What are the common properties of all finite fields expressible in first order logic (in the language of rings)? In 1968 James Ax, building on number theoretic results, answered this question and initiated the study of pseudofinite fields, the infinite models of the theory of finite fields. Ehud Hrushovski generalised Ax's theorem in two directions passing to richer structures: Finite fields equipped with an additive character and the algebraic closure of finite fields with a symbol for the Frobenius. In this talk, we will start by giving some basic notions from model theory as well as Ax's theorem and then show how to combine Hrushovski's results into one theory.
Let f_t(z)=z^2+t. For any rational number z, let S_z be the set of rational numbers t such that z is preperiodic for f_t. In this talk, we will discuss a uniform result regarding the sizes of S_z over rational numbers z. In order to do it, we will also discuss how to find the set of rational points on a specific curve of genus 4. This is a joint work with Michael Stoll.
By a theorem of Borel, any holomorphic map from a complex algebraic variety to the moduli space of abelian varieties (and more generally to an arithmetic variety) is in fact algebraic. A key input is to prove that any holomorphic map from a product of punctured disks to such an arithmetic variety does not have any essential singularities. In this talk, I'll discuss a p-adic analogue of these results. This is joint work with Ananth Shankar and Xinwen Zhu (with an appendix by Anand Patel).