“The average Mordell-Weil rank of elliptic surfaces over number fields”
Martedì 20 Gennaio 2026, ore 14:30 - Aula 2AB45 - Remke Kloosterman (Università degli Studi di Padova)
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 proving a conjecture of Alex Cowan. The proof consists of two parts, the first part relies on work by André and Maulik-Poonen on the jump loci of the Picard Number in flat families. This is sufficient to prove that the average Mordell-Weil in the family equals the Mordell-Weil rank of the generic fiber. The second part of the proof uses an argument involving quadratic twists in order to show that the generic Mordell-Weil rank of elliptic surfaces over a number field with fixed topological invariants equals zero.
