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Academy of Mathematics and Systems Science, CAS Colloquia & Seminars：Rational points and homogeneous spaces over function fields of curves
曲线函数场 有理点 齐次空间
2023/4/17
In the last decade, there has been a growing interest in the study of rational points on homogeneous spaces of linear algebraic groups defined over fields having good arithmetical properties other tha...
Academy of Mathematics and Systems Science, CAS Colloquia & Seminars：Number of rational points of difference varieties in finite difference fields
有限差分域 差分簇 有理点数
2023/4/13
A difference field is a field with a distinguished automorphism. Automorphisms of a finite field are powers of the Frobenius map. In this talk, I will discuss how to estimate the number of rational po...
Quadratic congruences on average and rational points on cubic surfaces
Quadratic congruences rational points Manin’s conjecture cubic surfaces universal torsors
2012/5/24
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.
Counting rational points over number fields on a singular cubic surface
rational points over number fields singular cubic surface Number Theory
2012/4/18
A conjecture of Manin predicts the distribution of K-rational points on certain algebraic varieties defined over a number field K. In recent years, a method using universal torsors has been successful...
On the $p$-adic closure of a subgroup of rational points on an Abelian variety
$p$-adic closure subgroup of rational points Abelian variety
2011/2/24
In 2007, B. Poonen (unpublished) studied the p{adic closure of a subgroup of rational points on a commutative algebraic group. More recently, J. Bellache asked the same question for the special case...
Density of rational points on elliptic curves and small transcendence degree
rational points elliptic curves
2010/11/19
In this paper we pursue two goals: (I) We show how Weil restrictions to real subfields can be fruitfully applied to improve transcendence results. (II) We elaborate (I) in the context of algebraic in...
Inhomogeneous cubic congruences and rational points on del Pezzo surfaces
Inhomogeneous cubic congruences rational points del Pezzo surfaces
2010/11/22
For given non-zero integers a,b,q we investigate the density of integer solutions (x,y) to the binary cubic congruence ax^2+by^3=0 (mod q) and use it to establish the Manin conjecture for a singular d...
Density of rational points on elliptic surfaces
Density of rational points elliptic surfaces
2010/12/9
Suppose V is a surface over a number field k that admits two elliptic fibrations. We show that for each integer d there exists an explicitly computable closed subset Z of V , not equal to V , such tha...
Rational points over finite fields for regular models of algebraic varieties of Hodge type $\geq 1$
Rational points algebraic varieties of Hodge type $\geq 1$
2010/11/26
Let R be a discrete valuation ring of mixed characteristics (0, p), with finite residue field k and fraction field K, let k′ be a finite extension of k, and let X be a regular, proper and flat R-schem...