Adelic Integrable Systems

TL;DR

This paper develops a unified adelic integrable system framework that incorporates real and p-adic hyperbolic plane problems, linking integrable evolutions with number-theoretic L-functions.

Contribution

It introduces a novel approach connecting integrable systems with number theory through adelic structures and L-functions, extending the Zakharov-Shabat system.

Findings

Unified real and p-adic systems into an adelic framework

L-functions appear in the S-matrix of the integrable system

Preservation of number-theoretic properties over time

Abstract

Incorporating the zonal spherical function (zsf) problems on real and $p$-adic hyperbolic planes into a Zakharov-Shabat integrable system setting, we find a wide class of integrable evolutions which respect the number-theoretic properties of the zsf problem. This means that at {\it all} times these real and $p$-adic systems can be unified into an adelic system with an $S$-matrix which involves (Dirichlet, Langlands, Shimura...) L-functions.