The average exponent of elliptic curves modulo p

Let E   be an elliptic curve defined over QQ. For a prime p of good reduction for E  , denote by epep the exponent of the reduction of E modulo p  . Under GRH, we prove that there is a constant CE∈(0,1)CE∈(0,1) such that∑p⩽xep=CELi(x2){1+OE((logx)4/3x1/6)} for all x⩾2x⩾2, where the implied constant depends on E at most. When E   has complex multiplication, the same asymptotic formula with a weaker error term OE(1/(logx)1/14) is established unconditionally. These improve some recent results of Freiberg and Kurlberg.