Problems on rational points and rational curves on algebraic varieties

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...

Geometry of Rational Curves on Algebraic Varieties

Geometry of Rational Curves on Algebraic Varieties

Counting 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)

Acquisition of Rational Points on Algebraic Curves

Rational Curves on Varieties