Algebra Seminar talk
2019-10-31 (Thursday!) 13:15
Descent on elliptic surfaces and arithmetic bounds for the Mordell-Weil rank
We introduce the use of p-descent techniques for elliptic surfaces over a perfect field of characteristic not 2 or 3. Under mild hypotheses, we obtain an upper bound for the rank of a non-constant elliptic surface. When p = 2, this bound is an arithmetic refinement of a well-known geometric bound for the rank deduced from Igusa’s inequality. This answers a question raised by Ulmer. We are also able to deduce some heuristics in favor of a conjecture by Silverman on the asymptotic behavior of the rank. This is joint work with Aaron Levin.