Root numbers and ranks in positive characteristic

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 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.