Is the geometry of a variety determined by the counts of rational points?

2012-10-04T18:16:42Z

In Diophantine Geometry: An introduction, Hindry and Silverman write "Geometry Determines Arithmetic" (pg. 2) and "Geometry Governs Arithmetic" (pg. 474).

On pg. 211 of the same book, the authors state the following theorem:

Let $k$ be a number field, let $C/k$ be a smooth curve of genus $g$, and assume that $C(k)$ is not empty. Then there are constants $a$ and $b$, which depend on $C/k$ and on the height used in the counting functions, such that

$N(C(k, T)) \sim aT^b$ if $g = 0$, here $a,b >0$

$N(C(k, T)) \sim a(log(T))^b$ if $g = 1$, here ($a > 0$ and $b \geq 0$)

$N(C(k, T)) \sim a$ if $g \geq 2$

Here $N(C(k, T))$ counts $k$ rational point of height $\leq T$.

Because one can find a rational point on a genus $0$ curve by passing to a finite extension of the base field and one can find a point of infinite order on a genus $1$ curve by passing to a finite extension of the base field, one sees that the topology of the curve is determined by the asymptotic numbers of rational points in number fields.

In the section with the heading on pg. 474, the authors state the Bombieri-Lang conjecture:

Let $X$ be a variety of general type defined over a number field $k$. Then there is a Zariski open subset $U$ of $X$ such that for all number fields $k'/k$, the set $U(k')$ is finite.

For curves $X$ this is just the Mordell conjecture (since a curve is of general type if and only if its genus is 2 or greater, and Zariski open subsets of curves are just complements of finite sets of points.)

The fact that the converse to the Mordell conjecture is true (in the sense that finiteness of rational points in all finite extensions implies that the curve is of genus 2) suggests that the same might be true for varieties of higher dimensions.

Is the converse to the Bombieri-Lang conjecture stated above true?

More generally,

Let $S$ be the set of varieties of dimension $d$ defined over number fields. To what extent is the geometry of an element of $S$ determined by the asymptotics of the functions that counts $k$ rational points for number fields $k$? (Possibly after passing to Zariski open subsets, etc.)

Is the geometry of a variety determined by the counts of rational points? - MathOverflow

Is the geometry of a variety determined by the counts of rational points?