Analytic Tate spaces and reciprocity laws

TL;DR

This paper introduces a functional analytic framework for Tate spaces, enabling the definition of symbols for analytic functions that satisfy Weil-type reciprocity laws in both complex and p-adic contexts.

Contribution

It develops a new category of topological vector spaces combining nuclear Frechet spaces and their duals, facilitating reciprocity law formulations.

Findings

Defined symbols for analytic functions satisfying reciprocity laws

Established the formalism in both complex and p-adic settings

Connected Tate space theory with Weil reciprocity laws

Abstract

We consider a functional analytic variant of the notion of Tate space, namely the category of those topological vector spaces which have a direct sum decomposition where one summand is nuclear Frechet space and the other is the dual of a nuclear Frechet. We show that, both in the complex and in the p-adic setting, one can use this formalism to define symbols for analytic functions which satisfy Weil-type reciprocity laws.