Daniele Garzoni, USC

Abstract: Let G be a linear algebraic group over a number field K (e.g., SLn), and let f: V → G be a cover of finite degree. Perform a long random walk on a Cayley graph of a fixed finitely generated subgroup of G(K) (e.g., SLn(Z)).  Should you expect to hit elements g whose fibre f-1(g) is K-irreducible?

After giving motivation, we will see that the answer is Yes, under some conditions on G and f. This represents a quantitative version of Hilbert's irreducibility theorem for linear algebraic groups. Joint work with Lior Bary-Soroker.