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Zbl 0872.14017
Caporaso, Lucia; Harris, Joe; Mazur, Barry
Uniformity of rational points.
(English)
[J] J. Am. Math. Soc. 10, No.1, 1-35 (1997). ISSN 0894-0347; ISSN 1088-6834/e

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:\par Uniformity conjecture. Let $K$ be a number field and $g\ge 2$ an integer. Then there is a number $B(K,g)$ such that for any smooth curve $X$ of genus $g$ defined over $K$, $\# X(K) \le B(K,g)$.\par (Here $X(K)$ 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. \par Weak Lang conjecture. If $X$ is a variety of general type defined over a number field $K$, then $X(K)$ is not Zariski dense.\par There is a stronger version of Lang's conjecture. \par 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.\par The results of arithmetic nature proved in this paper are as follows:\par 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 $B(K,g)$ such that no smooth curve of genus $g$ defined over $K$ has more than $B (K,g)$ rational points.\par If one further assumes the strong Lang conjecture, then the number $B (K,g)$ depends only on $g$ and not on $K$.\par Theorem 2. (Universal generic bound) The strong Lang conjecture implies that for any $g\ge 2$, there exists an integer $N(g)$ such that for any number field $K$ there are only finitely many smooth curves of genus $g$ defined over $K$ with more than $N(g)$ $K$-rational points.\par The main geometric result of the paper provides varieties of general type to which one can apply Lang's conjectures.\par 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, $X^n_B$ 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 $K$.\par 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 $\S 2$ to $\S 5$. Some examples are given: For instance, the asymptotic behaviour of $B(K,g)$ for fixed $K$ and varying $g$, and $N(g)$: $B(\bbfQ,g) \ge 8 \cdot g+12$; and $N(2)\ge 128$ and $N(3)\ge 72$.\par Then higher-dimensional cases are discussed. \par 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 $\Xi_X)$.\par 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 $n\gg 0$, $X^n_B$ 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.\par 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.\par Theorem 4. Assuming the geometric Lang conjecture and conjecture $(H)$, there is a number $D(d,k)$ 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 $\deg (\Xi_X) \le D(d,k)$.\par Here the total degree of a variety is the sum of the degrees of its irreducible components.\par Theorem 5. Assuming the weak Lang conjecture and conjecture $(H)$ for families of symmetric squares of curves, there exists for every integer $g$ and number field $K$ a number $B_q(g,K)$ such that no non-hyperelliptic, non-bielliptic curve $C$ of genus $g$ defined over $K$ has more than $B_q (g,K)$ points whose coordinates are quadratic over $K$. If we assume in addition the strong Lang conjecture, there exists for every integer $g$ a number $N_q (g)$ 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 $N_q (g)$ points whose coordinates are quadratic over $K$.
[N.Yui (Kingston/Ontario)]
MSC 2000:
*14G05 Rationality questions, rational points
14H25 Arithmetic ground fields (curves)
14H10 Families, algebraic moduli (curves)

Keywords: uniformity conjecture; Lang's conjectures; distribution of rational points

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