Félix Baril Boudreau (University of Luxembourg): Arithmetic rank bounds for abelian varieties over function fields

Abstract:
The set of rational points of an elliptic defined over a global field is a finitely generated Abelian
group by the Mordell-Weil theorem. Finding the precise rank of a given elliptic curve is difficult.
When the elliptic curve is defined over the function field K of a curve defined over a field k, it
follows from the work of Grothendieck, Ogg and Shafarevich that there is an upper bound on this
rank which depends on the genus of the curve and on bad reduction data. This bound is geometric
in nature in the sense that it holds for all finite constant extensions of the function field K. When k
is finite, Armand Brumer obtained in 1992 a bound depending on the arithmetic of that finite field.
Results by Joseph Silverman when k has characteristic 0 lead Douglas Ulmer to ask if there were
bounds of arithmetic nature in that situation as well. In 2019, Jean Gillibert and Aaron Levin ans-
wered this question by the affirmative.
In this talk, we present a joint work with Gillibert and Levin where we prove such arithmetic
bound existence for Abelian varieties in general. When specialized to elliptic curves, our result
substantially improves on the previous one.

Before this lecture, starting at 4:45 p.m., there will be coffee and tea in Room 1404.

Everyone is warmly invited.

Signed Dr. Sebastian Petersen