The average number of torsion points on elliptic curves

Given an elliptic curve E defined over a number field K, an integer m   and let E[m](F℘)E[m](F℘) be the set of m-torsion points of E   that are rational over F℘F℘ where F℘F℘ denotes the residue field at ℘. Our main goal is to compute the asymptotic average number of E[m](F℘)E[m](F℘) as the prime ℘ varies. The average number N(m)N(m) as a function of m   agrees with a divisor function. For the elliptic curves ED:Y2=X3−DXED:Y2=X3−DX, with D∈Z−{0}D∈Z−{0}, we compute the average number N(m)N(m) for all odd integers m.