Abstract

Varieties, if they are at all complicated, are expected to have very few rational points. This might mean “there are only finitely many rational points” or “the rational points are contained in a proper closed subvariety.” Statements like these are extremely difficult to prove in any degree of generality, with Faltings’ finiteness theorem for rational points on high-genus curves a notable exception. In this talk, I’ll explain how to prove theorems about “sparsity” of rational points, a weaker notion which asks that the number of such with height less than B grows more slowly than any power of B; it turns out that theorems of this kind can be proven for many varieties appearing as moduli spaces (though these still make up a very special subclass among the varieties we’d like to know about.) The two main ingredients are: 1) a classical trick allowing us to “trade” a single equation that’s hard to solve for many equations that are easier to solve; 2) theorems of Heath-Brown type which provide bounds on points of bounded height on varieties which are uniform in the sense that they barely depend on what variety you’re studying. These are very useful when proving theorems about varieties of which we know almost nothing. This is joint work with Brian Lawrence and Akshay Venkatesh.

Speaker

I've been at Wisconsin since the fall of 2005. My field is arithmetic algebraic geometry: my specific interests include rational points on varieties, asymptotic enumeration of number fields and other arithmetic objects, incidence problems and algebraic methods in combinatorial geometry, Galois representations attached to varieties and their fundamental groups, representation stability and FI-modules, the geometry of large data sets, non-abelian Iwasawa theory, pro-p group theory, automorphic forms, stable cohomology of moduli spaces, the complex of curves, Hilbert-Blumenthal abelian varieties, Q-curves, Serre's conjecture, the ABC conjecture, and Diophantine problems related to all of the above.

for more info, please visit the webpage:

https://people.math.wisc.edu/~ellenber/