On the number of rational points on certain elliptic curves

摘要

Let E be an elliptic curve defined over the rationals, with rational 2-torsion. We prove a uniform bound for the number of rational points on E of height less than or equal to B of the form #{P is an element of E(Q): H(P) less than or equal to B} less than or equal to c(epsilon) (max (H(E), B))(epsilon), valid for every fixed epsilon > 0 and a suitable positive computable constant c(epsilon). We give an application of this result to the counting of quadruples (P-1,P-2,P-3,P-4) of distinct primes that-do not exceed X and satisfy p(i)(2)Delta(jk) - p(j)(2)Delta(ik) + p(k)(2)Delta(ij) = 0 for all 1 less than or equal to i < j < k less than or equal to 4, where Delta(ij) are given integers. This is applied by Konyagin (in the paper [3], which is published simultaneously with the present one) to a problem on the large sieve by squares.