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Darmon cycles and the Kohnen-Shintani lifting


Author: Guhan Venkat
Journal: Trans. Amer. Math. Soc. 370 (2018), 4059-4087
MSC (2010): Primary 11F67, 11F37, 11F85, 11G40
DOI: https://doi.org/10.1090/tran/7077
Published electronically: February 28, 2018
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Abstract: Let $ \mathbf {f}(q)$ be a Coleman family of cusp forms of tame level $ N$. Let $ k_{0}$ be the classical weight at which the specialization of $ \mathbf {f}(q)$ is new. By the Kohnen-Shintani correspondence, we associate to every even classical weight $ k$, a half-integral weight form (for $ k \neq k_{0}$) $ g_{k} = \sum \limits _{D > 0} c(D, k)q^D \in S_{\frac {k+1}{2}}(\Gamma _{0}(4N))$ and $ g_{k_{0}} = \sum \limits _{D > 0} c(D, k)q^D \in S_{\frac {k+1}{2}}(\Gamma _{0}(4Np))$.

We first prove that the Fourier coefficients $ c(D, k)$ for $ k \in 2\mathbb{Z}_{> 0}$ can be interpolated by a $ p$-adic analytic function $ \tilde {c}(D, \kappa )$ with $ \kappa $ varying in a neighbourhood of $ k_{0}$ in the $ p$-adic weight space. For discriminants $ D$ such that $ \tilde {c}(D, k_{0}) = 0$, which we call Type II, we show that $ \frac {d}{d\kappa }[\widetilde {c}(D, \kappa )]_{k=k_{0}}$ is related to certain algebraic cycles associated to the motive $ \mathcal {M}_{k_{0}}$ attached to the space of cusp forms of weight $ S_{k_{0}}(\Gamma _{0}(Np))$. These algebraic cycles appear in the theory of Darmon cycles.


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Additional Information

Guhan Venkat
Affiliation: Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom
Address at time of publication: Département de Mathématiques et de Statistique Université Laval, Pavillion Alexandre-Vachon 1045 Avenue de la Médecine Québec, QC G1V 0A6, Canada
Email: guhan.venkat@gmail.com

DOI: https://doi.org/10.1090/tran/7077
Keywords: Darmon cycles, $p$-adic Abel-Jacobi map, half-integral weight modular forms, $p$-adic L-function, Coleman families
Received by editor(s): June 21, 2016
Received by editor(s) in revised form: September 15, 2016
Published electronically: February 28, 2018
Additional Notes: The author would like to thank Denis Benois, Adrian Iovita and Matteo Longo for their constant guidance throughout this project. The author would also like to thank the anonymous referee for suggesting improvements to an earlier draft. The work grew out of the author’s PhD thesis completed at the Université de Bordeaux and Università di degli Studi di Padova and was supported by the ALGANT-Doc commission.
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