Effective equidistribution of semisimple adelic periods and representations of quadratic forms
Manfred Einsiedler, Elon Lindenstrauss, Amir Mohammadi, Andreas Wieser
TL;DR
The paper develops an effective, quantitative theory for the equidistribution of semisimple adelic periods on compact quotients and applies it to a local-global principle for representations of integral quadratic forms. It combines dynamical methods (including the linearization technique and an effective closing lemma) with an effective Greenberg theorem, geometric invariant theory, and reduction theory to produce polynomially decaying error terms controlled by orbit complexity and ambient volume. The authors obtain an effective local-global statement for primitive representations in codimensions at least three and supply explicit bounds in terms of discriminants and minima, improving previous ineffective results. The work advances the quantitative understanding of adelic homogeneous dynamics and yields practical, explicit bounds for representations of quadratic lattices across number fields, with substantial implications for number theory and arithmetic geometry.
Abstract
We prove an effective equidistribution theorem for semisimple closed orbits on compact adelic quotients. The obtained error depends polynomially on the minimal complexity of intermediate orbits and the complexity of the ambient space. The proof uses dynamical arguments, property $(τ)$, Prasad's volume formula, an effective closing lemma, and a novel effective generation result for subgroups. The latter in turn relies on an effective version of Greenberg's theorem. We apply the above 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.