Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties

AbstractLet V be a K3 surface defined over a number field k. The Batyrev-Manin conjecture for V states that for every nonempty open subset U of V, there exists a finite set ZU of accumulating rational curves such that the density of rational points on U − ZU is strictly less than the density of rational points on ZU. Thus, the set of rational points of V conjecturally admits a stratification corresponding to the sets ZU for successively smaller sets U.In this paper, in the case that V is a Kummer surface, we prove that the Batyrev-Manin conjecture for V can be reduced to the Batyrev-Manin conjecture for V modulo the endomorphisms of V induced by multiplication by m on the associated abelian surface A. As an application, we use this to show that given some restrictions on A, the set of rational points of V which lie on rational curves whose preimages have geometric genus 2 admits a stratification of Batyrev-Manin type.

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Growth rate of ample heights and the Dynamical Mordell–Lang conjecture

International Journal of Number Theory ◽

10.1142/s1793042118501610 ◽

2018 ◽

Vol 14 (10) ◽

pp. 2673-2685

Author(s):

Kaoru Sano

Keyword(s):

Growth Rate ◽

Explicit Formula ◽

Number Fields ◽

Rational Points ◽

Positive Answer ◽

Projective Varieties ◽

Smooth Projective Varieties

We provide an explicit formula on the growth rate of ample heights of rational points under iteration of endomorphisms of smooth projective varieties over number fields. As an application, we give a positive answer to a variant of the Dynamical Mordell–Lang conjecture for pairs of étale endomorphisms, which is also a variant of the original one stated by Bell, Ghioca, and Tucker in their monograph.

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MIRROR SYMMETRY AND AN EXACT CALCULATION OF AN (N–2)-POINT CORRELATION FUNCTION ON A CALABI-YAU MANIFOLD EMBEDDED IN CPN−1

International Journal of Modern Physics A ◽

10.1142/s0217751x96000559 ◽

1996 ◽

Vol 11 (07) ◽

pp. 1217-1252 ◽

Cited By ~ 3

Author(s):

MASAO JINZENJI ◽

MASARU NAGURA

Keyword(s):

Correlation Function ◽

Yukawa Coupling ◽

Mirror Symmetry ◽

Rational Curves ◽

Holomorphic Curves ◽

Holomorphic Maps ◽

Point Function ◽

Point Correlation ◽

Point Correlation Function ◽

Zero Locus

We consider an (N–2)-dimensional Calabi-Yau manifold which is defined as the zero locus of the polynomial of degree N (of the Fermat type) in CPN−1 and its mirror manifold. We introduce an (N–2)-point correlation function (generalized Yukawa coupling) and evaluate it both by solving the Picard-Fuchs equation for period integrals in the mirror manifold and by explicitly calculating the contribution of holomorphic maps of degree 1 to the Yukawa coupling in the Calabi-Yau manifold using the method of algebraic geometry. In enumerating the holomorphic curves in the general-dimensional Calabi-Yau manifolds, we extend the method of counting rational curves on the Calabi-Yau three-fold using the Shubert calculus on Gr (2, N). The agreement of the two calculations for the (N–2)-point function establishes “the mirror symmetry at the correlation function level” in the general-dimensional case.

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A SIMPLE PROOF OF CHEBOTAREV’S DENSITY THEOREM OVER FINITE FIELDS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972718000448 ◽

2018 ◽

Vol 98 (2) ◽

pp. 196-202

Author(s):

STEVE MEAGHER

Keyword(s):

Conjugacy Class ◽

Finite Fields ◽

Simple Proof ◽

Function Fields ◽

Rational Points ◽

Density Theorem ◽

Chebotarev Density Theorem ◽

Projective Varieties

We present a simple proof of the Chebotarev density theorem for finite morphisms of quasi-projective varieties over finite fields following an idea of Fried and Kosters for function fields. The key idea is to interpret the number of rational points with a given Frobenius conjugacy class as the number of rational points of a twisted variety, which is then bounded by the Lang–Weil estimates.

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