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The one-dimensional case: elliptic curves

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Conjectures in Arithmetic Algebraic Geometry

Part of the book series: Aspects of Mathematics ((ASMA,volume 18))

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Abstract

The next class of objects we consider concerns one-dimensional varieties over the rational numbers, i.e. curves over ℚ. The simplest examples are the projective line ℝ and conics over ℚ. From the point of view of algebraic geometry these are equivalent objects and they are well understood. For our purposes we note that they are curves of genus zero and contain an infinite number of rational points (if any). More interesting are the elliptic curves over ℚ. These have been the subject of deep study from various points of view, algebraic, arithmetic and geometric. From the arithmetic point of view they give rise to some of the most intricate conjectures, the Birch & Swinnerton-Dyer Conjectures, which can be interpreted as the one-dimensional counterpart of Dedekind’s Class Number Formula. Also, more recently, a remarkable relation was found between elliptic curves and Fermat’s Last Theorem.

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Authors and Affiliations

  1. KMA, NL-4800 RG, Breda, The Netherlands

    Wilfred W. J. Hulsbergen

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  1. Wilfred W. J. Hulsbergen
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© 1994 Springer Fachmedien Wiesbaden

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Hulsbergen, W.W.J. (1994). The one-dimensional case: elliptic curves. In: Conjectures in Arithmetic Algebraic Geometry. Aspects of Mathematics, vol 18. Vieweg+Teubner Verlag, Wiesbaden. https://doi.org/10.1007/978-3-663-09505-7_3

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  • DOI: https://doi.org/10.1007/978-3-663-09505-7_3

  • Publisher Name: Vieweg+Teubner Verlag, Wiesbaden

  • Print ISBN: 978-3-663-09507-1

  • Online ISBN: 978-3-663-09505-7

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