EXPONENTIAL SUMS OVER POINTS OF ELLIPTIC CURVES WITH RECIPROCALS OF PRIMES

We consider exponential sums with x ‐coordinates of points qG and q −1 G 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 −1 G ). We obtain a new bound on exponential sums with q −1 G 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 g 1/ q for an integer g with gcd ( g , p )=1 have been estimated by J. Bourgain and I. E. Shparlinski.