scholarly journals Problems on rational points and rational curves on algebraic varieties

1993 ◽  
Vol 2 (1) ◽  
pp. 214-245 ◽  
Author(s):  
Y. I. Manin
Keyword(s):  
Rational Points ◽  
Rational Curves ◽  
Algebraic Varieties

scholarly journals A Reduction of the Batyrev-Manin Conjecture for Kummer Surfaces

2004 ◽  
Vol 47 (3) ◽  
pp. 398-406
Author(s):  
David McKinnon
Keyword(s):  
Number Field ◽  
Open Subset ◽  
Rational Points ◽  
Rational Curves ◽  
Abelian Surface ◽  
Kummer Surface ◽  
Geometric Genus ◽  
Finite Set ◽  
Genus 2 ◽  
Kummer Surfaces

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.

scholarly journals Counting in hyperbolic spikes: The diophantine analysis of multihomogeneous diagonal equations

2018 ◽  
Vol 2018 (737) ◽  
pp. 255-300
Author(s):  
Valentin Blomer ◽  
Jörg Brüdern
Keyword(s):  
Circle Method ◽  
Rational Points ◽  
Arithmetic Functions ◽  
New Method ◽  
Algebraic Varieties ◽  
Asymptotic Formulae ◽  
Versatile Tool

AbstractA method is described to sum multi-dimensional arithmetic functions subject to hyperbolic summation conditions, provided that asymptotic formulae in rectangular boxes are available. In combination with the circle method, the new method is a versatile tool to count rational points on algebraic varieties defined by multi-homogeneous diagonal equations.

The Number of Rational Points of a Family of Algebraic Varieties over Finite Fields

2017 ◽  
Vol 24 (04) ◽  
pp. 705-720 ◽  
Author(s):  
Shuangnian Hu ◽  
Junyong Zhao
Keyword(s):  
Number Theory ◽  
Normal Form ◽  
Finite Fields ◽  
Rational Points ◽  
System Of Equations ◽  
Algebraic Varieties ◽  
Natural Conditions ◽  
Characteristic P ◽  
Exponent Matrix ◽  
Positive Integers

Let 𝔽q stand for the finite field of odd characteristic p with q elements (q = pn, n ∈ ℕ) and [Formula: see text] denote the set of all the nonzero elements of 𝔽q. Let m and t be positive integers. By using the Smith normal form of the exponent matrix, we obtain a formula for the number of rational points on the variety defined by the following system of equations over [Formula: see text] where the integers t > 0, r0 = 0 < r1 < r2 < ⋯ < rt, 1 ≤ n1 < n2 <, ⋯ < nt and 0 ≤ j ≤ t − 1, bk ∊ 𝔽q, ak,i ∊ [Formula: see text] (k = 1, …, m, i = 1, …, rt), and the exponent of each variable is a positive integer. Further, under some natural conditions, we arrive at an explicit formula for the number of 𝔽q-rational points on the above variety. It extends the results obtained previously by Wolfmann, Sun, Wang, Hong et al. Our result gives a partial answer to an open problem raised in [The number of rational points of a family of hypersurfaces over finite fields, J. Number Theory 156 (2015) 135–153].

scholarly journals K3 surfaces, rational curves, and rational points

2010 ◽  
Vol 130 (7) ◽  
pp. 1470-1479 ◽  
Author(s):  
Arthur Baragar ◽  
David McKinnon
Keyword(s):  
K3 Surfaces ◽  
Rational Points ◽  
Rational Curves
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