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Author: William A. Stein
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Armand Brumer and Oisin McGuinness


Lacking a full exposition of the computations described in "The Behavior of the Mordell-Weil Group of Elliptic Curves", we give here various odds and ends, mostly unpublished, which describe pieces used.

  • "Positive Rank Examples of the Conjecture of Birch and Swinnerton-Dyer" by McGuinness, TeX version, DVI version. This 1989 unpublished manuscript determines (using an argument of Ogg) the 9 examples of the genus 1 curves of the form X0(p)/w, p a prime, and calculates the quantities appearing in the conjecture of Birch and Swinnerton-Dyer for these curves. The same calculations are carried out for the first rank 2 elliptic curve, which has conductor 389. It is incomplete; the appendixes outlining the period, height, and L-series calculations were not detailed.
  • Slides for a talk given by McGuinness at Rutgers May 11, 1990 are here. (These are mostly TeX-ed versions of those used by Brumer at MSRI.)
  • "The Archimedean Component of the Canonical Height on Elliptic Curves", by Brumer and McGuinness, TeX version, DVI version, examines the question of where on E(R) does the archimedean component of the height attain its minimum. The answer depends on the sign of the discriminant, i.e., on the number of components of E(R). This question was also examined, and answered, in a more general context by Shou-wu Zhang. This is a convenient place to read about the archimedean height. This 1990 manuscript was not finished, or published.

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