Root numbers and ranks

Опубликовано 6 сентября 2016, 5:44

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