Félix Baril Boudreau, University of Lethbridge
It is known since the works of Ogg (1962) and Shafarevich (1961) (under tameness assumptions), followed by Grothendieck (1968), that the rank of a given abelian variety over the function field of a curve is bounded by a quantity which depends on the genus of the base curve and on reduction data. This bound is "geometric" in nature. In particular, it holds if we replace the constant field by its algebraic closure.
Ulmer asked in 2004 if, for an elliptic curve, there was an arithmetic bound that could improve on the geometric one. This question recently obtained a positive answer (Gillibert and Levin, 2022).
In this talk, we present a similar arithmetic refinement of the geometric bound for higher-dimensional abelian varieties. When specialized to elliptic curves, we improve on Gillibert-Levin's bound. Time permitting, we will discuss some consequences of our result.
This is joint work with Jean Gillibert and Aaron Levin.
MC 5501