Root Numbers and Ranks in Positive Characteristic

For a global fieldK and an elliptic curve Eη overK(T ), Silverman’s specialization theorem implies rank(Eη(K(T ))) ≤ rank(Et(K)) for all but finitely many t ∈ P(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 nonisotrivial 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 6= 2, we construct an explicit 2-parameter family Ec,d of non-isotrivial elliptic curves over K(T ) (depending on arbitrary c, d ∈ κ×) such that, under the parity conjecture, each Ec,d has elevated rank. To Mike Artin on his 70th birthday