The average rank of elliptic $n$-folds Remke Kloosterman 46J1054C2019B10real function algebrasBass stable ranktopological stable rankextension of maps 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. Indiana University Mathematics Journal 2012 text pdf 10.1512/iumj.2012.61.4540 10.1512/iumj.2012.61.4540 en Indiana Univ. Math. J. 61 (2012) 131 - 146 state-of-the-art mathematics http://iumj.org/access/