Hot Topics: Interactions between Harmonic Analysis, Homogeneous Dynamics, and Number Theory: Effective equidistribution of semisimple adelic periods and representations of quadratic forms
Presenter
March 3, 2025
Keywords:
- Projection theorems
- Unipotent flows
- Effective equidistribution
- Diophantine approximation
- decoupling theory
MSC:
- 11F66 - Langlands $L$L-functions one variable Dirichlet series and functional equations
- 11J83 - Metric theory
- 22E30 - Analysis on real and complex Lie groups [See also 33C80
- 43-XX]
- 28A80 - Fractals [See also 37Fxx]
- 37A17 - Homogeneous flows [See also 22Fxx]
- 37C85 - Dynamics induced by group actions other than $\mathbb{Z}$\mathbb{Z} and $\mathbb{R}$\mathbb{R} and $\mathbb{C}$\mathbb{C} [See mainly 22Fxx
- and also 32M25
- 57R30
- 57Sxx]
- 42B15 - Multipliers for harmonic analysis in several variables
- 42B20 - Singular and oscillatory integrals (Calderón-Zygmund
- etc.)
Abstract
We discuss an effective equidistribution theorem for semisimple periodic orbits on compact adelic homogeneous spaces. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. We apply this equidistribution theorem to the problem of establishing a local-global principle for representations of integral quadratic forms, improving the codimension assumptions and providing effective bounds in a theorem of Ellenberg and Venkatesh. This is joint work with Manfred Einsiedler, Elon Lindenstrauss, and Amir Mohammadi.