This is the third post in a series begun here and continued here. The series presents one striking and comparatively accessible theorem associated with each of the 2026 Fields Medalists and the 2026 Abacus Medalist. This post concerns Fields Medalist Jacob Tsimerman and a basic quantitative question about elliptic curves: how many integer points can such a curve have?
Elliptic curves have appeared several times on this blog, for example here and here. In this post, an elliptic curve is given by an integral short Weierstrass equation where and The condition ensures that the curve is nonsingular. We also assume that the model is minimal in the following elementary sense: there is no prime for which both and . Its naive height is
We are interested in the number of integer points on . A fundamental theorem of Siegel (Siegel 1929) states that is finite. Siegel’s theorem, however, does not by itself provide a uniform bound for in terms of a simple invariant such as the discriminant.
In 1992, Schmidt (Schmidt 1992) proved that, for every , where depends only on . Helfgott and Venkatesh (Helfgott and Venkatesh 2006) substantially improved this result in 2006, showing that where Thus the number of integer points grows much more slowly than the square root of the discriminant.
This estimate remained the strongest known general bound for more than a decade. In 2020, Bhargava, Shankar, Taniguchi, Thorne, Tsimerman, and Zhao (Bhargava et al. 2020) nearly halved the exponent.
In other words, although an elliptic curve may have many integer points, their number is severely constrained by the arithmetic complexity of the curve. The theorem gives a uniform bound valid for every elliptic curve and improves the previously best exponent from approximately to approximately .