Abstract
Mock modular forms, which give the theoretical framework for Ramanujan’s enigmatic mock theta functions, play many roles in mathematics. We study their role in the context of modular parameterizations of elliptic curves \(E/\mathbb {Q}\). We show that mock modular forms which arise from Weierstrass ζ-functions encode the central L-values and L-derivatives which occur in the Birch and Swinnerton-Dyer Conjecture. By defining a theta lift using a kernel recently studied by Hövel, we obtain canonical weight 1/2 harmonic Maass forms whose Fourier coefficients encode the vanishing of these values for the quadratic twists of E. We employ results of Bruinier and the third author, which builds on seminal work of Gross, Kohnen, Shimura, Waldspurger, and Zagier. We also obtain p-adic formulas for the corresponding weight 2 newform using the action of the Hecke algebra on the Weierstrass mock modular form.
2010 Mathematics Subject Classification: 11F37; 11G40; 11G05; 11F67
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Acknowledgements
The first author is supported by the DFG Research Unit FOR 1920 “Symmetry, Geometry and Arithmetic”. The second three authors thank the generous support of the National Science Foundation, and the third author also thanks the Asa Griggs Candler Fund. The fourth author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative.
The authors thank Jan Bruinier and Pavel Guerzhoy for helpful discussions. We also thank Stephan Ehlen for his numerical calculations in this paper, his corrections and many fruitful conversations. Furthermore, we are grateful to the referee for their many useful suggestions which improved the exposition of this paper.
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Alfes, C., Griffin, M., Ono, K. et al. Weierstrass mock modular forms and elliptic curves. Res. number theory 1, 24 (2015). https://doi.org/10.1007/s40993-015-0026-2
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DOI: https://doi.org/10.1007/s40993-015-0026-2