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Uniformity of rational points. (English) Zbl 0872.14017

Let $K$ be a number field. A question addressed in this paper is the following: Given a family $f:X\to B$ of curves defined over $K$, how does the set of $K$-rational points of the fibres vary with $b\in B$, and in particular, how does its cardinality behave as a function of $b$? This is equivalent to the following conjecture:

Uniformity conjecture. Let $K$ be a number field and $g\ge 2$ an integer. Then there is a number $B\left(K,g\right)$ such that for any smooth curve $X$ of genus $g$ defined over $K$, $#X\left(K\right)\le B\left(K,g\right)$.

(Here $X\left(K\right)$ denote the set of $K$-rational points of $X$.)– The main results of this paper is that the uniformity conjecture holds true if one assumes the validity of Lang’s conjectures about the distribution of rational points on higher dimensional varieties over number fields. First recall Lang’s conjectures.

Weak Lang conjecture. If $X$ is a variety of general type defined over a number field $K$, then $X\left(K\right)$ is not Zariski dense.

There is a stronger version of Lang’s conjecture.

Strong Lang conjecture. Let $X$ be any variety of general type defined over a number field $K$. There exists a proper closed subvariety ${\Xi }\subset X$ such that for any number field $L$ containing $K$, the set of $L$-rational points of $X$ lying outside ${\Xi }$ is finite.

The results of arithmetic nature proved in this paper are as follows:

Theorem 1. (Uniform bound) If the weak Lang conjecture is true, then for every number field $K$ and integer $g\ge 2$, there exists an integer $B\left(K,g\right)$ such that no smooth curve of genus $g$ defined over $K$ has more than $B\left(K,g\right)$ rational points.

If one further assumes the strong Lang conjecture, then the number $B\left(K,g\right)$ depends only on $g$ and not on $K$.

Theorem 2. (Universal generic bound) The strong Lang conjecture implies that for any $g\ge 2$, there exists an integer $N\left(g\right)$ such that for any number field $K$ there are only finitely many smooth curves of genus $g$ defined over $K$ with more than $N\left(g\right)$ $K$-rational points.

The main geometric result of the paper provides varieties of general type to which one can apply Lang’s conjectures.

Theorem 3. (Correlation) Let $f:X\to B$ be a proper morphism of integral varieties, whose general fiber is a smooth curve of genus at least 2. Then for $n$ sufficiently large, ${X}_{B}^{n}$ admits a dominant rational map $h$ to a variety of general type $W$. Moreover, if $X$ is defined over the number field $K$, then $W$ and $h$ are also defined over $K$.

Theorem 1 is proved assuming the weak Lang conjecture and theorem 3, and similarly theorem 2 is proved assuming the strong Lang conjecture and theorem 3. The proof of theorem 3 (correlation) forms the core of the paper taking up $§2$ to $§5$. Some examples are given: For instance, the asymptotic behaviour of $B\left(K,g\right)$ for fixed $K$ and varying $g$, and $N\left(g\right)$: $B\left(ℚ,g\right)\ge 8·g+12$; and $N\left(2\right)\ge 128$ and $N\left(3\right)\ge 72$.

Then higher-dimensional cases are discussed.

Geometric Lang conjecture. If $X$ is any variety of general type, then the union of all irreducible positive-dimensional subvarieties of $X$ not of general type is a proper, closed subvariety ${\Xi }\subset X$ (which is called Langian exceptional locus of $X$ and denoted by ${{\Xi }}_{X}\right)$.

Conjecture (H). (Correlation in higher dimensional cases) Let $f:X\to B$ be an arbitrary morphism of integral varieties, whose general fiber is an integral variety of general type. Then for $n\gg 0$, ${X}_{B}^{n}$ admits a dominant rational map $h$ to a variety $W$ of general type such that the restriction of $h$ to a general fiber of $f$ is generically finite.

One may ask: how the subvarieties ${\Xi }\subset X$ vary with parameters? If one is given a family $f:X\to B$ of varieties of general type, what can one say about the exceptional subvarieties ${{\Xi }}_{b}={{\Xi }}_{{X}_{b}}$ of the fibres? This is answered in the following theorem.

Theorem 4. Assuming the geometric Lang conjecture and conjecture $\left(H\right)$, there is a number $D\left(d,k\right)$ such that for all projective varieties $X$ of degree $d$ or less and dimension $k$ or less, the total degree of the Langian exceptional locus is $deg\left({{\Xi }}_{X}\right)\le D\left(d,k\right)$.

Here the total degree of a variety is the sum of the degrees of its irreducible components.

Theorem 5. Assuming the weak Lang conjecture and conjecture $\left(H\right)$ for families of symmetric squares of curves, there exists for every integer $g$ and number field $K$ a number ${B}_{q}\left(g,K\right)$ such that no non-hyperelliptic, non-bielliptic curve $C$ of genus $g$ defined over $K$ has more than ${B}_{q}\left(g,K\right)$ points whose coordinates are quadratic over $K$. If we assume in addition the strong Lang conjecture, there exists for every integer $g$ a number ${N}_{q}\left(g\right)$ such that for any number field $K$ there are only finitely many non-hyperelliptic, non-bielliptic curves of genus $g$ defined over $K$ that have more than ${N}_{q}\left(g\right)$ points whose coordinates are quadratic over $K$.

##### MSC:
 14G05 Rational points 14H25 Arithmetic ground fields (curves) 14H10 Families, algebraic moduli (curves)