Elliptic curve with discriminant D has at most O(|D|^(0.2007…+e)) integer points

Number theory 1 Minute

You need to know: Notations: {\mathbb Z} for the set of integers, {\mathbb Q} for the set of rational numbers.

Background: Consider equation y^2 = x^3+ax+b, where a,b \in {\mathbb Z} are such that \Delta_{a,b}=-16(4a^3+27b^2)\neq 0. A theorem of Siegel implies that this equation has only finitely many integer solutions x,y \in {\mathbb Z}. Denote N_{a,b} the number of such solutions.

The Theorem: On 11th May 2004, Harald Helfgott and Akshay Venkatesh submitted to arxiv a paper in which they proved that for every sufficiently small \epsilon>0 there is a constant C_\epsilon<\infty, such that N_{a,b} \leq C_\epsilon |\Delta_{a,b}|^{\beta+\epsilon}, where \beta=\frac{4\sqrt{3}\log(2+\sqrt{3})-6\log 2 -3\log 3}{12\log 2}=0.2007....

Short context: The equation y^2 = x^3+ax+b, with a,b \in {\mathbb Z} and \Delta_{a,b}=-16(4a^3+27b^2)\neq 0 is called non-singular elliptic curve over {\mathbb Q} in Weierstrass form with integer coefficients. Studying integer and rational points on such curve (that is, integer and rational solutions of the equation) is one of the important research directions in number theory. In 1992, Schmidt proved that N_{a,b} \leq C_\epsilon |\Delta_{a,b}|^{1/2+\epsilon}. The Theorem improves this bound significantly. The bound in the Theorem remained unimproved for over decade, until Bhargava et.al. proved that N_{a,b} \leq C_\epsilon |\Delta_{a,b}|^{0.1117...+\epsilon}.

Links: Free arxiv version of the original paper is here, journal version is here.

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Published by Bogdan Grechuk

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