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CM cycles on Shimura curves, and p-adic L-functions

Part of: Arithmetic algebraic geometry Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  15 May 2012

Marc Masdeu
Marc Masdeu*
Affiliation:
Columbia University, Rm 415, MC 4441, 2990 Broadway, New York, NY 10027, USA (email: [email protected])
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Abstract

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Let f be a modular form of weight k≥2 and level N, let K be a quadratic imaginary field and assume that there is a prime p exactly dividing N. Under certain arithmetic conditions on the level N and the field K, one can attach to this data a p-adic L-function Lp (f,K,s) , as done by Bertolini–Darmon–Iovita–Spieß in [Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting, Amer. J. Math. 124 (2002), 411–449]. In the case of p being inert in K, this analytic function of a p-adic variable s vanishes in the critical range s=1,…,k−1 , and one may be interested in the values of its derivative in this range. We construct, for k≥4 , a Chow motive endowed with a distinguished collection of algebraic cycles which encode these values, via the p-adic Abel–Jacobi map. Our main result generalizes the result obtained by Iovita and Spieß in [Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math. 154 (2003), 333–384], which gives a similar formula for the central value s=k/2 . Even in this case our construction is different from the one found by Iovita and Spieß.

Keywords

CM cyclesShimura curvep-adic integrationanti-cyclotomic p-adic L-function

MSC classification

Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture 11F11: Holomorphic modular forms of integral weight 11G18: Arithmetic aspects of modular and Shimura varieties
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 4 , July 2012 , pp. 1003 - 1032
Copyright
Copyright © Foundation Compositio Mathematica 2012

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