IUMJ

Title: The average rank of elliptic $n$-folds

Authors: Remke Kloosterman

Issue: Volume 61 (2012), Issue 1, 131-146

Abstract:

Let $V/\mathbf{F}_q$ be a variety of dimension at least $2$. We show that the density of elliptic curves $E/\mathbf{F}_q(V)$ with positive rank is zero if $V$ has dimension at least $3$ and is at most $1-\zeta_V(3)^{-1}$ if $V$ is a surface.