For any finite field k with odd characteristic and F = k(u), we construct a non-isotrivial elliptic curve over F(t) such that all of its F-fibers have root number 1 (and hence even rank, under BSD) whereas the generic fiber has Mordell-Weil group with rank 1. The proof involves a mixture of arithmetic and geometric specialization arguments, and an amusing application of the Lang-Neron theorem. Non-isotrivial families with such a parity discrepancy are not expected to exist over Q, but the argument over Q rests on a standard conjecture in analytic number theory whose function field analogue admits surprising counterexamples (especially mysterious in characteristic 2).
If you saw me give a talk on this topic before last September then you may wish to considering seeing it again since the technique at that time rested on a non-vanishing calculation that turned out to be incorrect and the salvaged proof rests on an entirely different geometric idea.
This is joint work with K. Conrad and H. Helfgott.