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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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A $p$-adic analogue of the conjecture of Birch and Swinnerton-Dyer for modular abelian varieties
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by Jennifer S. Balakrishnan, J. Steffen Müller and William A. Stein PDF
Math. Comp. 85 (2016), 983-1016 Request permission


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
  • MR Author ID: 910890
  • Email:
  • J. Steffen Müller
  • Affiliation: Institut für Mathematik, Carl von Ossietzky Universität Oldenburg, 26111 Oldenburg, Germany
  • MR Author ID: 895560
  • Email:
  • William A. Stein
  • Affiliation: Department of Mathematics, University of Washington, Box 354350, Seattle, Washington 98195
  • MR Author ID: 679996
  • Email:
  • 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.
  • © Copyright 2015 American Mathematical Society
  • Journal: Math. Comp. 85 (2016), 983-1016
  • MSC (2010): Primary 11G40, 11G50, 11G10, 11G18
  • DOI:
  • MathSciNet review: 3434891