The average Mordell--Weil rank of elliptic surfaces over number fields

Abstract: Let $K$ be a number field  and let $n$ be a non-negative integer. In this talk we determine the average (arithmetic) Mordell--Weil rank of elliptic surfaces over K with base curve P^1 and geometric genus n. Hereby we prove a conjecture of Alex Cowan, which in turn was inspired by Nagao's conjecture and subsequent work.

The proof consists of two parts, the first part relies on work by Andr\'e and Maulik--Poonen on the jump loci of the Picard Number in flat families. This is sufficient to prove that the average Mordell--Weil rank equals the generic Mordell--Weil rank. A second argument yields that the generic Mordell--Weil rank of elliptic surfaces over a number field with fixed topological equals.
 

Spiegelgasse 5, Seminarraum 05.001