# Exponential sums over points of elliptic curves with reciprocals of primes

Ostafe, Alina; Shparlinski, Igor E (2012). Exponential sums over points of elliptic curves with reciprocals of primes. Mathematika, 58(1):21-33.

## Abstract

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 toN (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

## Abstract

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 toN (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

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