(joint with Brian Lawrence and Akshay Venkatesh) One expects, speaking vaguely, that, among varieties of a certain type over Q, there are only finitely many with good reduction away from a specified set S. Such statements are often called “Shafarevich conjectures” -- when the varieties are abelian varieties, this actually was conjectured by Shafarevich, and is no longer a conjecture. We cannot prove this. But we do prove that, in some degree of generality, the ones with good reduction away from S are *sparse*, in the sense that the number of such with height less than B grows more slowly than any power of B. The key tool is the use of theorems of Heath-Brown type which provide bounds on points of bounded height on varieties which are uniform in the variety. One way to summarize what we prove is that varieties with large fundamental group have sparse integral points (which conforms with a story we know very well for curves.) Because people at Penn love the section conjecture, and so do I, I will explain how the whole story can be thought of in terms of low-height points repelling each other in the profinite topology afforded by thinking of points as Galois-theoretic sections.

### Algebra Seminar

Monday, March 28, 2022 - 2:30pm

#### Jordan Ellenberg

University of Wisconsin