Published online by Cambridge University Press: 13 July 2011
We consider exponential sums with x-coordinates of points qG and q−1G where G is a point of order T on an elliptic curve modulo a prime p and q runs through all primes up to N (with gcd (q,T)=1 in the case of the points q−1G). We obtain a new bound on exponential sums with q−1G and correct an imprecision in the work of W. D. Banks, J. B. Friedlander, M. Z. Garaev and I. E. Shparlinski on exponential sums with qG. We also note that similar sums with g1/q for an integer g with gcd (g,p)=1 have been estimated by J. Bourgain and I. E. Shparlinski.