Abstracts for Conference on "Arithmetic Statistics, Automorphic Forms and Ergodic Methods"

Low-lying horocycles are known to equidistribute on the modular curve. Here we consider the joint distribution of two low-lying horocycles of different speeds in the product of two modular curves and show equidistribution under certain necessary diophantine conditions. The techniques involve a combination of arithmetic, automorphic and ergodic techniques. This is joint work with Philippe Michel.

We reprove the main equidistribution instance in the Ferrero-Washington proof of the vanishing of cyclotomic Iwasawa $\mu$-invariant, based on the ergodicity of a certain $p$-adic skew-product extension dynamical system that can be identified with Bernoulli shift (joint with Bharathwaj Palvannan).

The roots of quadratic congruences are a classical object of study, and largely due to applications in analytic number theory, their statistical distribution is well-studied. On the other hand, very little is known about roots of higher degree congruences beyond their equidistribution. We discuss some recent results (joint with Jens Marklof and Zonglin Li) in the quadratic case and some results related to the cubic case.

Quantum modular forms are functions $f$ defined on the rationals whose period functions, such as $\psi(x):= f(x) - x^{-k} f(-1/x)$ (for level 1), satisfy some continuity properties. In the case of $k=0$, $f$ can be interpreted as a Birkhoff sum associated with the Gauss map. In particular, under mild hypotheses on $\psi$, one can show convergence to a stable law. If $k$ is non-zero, the situation is rather different and we can show that mild conditions on $\psi$ imply that $f$ itself has to exhibit some continuity property.