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Height uniformity for algebraic points on curves. (English) Zbl 1031.11041
G. Faltings proved in [Invent. Math. 73, 349–366 (1983); Erratum ibid. 75, 381 (1984); Zbl 0588.14026)] that every non-singular projective curve of genus at least two defined over a number field has only finitely many rational points, and it is known that there are only finitely many points of bounded degree over an arbitrary number field on some curves of genus at least two. Based on these results, we can ask two natural question: How many are there and how big are they ? The first question is answered in [L. Caporaso, J. Harris and B. Mazur, J. Am. Math. Soc. 10, 1–35 (1997; Zbl 0872.14017)]. Assuming the (weak) Lang conjecture, they proved that there exists a uniform upper bound (depending on the genus \(g\) and the number field \(k\)) for the number of \(k\)-rational points for curves defined over \(k\) and genus \(g\geq 2\).
In this paper, the author obtains an analogous result for the second question. More precisely: Let \(d\) and \(e\) be two integers bigger than or equal to one. Let \(\pi: X \to Y\) be a family of curves of genus bigger than or equal to \(2\) (that is; \(X\) and \(Y\) are nonsingular projective varieties and the generic fiber is a nonsingular projective curve of genus at least \(2\)). \(X\), \(Y\) and \(\pi\) are assumed to be defined over \(k\), and we choose an arbitrary height \(h\) on \(X\) and a height \(h_Y\) on \(Y\) associated to an ample divisor satisfying \(h_Y\geq 1\). With the above notations and the additional hypothesis that all the one-dimensional nonsingular fibers of \(\pi\) have only finitely many algebraic points of degree at most \(e\) over an arbitrary number field, and assuming the Vojta conjecture [P. Vojta, Diophantine approximations and value distribution theory. Lecture Notes in Mathematics, 1239. Berlin: Springer (1987; Zbl 0609.14011)] for varieties of dimension at most \(d\cdot(e+ \dim(Y))\), there is a constant \(c>0\) such that \[ h(P)\leq c \cdot h_Y(\pi(P)), \] where \(P\) is an algebraic point of degree at most \(e\) over \(k(\pi(P))\), \(\pi(P)\) has degree at most \(d\) over \(k\) and the fiber of \(\pi\) over the point \(\pi(P)\) is a nonsingular projective curve of genus at least two. Moreover, the constant \(c\) is independent of \(P\).
The main tools used in the proof are the Vojta conjecture and the dominance of a height associated to an ample divisor [S. Lang, Fundamentals of Diophantine geometry. New York: Springer (1983; Zbl 0528.14013)].
As an application, the so-called height zeta-function is introduced. It is associated to the families studied in the theorem (assuming, for convenience, that \(e=d=1\)), which is defined for \(s\) a complex number with \(\text{Re}(s)\geq 0\). It is proved that it defines an analytic function for \(s\) as above.

MSC:
11G35 Varieties over global fields
11G50 Heights
14G05 Rational points
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
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