Congruent numbers and Heegner points
By Ye Tian. 2012
75 citations recorded by the source corpus.
A positive integer is called a congruent number if it is the area of a rightangled triangle, all of whose sides have rational length. The problem of determining which positive integers are congruent is buried in antiquity (see Chapter 9 of Dickson [6]), with it long being known that the numbers 5, 6, and 7 are congruent. Fermat proved that 1 is not a congruent number, and similar arguments show that also 2 and 3 are not congruent numbers. No algorithm has ever been proven for infallibly deciding whether a given integer n ≥ 1 is congruent. The reason for this is that it can easily be seen that an integer n ≥ 1 is congruent if and only if there exists a point (x, y), with x and y rational numbers and y = 0, on the elliptic curve ny2 = x3− x. Moreover, assuming n to be square free, a classical calculation of root numbers shows that the complex L-function of this curve has zero of odd order at the center of its critical strip precisely when n lies in one of the residue classes of 5, 6, or 7 modulo 8. Thus, in particular, the unproven conjecture of Birch and Swinnerton-Dyer predicts that every positive integer lying in the residue classes of 5, 6, and 7 modulo 8 should be a congruent number. The aim of this paper is to prove the following partial results in this direction.
Subfields: Algebra and Number Theory, Geometry and Topology, Theoretical Computer Science
View Paper: https://www.semanticscholar.org/paper/99c9700b2bf5e02b541cc2845dea1ae6fda2601e
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In Algebra and Number Theory
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Citations in the corpus
- Selmer groups and the indivisibility of Heegner points (2014, unknown): “Tian for the congruent number elliptic curves [39, 40].”
- Automorphic period and the central value of Rankin--Selberg L-function (2012, background): “ [46],[47],[44]) play an important role in the spectacular development in application to the Birch and Swinnerton-Dyer conjecture for elliptic curves in the past thirty years. More recently, Y. Tian ([39]) applies both formulae together to a classical Diophantine question and proves the infinitudes o”