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Heegner points and Jochnowitz congruences on Shimura curves


Author: Stefano Vigni
Journal: Proc. Amer. Math. Soc. 142 (2014), 4113-4126
MSC (2010): Primary 11G05, 11G40
DOI: https://doi.org/10.1090/S0002-9939-2014-12188-5
Published electronically: August 14, 2014
MathSciNet review: 3266982
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Abstract: Given an elliptic curve $ E$ over $ \mathbb{Q}$, a suitable imaginary quadratic field $ K$ and a quaternionic Hecke eigenform $ g$ of weight $ 2$ obtained from $ E$ by level raising such that the sign in the functional equation for $ L_K(E,s)$ (respectively, $ L_K(g,1)$) is $ -1$ (respectively, $ +1$), we prove a ``Jochnowitz congruence'' between the algebraic part of $ L'_K(E,1)$ (expressed in terms of Heegner points on Shimura curves) and the algebraic part of $ L_K(g,1)$. This establishes a relation between Zhang's formula of Gross-Zagier type for central derivatives of $ L$-series and his formula of Gross type for special values. Our results extend to the context of Shimura curves attached to division quaternion algebras previous results of Bertolini and Darmon for Heegner points on classical modular curves.


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Stefano Vigni
Affiliation: Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy
Email: vigni@dima.unige.it

DOI: https://doi.org/10.1090/S0002-9939-2014-12188-5
Keywords: Heegner points, Shimura curves, $L$-functions, Jochnowitz congruences
Received by editor(s): May 17, 2012
Received by editor(s) in revised form: February 5, 2013
Published electronically: August 14, 2014
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2014 American Mathematical Society