Distribution of Rational Points on Toric Varieties – A Multi-Height Approach

Abstract: Manin's conjecture was verified by Victor Batyrev and Yuri Tschinkel for toric varieties. Emmanuel Peyre has proposed two notions, "freeness" and "all the heights" approach to delete accumulating subvarieties in "Libert\'e et accumulation" and "Beyond heights: slopes and distribution of rational points". Based on the all the heights approach, in this talk, we will explain a multi-height variant of the Batyrev-Tschinkel theorem where one considers working at {\em height boxes}, instead of a single height function, as a way to get rid of accumulating subvarieties. This is our main result: Let $X$ be an arbitrary toric variety over a number field $F$, and let $H_i$, $1 \leq i \leq r$, be height functions associated to the generators of the cone of effective divisors of $X$. Fix positive real numbers $a_i$, $1 \leq i \leq r$. Then the number of rational points $P \in X(F)$ such that for each $i$, $H_i(P) \leq B^{a_i}$ as $B$ gets large is equal to $C B^{a_1 + \dots + a_r} + O(B^{a_1 + \dots + a_r-\epsilon})$ for an $\epsilon >0$. Our result is a first example of a large family of varieties along the lines of Peyre's idea.

algebraic geometrycombinatoricsdynamical systemsgeneral topologynumber theory

Audience: researchers in the topic

ZORP (zoom on rational points)

Series comments: 2 talks on a Friday, roughly once per month.

Online coffee break in between.