Counting rational points on varieties with large fundamental group

A nonsingular projective curve of genus at least 2 on a number field admits only finitely many rational points. Elliptic curves might have infinitely many rational points (as the projective line does), but “way less” than the projective line. In a joint work with Y. Brunebarbe, inspired by a recent result of Ellenberg-Lawrence-Venkatesh, we prove an analogous statement in higher dimension: projective varieties with large fundamental group in the sense of Kollár have “way less” rational points than Fano varieties.