The distribution and growth of the elementary divisors of the reductions of an elliptic curve over a function field

Let K be a global field of characteristic p⩾5 and let E/K be a non-isotrivial elliptic curve. For places v of K of good reduction for E, let Ev/kv be the reduction of E modulo v. Then Ev(kv)≃Z/dvZ×Z/dvevZ for positive integers dv, ev, uniquely determined by E and v. We study the distribution of dv and the growth of dvev as v varies over places of degree n and n→∞.