Root numbers and ranks in positive characteristic

Abstract

For a global field K and an elliptic curve E η over K ( T ) , Silverman's specialization theorem implies rank ( E η ( K ( T ) ) ) ⩽ rank ( E t ( K ) ) for all but finitely many t ∈ P 1 ( K ) . If this inequality is strict for all but finitely many t, the elliptic curve E η 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.