Activities This Week

Given an integral vector, there are several geometric and arithmetic objects one can attach to it. For example, its direction (as a point on the unit sphere), the lattice obtained by projecting the integers to the othonormal hyperplane to the vector, and the vector of residues modulo a prime p to name a few. In this talk I will discuss results pertaining to the statistical properties of these objects as we let the integral vector vary in natural ways.

אנסה להסביר כיצד נקודת מבט הסתברותית עוזרת לקבל באופן אלמנטרי תובנות שונות בתורת המספרים.

If f is a cuspidal modular eigenform of weight k>1, Ribet proved that its associated p-adic Galois representation is irreducible for all primes. More generally, it is conjectured that the p-adic Galois representations associated to cuspidal automorphic representations of GL(n) should always be irreducible.

In this talk, I will prove a version of this conjecture for low weight, genus 2 Siegel modular forms. These two-dimensional analogues of weight 1 modular forms are, conjecturally, the automorphic objects that correspond to abelian surfaces.

A topological Markov shift is the set of two sided inifinite paths in a finite directed graph endowed with the product topology and with the left shift acting on this space. The automorphisms of the space are the shift commuting self-homeomorphisms. Wagoner realized the automorphism group of a topological Markov shift as the fundamental group of a certain CW complex. This construction has been crucial in many results regarding automorphisms and isomorphism in symbolic dynamics. We give a simplified construction of this complex, which also works in more general contexts, and sketch some applications.