×

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 \(g\geq 2\) an integer. Then there is a number \(B(K,g)\) such that for any smooth curve \(X\) of genus \(g\) defined over \(K\), \(\# X(K) \leq B(K,g)\).
(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.
Weak Lang conjecture. If \(X\) is a variety of general type defined over a number field \(K\), then \(X(K)\) 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 \(g\geq 2\), 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.
If one further assumes the strong Lang conjecture, then the number \(B (K,g)\) depends only on \(g\) and not on \(K\).
Theorem 2. (Universal generic bound) The strong Lang conjecture implies that for any \(g\geq 2\), 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.
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, \(X^n_B\) 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 \(K\).
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 \(N(g)\): \(B(\mathbb{Q},g) \geq 8 \cdot g+12\); and \(N(2)\geq 128\) and \(N(3)\geq 72\).
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 \(\Xi_X)\).
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.
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 \((H)\), 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 \(\deg (\Xi_X) \leq D(d,k)\).
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 \((H)\) 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 \(K\).

MSC:

14G05 Rational points
14H25 Arithmetic ground fields for curves
14H10 Families, moduli of curves (algebraic)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] D.Abramovich. On the number of stably integral points on an elliptic curve. preprint. · Zbl 0898.11020
[2] D.Abramovich. Uniformité des points rationnels des courbes algébriques sur les extensions quadratiques et cubiques. preprint. · Zbl 0874.14011
[3] Dan Abramovich and Joe Harris, Abelian varieties and curves in \?_{\?}(\?), Compositio Math. 78 (1991), no. 2, 227 – 238. · Zbl 0748.14010
[4] S. Ju. Arakelov, Families of algebraic curves with fixed degeneracies, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 1269 – 1293 (Russian).
[5] E. Arbarello, M. Cornalba, P. A. Griffiths, and J. Harris, Geometry of algebraic curves. Vol. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 267, Springer-Verlag, New York, 1985. · Zbl 0559.14017
[6] Edward Bierstone and Pierre D. Milman, A simple constructive proof of canonical resolution of singularities, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 11 – 30. · Zbl 0743.14012 · doi:10.1007/978-1-4612-0441-1_2
[7] F. A. Bogomolov, Families of curves on a surface of general type, Dokl. Akad. Nauk SSSR 236 (1977), no. 5, 1041 – 1044 (Russian).
[8] L.Caporaso, J.Harris, B.Mazur. How many rational points can a curve have? Proceedings of the Texel Conference, Progress in Math. vol. 129, Birkhauser Boston, 1995, p. 13-31. CMP 96:04
[9] L.Caporaso, J.Harris, B.Mazur. Uniformity of rational points. Preliminary version of this paper, available by anonymous ftp from math.harvard.edu. · Zbl 0872.14017
[10] Schémas en groupes. I: Propriétés générales des schémas en groupes, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 151, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. II: Groupes de type multiplicatif, et structure des schémas en groupes généraux, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 152, Springer-Verlag, Berlin-New York, 1970 (French). Schémas en groupes. III: Structure des schémas en groupes réductifs, Séminaire de Géométrie Algébrique du Bois Marie 1962/64 (SGA 3). Dirigé par M. Demazure et A. Grothendieck. Lecture Notes in Mathematics, Vol. 153, Springer-Verlag, Berlin-New York, 1970 (French).
[11] Lawrence Ein, Subvarieties of generic complete intersections. II, Math. Ann. 289 (1991), no. 3, 465 – 471. · Zbl 0746.14019 · doi:10.1007/BF01446583
[12] Renée Elkik, Singularités rationnelles et déformations, Invent. Math. 47 (1978), no. 2, 139 – 147 (French). · Zbl 0363.14002 · doi:10.1007/BF01578068
[13] Gerd Faltings, The general case of S. Lang’s conjecture, Barsotti Symposium in Algebraic Geometry (Abano Terme, 1991) Perspect. Math., vol. 15, Academic Press, San Diego, CA, 1994, pp. 175 – 182. · Zbl 0823.14009
[14] A. Grothendieck, Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents. I, Inst. Hautes Études Sci. Publ. Math. 11 (1961), 167 (French). · Zbl 0122.16102
[15] Robin Hartshorne, Ample subvarieties of algebraic varieties, Lecture Notes in Mathematics, Vol. 156, Springer-Verlag, Berlin-New York, 1970. Notes written in collaboration with C. Musili. · Zbl 0208.48901
[16] B.Hassett. Correlation for surfaces of general type. preprint. · Zbl 0874.14030
[17] Heisuke Hironaka, Idealistic exponents of singularity, Algebraic geometry (J. J. Sylvester Sympos., Johns Hopkins Univ., Baltimore, Md., 1976) Johns Hopkins Univ. Press, Baltimore, Md., 1977, pp. 52 – 125. · Zbl 0496.14011
[18] János Kollár, Subadditivity of the Kodaira dimension: fibers of general type, Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, Amsterdam, 1987, pp. 361 – 398. · Zbl 0666.14003
[19] János Kollár, Projectivity of complete moduli, J. Differential Geom. 32 (1990), no. 1, 235 – 268. · Zbl 0684.14002
[20] Serge Lang, Hyperbolic and Diophantine analysis, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 2, 159 – 205. · Zbl 0602.14019
[21] E.Looijenga. Smooth Deligne-Mumford compactifications by means of Prym levels structires. J.AlgGeom. 3 (1992) p.283-293. · Zbl 0814.14030
[22] S.Lu, M. Miyaoka. preprint.
[23] David Mumford, Stability of projective varieties, Enseignement Math. (2) 23 (1977), no. 1-2, 39 – 110. · Zbl 0363.14003
[24] David Mumford and John Fogarty, Geometric invariant theory, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 34, Springer-Verlag, Berlin, 1982. · Zbl 0504.14008
[25] Herbert Popp, Modulräume algebraischer Mannigfaltigkeiten, Classification of algebraic varieties and compact complex manifolds, Springer, Berlin, 1974, pp. 219 – 242. Lecture Notes in Math., Vol. 412 (German). · Zbl 0299.14009
[26] Herbert Popp, Moduli theory and classification theory of algebraic varieties, Lecture Notes in Mathematics, Vol. 620, Springer-Verlag, Berlin-New York, 1977. · Zbl 0359.14005
[27] Miles Reid, Canonical 3-folds, Journées de Géometrie Algébrique d’Angers, Juillet 1979/Algebraic Geometry, Angers, 1979, Sijthoff & Noordhoff, Alphen aan den Rijn — Germantown, Md., 1980, pp. 273 – 310.
[28] E.Viehweg. Weak positivity and the additivity of the Kodaira dimension for certain fibre spaces. Adv. Stud. in Pure Math. 1 (1983) p. 329-353. · Zbl 0513.14019
[29] E.Viehweg. Canonical divisors and the additivity of the Kodaira dimension for morphisms of relative dimension one. Compositio Math. 35, Fasc 2 (1977) p. 197-223. · Zbl 0357.14014
[30] E.Viehweg. Rational singularities of higher dimensional schemes. Proc. AMS. 63 n.1 (1977) p.6-8. · Zbl 0352.14003
[31] Paul Vojta, Diophantine approximations and value distribution theory, Lecture Notes in Mathematics, vol. 1239, Springer-Verlag, Berlin, 1987. · Zbl 0609.14011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.