Edgar Assing - On the unipotent mixing conjecture

Abstract

It is a classical fact that low-lying horocycles equidistribute in the modular curve. Going a step further, one can consider shifted pairs of low-lying horocycles and ask whether they equidistribute simultaneously. This question was recently addressed by Blomer and Michel using a beautiful mix of tools from number theory and dynamical systems. In this talk, I will explain the work of Blomer and Michel and discuss some extensions thereof.

Introduction

We look at $S{L}_2(\mathbb{Z})$ with $SO(2)$ quotiented out. (The hyperbolic plane?)

Low lying horocycles (parallel to the axis) are well studied. We look at the Fundamentalbereich of $\Gamma$.

Question: Do low-lying horocycles localize or spread out over the fundamental domain?

On a low-lying horocycle we pick $q\in \mathbb{N}$ points with a (hyperbolical?) distance of $1$. As $q$ goes to infinity $\frac{1}{q}]su{m}_{x\in H_q}f(x)$ goes to something about the volume. (Whatever $f$ we pick, I think)

A nice paper proving this is Burnn-Shapira-Yu 2022.

We do some stuff I missed.

We move to a simultaneous equidistant problem. We define a set $H_{q,b}$ dependet on $q,b$ which is a collection of pairs with both parts sitting in $\Gamma \backslash S{L}_2(\mathbb{Z})$. The question is, does this equidistribute and how does it look like?

We can define a lattice based on $b$ and $q$, then we take the minimum of two points in the lattice. This minimum then gives us a condition for when the points are equidistant. Einsiedler-Lindenstauss-Michaael 2018 already gave some examples for equidistribution. A continuous version of this result was given by Burris.

The last part of the talk gave the proof for this. I honestly couldn’t care to listen.

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