Abstract
Let \(E_{/_\mathbb{Q }}\) be an elliptic curve of conductor \(Np\) with \(p\not \mid N\) and let \(f\) be its associated newform of weight \(2\). Denote by \(f_\infty \) the \(p\)-adic Hida family passing though \(f\), and by \(F_\infty \) its \(\varLambda \)-adic Saito–Kurokawa lift. The \(p\)-adic family \(F_\infty \) of Siegel modular forms admits a formal Fourier expansion, from which we can define a family of normalized Fourier coefficients \(\{\widetilde{A}_T(k)\}_T\) indexed by positive definite symmetric half-integral matrices \(T\) of size \(2\times 2\). We relate explicitly certain global points on \(E\) (coming from the theory of Darmon points) with the values of these Fourier coefficients and of their \(p\)-adic derivatives, evaluated at weight \(k=2\).
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Acknowledgments
We would like to thank: H. Kawamura for kindly contributing his expertise on the Saito–Kurokawa lifting, and for immediately pointing out a mistake in a preliminary version of this paper; H. Katsurada for mentioning [12] and thus providing the correct tool the rectify the above mistake; T. Ibukiyama for sending us the preprint [12] before its public distribution; and the Hakuba conference organizers of 2007 and 2008 for giving the second author the chance to learn about explicit formulae for classical automorphic forms from the experts. This work was entirely completed while the second author (and for a short stay, the first author) enjoyed the hospitality of the Max Planck Institute für Mathematik in Bonn. Finally, we would like to thank the referee for his/her careful reading of the manuscript and for valuable suggestions and comments.
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Longo, M., Nicole, MH. The Saito–Kurokawa lifting and Darmon points. Math. Ann. 356, 469–486 (2013). https://doi.org/10.1007/s00208-012-0846-5
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DOI: https://doi.org/10.1007/s00208-012-0846-5