Shimura Curves and Special Values of p-adic L-functions

We construct "generalized Heegner cycles" on a variety fibered over a Shimura curve, defined over a number field. We show that their images under the p-adic Abel-Jacobi map coincide with the values (outside the range of interpolation) of a p-adic L-function L-p(f, chi) which interpolates special values of the Rankin-Selberg convolution of a fixed newform f and a theta-series theta(chi) attached to an unramified Hecke character of an imaginary quadratic field K. This generalizes previous work of Bertolini, Darmon, and Prasanna, which demonstrated a similar result in the case of modular curves. Our main tool is the theory of Serre-Tate coordinates, which yields p-adic expansions of modular forms at CM points, replacing the role of q-expansions in computations on modular curves.


Published in:
International Mathematics Research Notices, 12, 4177-4241
Year:
2015
Publisher:
Oxford, Oxford Univ Press
ISSN:
1073-7928
Laboratories:




 Record created 2015-09-28, last modified 2018-03-17


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