Root numbers and ranks over global function fields
by
Brian Conrad
University of Michigan

Standard conjectures in analytic number theory imply that for a 1-parameter family of elliptic curves over Q satisfying the reasonable geometric requirements of nonisotriviality and the existence of a multiplicative fiber in the family, the root numbers of the fibers average to 0. The relevant standard conjectures have analogues over global function fields, and one such analogue is provably false.

In contrast with the (conjectural) situation over Q, in every characteristic > 3 we construct many 1-parameter families of elliptic curves over k(u) (for finite k) that satisfy the above geometric hypotheses yet have odd-rank fiber over the generic point and have root number 1 (and hence, positive even rank under the Birch and Swinnerton-Dyer Conjecture) for all smooth k(u)-fibers in the family. The rank of the generic fiber is computed by a mixture of cohomological and geometric methods, resting on the Chebotarev Density Theorem and a closed study of Neron models.

This is joint work with K. Conrad and H. Helfgott.

Date received: June 8, 2004