Effective quantum unique ergodicity for Hecke-Maass newforms and Landau-Siegel zeros

TL;DR

This paper demonstrates that Landau-Siegel zeros either do not exist or imply effective quantum unique ergodicity for certain automorphic forms, showing a zero repulsion phenomenon among L-functions.

Contribution

It establishes a link between the non-existence of Landau-Siegel zeros and effective quantum unique ergodicity for Hecke-Maass forms, providing a new zero repulsion result.

Findings

Landau-Siegel zeros do not exist or imply effective QUE for Hecke-Maass forms

Landau-Siegel zeros repel zeros of other automorphic L-functions from the line Re(s)=1

Effective convergence rates are achieved in QUE under these conditions

Abstract

We show that Landau-Siegel zeros for Dirichlet $L$-functions do not exist or quantum unique ergodicity for $\mathrm{GL}_2$ Hecke-Maass newforms holds with an effective rate of convergence. This follows from a more general result: Landau-Siegel zeros of Dirichlet $L$-functions repel the zeros of all other automorphic $L$-functions from the line $\mathrm{Re}(s)=1$.