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Introduction to the Conjectures of Birch and Swinnerton-Dyer

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The Computational and Theoretical Aspects of Elliptic Curves

Part of the book series: Mathematical Lectures from Peking University ((MLPKU))

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Abstract

The aim of these lectures is to introduce the Birch and Swinnerton-Dyer conjectures in its entirety. One part of these conjectures predicts the equality of two different ‘ranks’ associated to an elliptic curve defined over a number field. These are the so called algebraic and analytic ranks. The other part of the conjectures is an exact formula expressing the leading coefficient of a certain power series associated to the elliptic curve in terms of various important and mysterious arithmetic invariants. The approach we shall take is to define and provide a brief introduction to these arithmetic invariants, thereby providing a compact introduction to this conjecture.

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Author information

Authors and Affiliations

  1. IIT Kanpur, Kanpur, India

    Sudhanshu Shekhar

  2. University of British Columbia, Vancouver, BC, Canada

    R. Sujatha

Authors
  1. Sudhanshu Shekhar
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  2. R. Sujatha
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Correspondence to Sudhanshu Shekhar .

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Editors and Affiliations

  1. Capital Normal University, Beijing, China

    Zhibin Liang

  2. Department of Mathematical Sciences, Indian Institute of Science Education and Research Mohali, Sahibzada Ajit Singh Nagar, Punjab, India

    Chandrakant Aribam

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Shekhar, S., Sujatha, R. (2019). Introduction to the Conjectures of Birch and Swinnerton-Dyer. In: Liang, Z., Aribam, C. (eds) The Computational and Theoretical Aspects of Elliptic Curves. Mathematical Lectures from Peking University. Springer, Singapore. https://doi.org/10.1007/978-981-13-6664-2_1

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