Counting rational points on algebraic varieties
نویسندگان
چکیده
منابع مشابه
Counting Rational Points on Algebraic Varieties
In these lectures we will be interested in solutions to Diophantine equations F (x1, . . . , xn) = 0, where F is an absolutely irreducible polynomial with integer coefficients, and the solutions are to satisfy (x1, . . . , xn) ∈ Z. Such an equation represents a hypersurface in A, and we may prefer to talk of integer points on this hypersurface, rather than solutions to the corresponding Diophan...
متن کاملCounting Rational Points on Algebraic Varieties
For any N ≥ 2, let Z ⊂ P be a geometrically integral algebraic variety of degree d. This paper is concerned with the number NZ(B) of Q-rational points on Z which have height at most B. For any ε > 0 we establish the estimate NZ(B) = Od,ε,N (B ), provided that d ≥ 6. As indicated, the implied constant depends at most upon d, ε and N . Mathematics Subject Classification (2000): 11G35 (14G05)
متن کاملCounting Rational Points on Ruled Varieties
In this paper, we prove a general result computing the number of rational points of bounded height on a projective variety V which is covered by lines. The main technical result used to achieve this is an upper bound on the number of rational points of bounded height on a line. This upper bound is such that it can be easily controlled as the line varies, and hence is used to sum the counting fu...
متن کاملAlgebraic varieties with many rational points
sometimes we want to understand a typical equation, i.e., a general equation in some family. To draw inspiration (and techniques) from different branches of algebra it is necessary to consider solutions with values in other rings and fields, most importantly, finite fields Fq, finite extensions of Q, or the function fields Fp(t) and C(t). While there is a wealth of ad hoc elementary tricks to d...
متن کاملCounting rational points on ruled varieties over function fields
Let K be the function field of an algebraic curve C defined over a finite field Fq. Let V ⊂ PK be a projective variety which is a union of lines. We prove a general result computing the number of rational points of bounded height on V/K. We first compute the number of rational points on a general line defined over K, and then sum over the lines covering V . Mathematics Subject Classification: 1...
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ژورنال
عنوان ژورنال: Duke Mathematical Journal
سال: 2006
ISSN: 0012-7094
DOI: 10.1215/s0012-7094-06-13236-2