Geodesic
Heegner divisors, $L$-functions and harmonic weak Maass forms
Jan Bruinier1 ; Ken Ono2
1 Fachbereich Mathematik<br/>Technische Universität Darmstadt<br/>Schlossgartenstrasse 7<br/>D–64289 Darmstadt<br/>Germany
2 Department of Mathematics<br/>University of Wisconsin<br/>Madison, WI 53706<br/>United States<br/>and <br/>Department of Mathematics and Computer Science<br/>Emory University<br/>Atlanta, Georgia 30322<br/>United States
Annals of mathematics, Tome 172 (2010) no. 3, pp. 2135-2181.

Voir la notice de l'article provenant de la source Annals of Mathematics website

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.
DOI : 10.4007/annals.2010.172.2135

Jan Bruinier 1 ; Ken Ono 2

1 Fachbereich Mathematik<br/>Technische Universität Darmstadt<br/>Schlossgartenstrasse 7<br/>D–64289 Darmstadt<br/>Germany
2 Department of Mathematics<br/>University of Wisconsin<br/>Madison, WI 53706<br/>United States<br/>and <br/>Department of Mathematics and Computer Science<br/>Emory University<br/>Atlanta, Georgia 30322<br/>United States 
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Jan Bruinier; Ken Ono. Heegner divisors, $L$-functions and harmonic weak Maass forms. Annals of mathematics, Tome 172 (2010) no. 3, pp. 2135-2181. doi : 10.4007/annals.2010.172.2135. http://geodesic.mathdoc.fr/articles/10.4007/annals.2010.172.2135/

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