J un 2 00 5 ROOT NUMBERS AND RANKS IN POSITIVE CHARACTERISTIC

For a global field K and an elliptic curve Eη over K(T), Silverman's specialization theorem implies rank(Eη(K(T))) ≤ rank(Et(K)) for all but finitely many t ∈ P 1 (K). If this inequality is strict for all but finitely many t, the elliptic curve Eη is said to have elevated rank. All known examples of elevated rank for K = Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, an analogue of one of these additional conjectures is false. Inspired by this, for the rational function field K = κ(u) over any finite field κ with characteristic = 2, we construct an explicit 2-parameter family E c,d of non-isotrivial elliptic curves over K(T) (depending on arbitrary c, d ∈ κ ×) such that, under the parity conjecture, each E c,d has elevated rank.