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Mathematics of Computation

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A $ p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties

Authors: Jennifer S. Balakrishnan, J. Steffen Müller and William A. Stein
Journal: Math. Comp. 85 (2016), 983-1016
MSC (2010): Primary 11G40, 11G50, 11G10, 11G18
Published electronically: August 12, 2015
MathSciNet review: 3434891
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Abstract: Mazur, Tate, and Teitelbaum gave a $ p$-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the $ p$-adic $ L$-function of a modular abelian variety and showing that it satisfies the appropriate interpolation property. This relies on a careful normalization of the $ p$-adic $ L$-function, which we achieve by a comparison of periods. Our generalization agrees with the conjecture of Mazur, Tate, and Teitelbaum in dimension 1 and the classical Birch and Swinnerton-Dyer conjecture formulated by Tate in rank 0. We describe the theoretical techniques used to formulate the conjecture and give numerical evidence supporting the conjecture in the case when the modular abelian variety is of dimension 2.

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Additional Information

Jennifer S. Balakrishnan
Affiliation: Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom

J. Steffen Müller
Affiliation: Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany

William A. Stein
Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195

Received by editor(s): September 19, 2014
Published electronically: August 12, 2015
Additional Notes: The first author was supported by NSF grant DMS-1103831.
The second author was supported by DFG grants STO 299/5-1 and KU 2359/2-1.
The third author was supported by NSF Grants DMS-1161226 and DMS-1147802.
Article copyright: © Copyright 2015 American Mathematical Society