Title: Hilbert's Irreducibility Theorem via random walks
Abstract: Let G be a linear algebraic group over a number field K, and let f: V --> G be a cover of finite degree, at least two. We wish to show that "many" points of G(K) do not lift to points of V(K), and that "many" points of G(K) have K-irreducible fibre. This is analogous to Hilbert's Irreducibility Theorem and, of course, depends on the meaning of the word "many".
We will focus on the following model: Fix any finitely generated Zariski dense subgroup H of G(K), and perform a random walk on a Cayley graph of H. We will see that, under suitable necessary conditions on G and f, almost surely a long random walk hits elements having the above property. We will also see examples and variations of this result.
Joint work with Lior Bary-Soroker.