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EXPONENTIAL SUMS OVER POINTS OF ELLIPTIC CURVES WITH RECIPROCALS OF PRIMES

Part of: Exponential sums and character sums Arithmetic algebraic geometry Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  13 July 2011

Alina Ostafe  and
Igor E. Shparlinski
Alina Ostafe
Affiliation:
Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190 CH-8057, Zürich, Switzerland (email: alina.ostafe@math.uzh.ch)
Igor E. Shparlinski
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia (email: igor.shparlinski@mq.edu.au)
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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 to N (with gcd (q,T)=1in 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)=1have been estimated by J. Bourgain and I. E. Shparlinski.

MSC classification

Secondary: 11L07: Estimates on exponential sums 11T23: Exponential sums 11G20: Curves over finite and local fields

Information

Type
Research Article
Information
Mathematika , Volume 58 , Issue 1 , January 2012 , pp. 21 - 33
Copyright
Copyright © University College London 2012

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