On the Sum-Product Problem on Elliptic Curves
by
Omran Ahmadi
University of Waterloo
Coauthors: Igor Shparlinski

Let \E be an ordinary elliptic curve over a finite field \F_q of q elements and x(Q) denote the x-coordinate of a point Q = (x(Q), y(Q)) on \E. Given an \F_q-rational point P of order T, we show that for any subsets \cA, \cB of the unit group of the residue ring modulo T, at least one of the sets

{x(aP) + x(bP) : a ∈ \cA, b ∈ \cB}    and    {x(abP) : a ∈ \cA, b ∈ \cB}

is large. This question is motivated by a series of recent results on the sum-product problem over finite fields and other algebraic structures.

Date received: June 15, 2008