Lubin–Tate theory

Lubin–Tate theory
Name	Lubin–Tate theory
Field	Number theory
Introduced	1965
Main contributors	Jonathan Lubin, John Tate
Related	Local class field theory, Formal group laws, Deformation theory

Lubin–Tate theory.

Lubin–Tate theory is a framework in algebraic number theory connecting local field arithmetic with one-dimensional formal group laws to explicitly construct abelian extensions and study deformation spaces, linking ideas from local class field theory, Galois theory, and algebraic geometry. It was developed by Jonathan Lubin and John Tate and has influenced work of many mathematicians in areas spanning p-adic Hodge theory, Iwasawa theory, and the theory of moduli spaces, interacting with research by figures associated to institutions such as the Institute for Advanced Study and universities like Harvard University and University of Chicago.

Introduction

Lubin–Tate theory arose in the context of explicit constructions for the maximal abelian extension of a nonarchimedean local field such as a finite extension of Q_p, refining concepts from Kummer theory and explicit class field methods used by mathematicians around Emil Artin and Helmut Hasse. The theory uses endomorphisms of one-dimensional formal groups defined over the ring of integers of a local field to generate towers of extensions governed by Galois groups and automorphisms similar to the cyclotomic constructions used by Srinivasa Ramanujan's successors. The approach complements cohomological techniques developed by researchers at places like École Normale Supérieure and Princeton University and connects to the work of Alexander Grothendieck on deformation problems.

Formal groups and formal group laws

Formal groups are specified by formal power series satisfying associativity and identity analogues first studied in the context of Elliptic curve formal groups and by contributors associated to University of Cambridge and University of Bonn. A one-dimensional formal group law over a complete discrete valuation ring ties to endomorphism rings resembling structures explored by figures at Max Planck Institute for Mathematics and interacts with concepts from Jacob Lurie's circle and influences research at Stony Brook University. Formal group laws give local analytic parametrizations analogous to coordinate charts in the work of André Weil and Oscar Zariski, and their study drew on methods familiar from Dieudonné theory and the classification problems tackled by scholars at Institut des Hautes Études Scientifiques.

Lubin–Tate formal groups and modules

A Lubin–Tate formal group is a one-dimensional formal group law over the ring of integers O_K of a local field K together with a chosen endomorphism analogous to multiplication by a uniformizer; this construction builds explicit O_K-module structures and provides local analogues of cyclotomic units used in Kummer theory. The theory formalizes choices akin to uniformizers studied in research at University of California, Berkeley and uses techniques related to the study of Tate modules that were advanced by researchers at University of Illinois Urbana–Champaign and Columbia University. Lubin–Tate modules yield endomorphism rings connected to local reciprocity maps derived from the work of John Tate and colleagues at institutions like Massachusetts Institute of Technology.

Local class field theory via Lubin–Tate

Lubin–Tate theory gives an explicit description of the local reciprocity map of local class field theory by associating to each uniformizer a tower of abelian extensions whose Galois group is canonically isomorphic to the multiplicative group of the local field, echoing constructions dating to Richard Dedekind and Kurt Hensel. The explicit reciprocity connects to the reciprocity laws developed in the tradition of David Hilbert and was influential for later work by mathematicians at ETH Zurich and University of Bonn. Through these constructions, Lubin–Tate theory complements cohomological approaches of researchers at University of Cambridge and Yale University and interacts with explicit reciprocity computations undertaken by scholars at Kyoto University and University of Tokyo.

Extensions and Galois groups constructed by Lubin–Tate

Given a Lubin–Tate formal group, adjoining its torsion points produces totally ramified abelian extensions whose Galois groups are described via endomorphism actions, a mechanism employed in analogues of cyclotomic extensions studied by investigators at University of Oxford and University of Paris. These towers are instrumental in the study of local Galois representations, influencing the work of mathematicians at Harvard University and Princeton University, and tie into the structure theory of pro-p groups examined at University of Michigan and University of Texas at Austin. The explicit nature of the constructed extensions allowed refinements of norm-compatible systems used in studies at University of Washington and University of Wisconsin–Madison.

Deformation theory and moduli of formal groups

Deformations of one-dimensional formal group laws, as developed in Lubin–Tate theory, lead to universal deformation rings and to moduli spaces that were precursors to broader deformation theories advanced by Grothendieck and later examined in the work of Barry Mazur and researchers at Harvard University and Columbia University. The formal moduli spaces relate to the study of local models in Shimura variety theory pursued at Université Paris-Saclay and University of British Columbia, and their structure interfaces with the development of p-adic period mappings by groups at Institut de Mathématiques de Jussieu and IHÉS. These deformation spaces underpin aspects of the local Langlands correspondence studied by contributors at Institute for Advanced Study and research teams at Rutgers University.

Applications and generalizations

Lubin–Tate theory has applications across Iwasawa theory, p-adic Hodge theory, and explicit reciprocity laws, influencing progress at institutions like Mathematical Sciences Research Institute and University of Chicago and informing computational approaches used by teams at Microsoft Research and Simons Foundation-funded collaborations. Generalizations include higher-dimensional formal groups, links to Drinfeld module theory developed at Institute of Mathematics of the Polish Academy of Sciences, and interactions with nonabelian extensions studied by groups at Imperial College London and University of Cambridge. The framework continues to inform modern research programs in arithmetic geometry and representation theory pursued internationally at places such as Rice University, Brown University, Duke University, and University of California, Los Angeles.

Category:Number theory