Counting rational points on smooth cubic curves

Abstract

We use a global version of Heath-Brown's p−adic determinant method developed by Salberger to give upper bounds for the number of rational points of height at most B on non-singular cubic curves defined over Q. The bounds are uniform in the sense that they only depend on the rank of the corresponding Jacobian.

Uniform bounds for rational points on complete intersections of two quadric surfaces

“…Let Y be a subvariety of P 3 × P 3 , D ⊂ Y be an effective divisor and L be the restriction to Y of the line bundle O P 3 ×P 3 (a, b). There is then a short exact sequence 0 → L(−D) → L → L| D → 0 (15) of sheaves on Y . From the long exact cohomology sequence associated to (15), it follows that if the cohomology group H 1 (Y, L(−D)) vanishes, then the restriction of global sections…”

“…The proof follows the same strategy as in the paper [7] on non-singular cubic curves where the authors combine Heath-Brown's p-adic determinant method in [6] with descent theory. But we will follow the approach in [15] and replace the p-adic determinant method by Salberger's global determinant method [13]. Taking m = 1 + [ √ log B] we immediately obtain the following result.…”