Riemannian structures and point-counting

TL;DR

This paper links geometric properties of Riemannian metrics on complex varieties to the distribution of rational points over finite fields, establishing conditions under which point counts grow infinitely.

Contribution

It introduces a new estimate connecting Riemannian curvature and diameter with finite field point counts for complex varieties and their characteristic zero analogues.

Findings

Point counts over finite fields grow unbounded when Riemannian curvature and diameter satisfy certain conditions.

Established a general theorem estimating finite field points based on geometric Riemannian data.

Proved a characteristic zero analogue relating algebraic endomorphisms to Riemannian structures.

Abstract

Suppose $(X_n)$ is a sequence of positive-dimensional smooth projective complete intersections over $\mathbb{F}_q$ with dimensions bounded from above and with characteristic zero lifts $(\tilde{X}_n)$ to smooth projective geometrically connected varieties. Suppose each complex variety $\tilde{X}^{an}_n$ has (underlying real manifold equipped with) a Riemannian metric $g_n$ of sectional curvature at least $-\kappa_n^2$, $\kappa_n\geq 0$, and diameter at most $D_n$. In this note, we show that if \[\lim_{n\rightarrow\infty}\min\{d:X_n(\mathbb{F}_{q^d})\neq\emptyset\}=+\infty,\] then $\kappa_nD_n\rightarrow+\infty$. We deduce this theorem by proving a more general theorem estimating the number of points over finite fields of the above varieties in terms of the sectional curvature and diameter of Riemannian structures on the analytification of characteristic zero lifts. We also prove a version of this estimate for non-projective varieties equipped with complete metrics of non-negative sectional curvature. Finally, we prove a characteristic zero analogue of the above estimate by relating fixed-points of algebraic endomorphisms of smooth projective complex complete intersections to Riemannian structures on them.