Counting rational points on cubic curves

Counting Rational Points on Cubic Hypersurfaces

Let X ⊂ P be a geometrically integral cubic hypersurface defined over Q, with singular locus of dimension 6 dimX − 4. Then the main result in this paper is a proof of the fact that X(Q) contains Oε,X(B ) points of height at most B.

Counting Rational Points on Cubic Hypersurfaces: Corrigendum

R0<b162R0 gcd(b1, N )1/2 R 0 (HP) . The second line is false and in fact one has M1 = 1 in Proposition 3. The author is very grateful to Professor Hongze Li for drawing his attention to this flaw. The error can be fixed by introducing an average over b1 into the statement of Proposition 3. This allows us to recover the main theorem in [1], and also [2, Lemma 11], via the following modification....

Theory L . Caporaso COUNTING RATIONAL POINTS ON ALGEBRAIC CURVES

We describe recent developments on the problem of finding examples of algebraic curves of genus at least 2 having the largest possible number of rational points. This question is related to the Conjectures of Lang on the distribution of rational points on the varieties of general type.

Counting Rational Points on Curves over Finite Fields (Extended Abstract)

We consider the problem of counting the number of points on a plane curve, given by a homogeneous polynomial F E Fp[x, y, 21, which is rational over the ground field IFp. More precisely, we show that if we are given a projective plane curve C of degree n, and if C has only ordinary multiple points, then one can compute the number of IFp-rational points on C in randomized time (logp)" where A = ...

Rational points on curves