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Studying the Birch and Swinnerton-Dyer conjecture for modular abelian varieties using Magma

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Discovering Mathematics with Magma

Part of the book series: Algorithms and Computation in Mathematics ((AACIM,volume 19))

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Abstract

In this paper we describe the Birch and Swinnerton-Dyer conjecture in the case of modular abelian varieties and how to use Magma to do computations with some of the quantities that appear in the conjecture. We assume the reader has some experience with algebraic varieties and number theory, but do not assume the reader has proficiency working with elliptic curves, abelian varieties, modular forms, or modular symbols. The computations give evidence for the Birch and Swinnerton- Dyer conjecture and increase our explicit understanding of modular abelian varieties.

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, MA, USA

    William Stein

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  1. William Stein
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Editors and Affiliations

  1. Vakgroep Wiskunde, Katholieke Univ. Nijmegen, 9010, 6500, Nijmegen, GL, Netherlands

    Wieb Bosma Dr.

  2. Department Pure Mathematics, University Sydney, F07, 2006, Sydney, Australia

    John Cannon Dr.

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Stein, W. (2006). Studying the Birch and Swinnerton-Dyer conjecture for modular abelian varieties using Magma. In: Bosma, W., Cannon, J. (eds) Discovering Mathematics with Magma. Algorithms and Computation in Mathematics, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-37634-7_4

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