AAG Seminar: Skolem's conjecture and Higher reciprocity law for the polynomial ring over ultra-finite fields, using model theory

Abstract: In 1938, Skolem conjectured that the group $\text{SL}_2(\mathbb{Z})$ can be parametrized using a polynomial mapping, i.e., there exists a polynomial mapping $P : \mathbb{Z}^n \to \text{SL}_2(\mathbb{Z})$ for some positive integer $n$ such that $\text{SL}_2(\mathbb{Z}) = P(\mathbb{Z}^n)$. Vaserstein (Annals of Math, 2010) confirmed Skolem's conjecture for $n  = 46$. One of the key ingredients in his proof is quadratic reciprocity law, combining with classical algebraic K-theory. Larsen and I (2021) prove a generalization of Skolem's conjecture over the ring of integers of an arbitrary number field, also using reciprocity law. In this talk, I will discuss my recent work, using mainly ultraproducts in model theory to prove a certain higher reciprocity law for the polynomial ring over an ultraproduct of distinct finite fields. Such a higher reciprocity law will be then used to prove a variation of Skolem's conjecture for rational function fields $K[x]$, where $K$ is an (possibly infinite degree) algebraic extension of the rationals. If time permits, I will talk about the applications of such a higher reciprocity law to Hilbert tenth problem and the Hasse principle.

Host: Statistics and Data Science Department, Washington University in St. Louis