QUOTIENTS OF ELLIPTIC CURVES OVER FINITE FIELDS

Fix a prime l, and let F-q be a finite field with q equivalent to 1 (mod l) elements. If l > 2 and q >>(l) 1, we show that asymptotically (l - 1)(2)/2l(2) of the elliptic curves E/F-q with complete rational l-torsion are such that E/< P > does not have complete rational l-torsion for any point P is an element of E(F-q) of order l. For l = 2 the asymptotic density is 0 or 1/4, depending whether q equivalent to 1 (mod 4) or 3 (mod 4). We also show that for any l, if E/F-q has an F-q-rational point R of order l(2), then E/< lR > always has complete rational l-torsion.