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On Computing the Rank of Elliptic Curves Jeff Achter
 

Summary: On Computing the Rank of Elliptic Curves
Jeff Achter
May 1992
Where man looks up, and proud to claim
His rank within the social frame,
Sees a grand system round him roll,
Himself its centre, sun and soul!
Far from the shocks of Europe; far
From every wild, elliptic star.
-- Thomas Moore, Epistle II.
The theory of elliptic curves is singular in the way it lies at the intersection of so many branches
of mathematics. One can arrive at it by studying the sums of cubes, the theory of doubly-periodic
meromorphic functions, or the Jacobians of varieties. One can use the resulting theory to answer
the original questions, to test more general theorems of algebraic geometry, or to design public-key
cryptosystems.
One of the fundamental arithmetic results is the Mordell-Weil theorem, which says that the
group of K-rational points on an abelian variety, and in particular the group of Q-rational points
on an elliptic curve, is finitely generated. The goal of this paper is to outline the proof of this
theorem for elliptic curves with a rational 2-torsion point; to show how the proof yields a par-
tially effective algorithm for computing the rank of the group; and to discuss the results of this

  

Source: Achter, Jeff - Department of Mathematics, Colorado State University

 

Collections: Environmental Sciences and Ecology; Mathematics