials of the third kind with twistedHeegner divisor by suitably generalizing the Borcherds lift to harmonic weakMaass forms. The connection with periods, Fourier coefficients, derivatives of$L$-functions, and points in the Jacobian of modular curves is obtained byanalyzing the properties of these differentials using works of Scholl,Waldschmidt, and Gross and Zagier.



Author: Jan H. Bruinier, Ken Ono

Source: https://arxiv.org/







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