Generalised Integrality and Applications to Number Theory

In this proposal semi-integral points refer to notions of rational points on algebraic varieties that satisfy an integrality condition with respect to a weighted boundary divisor. They were first introduced by Campana and by Darmon.

Campana points have recently risen to the attention of the number theory community thanks to a Manin type conjecture in the recent work of Pieropan, Smeets, Tanimoto and Várilly-Alvarado.

Semi-integral points provide both an intermediate notion and a generalisation of the notions of rational and integral points, thereby unifying the two theories.

This proposal concerns the existence of semi-integral points and the density of orbifold pairs in general families having semi-integral points.

The aims of this proposal are to determine good upper bounds for the density of orbifold pairs in a general family that have semi-integral points (WP1) and to compute obstructions to the existence of semi-integral points (and hence to integral points) in key examples corresponding to long-lasting questions in number theory (WP2).The approach will combine a variety of techniques from analytic number theory, algebraic geometry and arithmetic statistics.

For (WP1), the experienced researcher and the supervisor will develop a criterion to detect local semi-integral points together with a sieve method to estimate the number of everywhere locally soluble varieties in the family.

For (WP2), the research team will develop a Brauer-Manin obstruction theory for semi-integral points to compute failures of the integral Hasse principle in fundamental examples and handle classical Diophantine problems such as the existence of integral points on diagonal cubic surfaces and the non-existence of consecutive powerful numbers.