p-adic analysis
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| p-adic analysis | |
|---|---|
| Name | p-adic analysis |
| Discipline | Mathematics |
| Subdiscipline | Number theory; Algebraic geometry; Analysis |
| Notable persons | Kurt Hensel, Jean-Pierre Serre, André Weil, Kurt Mahler, John Tate, Bernard Dwork, Serge Lang, Jean-Michel Fontaine, Alexandre Grothendieck, Haruzo Hida, Barry Mazur, Gerd Faltings, Ken Ribet, Pierre Deligne |
p-adic analysis is the study of analysis carried out over fields complete with respect to non-Archimedean absolute values induced by primes. It develops analogues of real and complex analytic concepts—such as limits, continuity, power series, integration, and differential equations—within the context of Kurt Hensel's constructions and subsequent arithmetic frameworks by figures like John Tate and André Weil. The subject interacts deeply with arithmetic geometry, representation theory, and algebraic number theory via tools introduced by Jean-Pierre Serre, Bernard Dwork, and Alexandre Grothendieck.
Introduction
The modern subject traces to Kurt Hensel's 1890s introduction of completions of the rationals at primes and owes development to contributions from Jean-Pierre Serre, John Tate, and André Weil. Foundational work by Bernard Dwork on zeta functions, Kurt Mahler on interpolation, and the cohomological frameworks of Alexandre Grothendieck and Pierre Deligne integrated p-adic methods into Gerd Faltings's and Ken Ribet's advances in arithmetic geometry. Institutional centers such as Institute for Advanced Study, École Normale Supérieure, and Princeton University have hosted many contributors, including Barry Mazur and Haruzo Hida.
p-adic Numbers and Valuations
p-adic fields arise by completing Carl Friedrich Gauss's rationals with respect to a non-Archimedean valuation associated to each prime, following Kurt Hensel. Key constructions include the field Q_p, its ring of integers, and extensions analyzed by Jean-Pierre Serre and John Tate. Ramification theory, local class field theory, and Galois representations were shaped by Emil Artin, Helmut Hasse, and later by Jean-Michel Fontaine and Pierre Deligne in the study of potentially semi-stable representations. Important invariants and notions—valuation rings, residue fields, inertia groups, and decomposition groups—play roles in works by André Weil and Serge Lang.
Topology and Metric Properties
Completeness, ultrametric inequalities, and totally disconnected topologies characterize p-adic spaces studied by Kurt Hensel and elaborated by Jean-Pierre Serre. Balls are both open and closed, and compactness criteria tie to Tate's uniformization and local analytic geometry developed by John Tate and Alexandre Grothendieck. Rigid analytic spaces and Berkovich spaces emerged from problems considered by John Tate, Vladimir Berkovich, and Pierre Deligne to provide workable notions of connectedness and analyticity in non-Archimedean settings, influencing later work by Gerd Faltings.
Analytic Functions over p-adic Fields
Power series, analytic continuation, and Weierstrass preparation-type results in the non-Archimedean context were advanced by Kurt Mahler, Bernard Dwork, and John Tate. Coleman integration and p-adic modular forms studied by Robert Coleman and Haruzo Hida connect analytic functions on rigid spaces to arithmetic of Barry Mazur and Ken Ribet. Theories of p-adic L-functions, pioneered by Kubota–Leopoldt and extended by Serge Lang and Haruzo Hida, relate values of analytic functions to special values and congruences in works of Pierre Deligne and Gerd Faltings.
Integration and Measures
p-adic measures, Haar measures on local fields, and integration techniques were systematized in the work of John Tate, Serge Lang, and Bernard Dwork. Applications include p-adic zeta functions, Iwasawa theory as developed by Kenkichi Iwasawa, and p-adic families of modular forms analyzed by Haruzo Hida and Barry Mazur. Measures on rigid analytic spaces and distributions in the sense of John Tate underpin constructions by Pierre Deligne and Jean-Pierre Serre linking cohomology theories to special values of L-functions.
p-adic Differential Equations and Dynamical Systems
Differential equations over non-Archimedean fields, differential modules, and Frobenius structures were studied by Bernard Dwork, Gerald Christol, and Philippe Robba, contributing to p-adic cohomology theories used by Pierre Deligne and Gerd Faltings. p-adic dynamical systems, iteration of rational functions, and arboreal Galois representations intersected with work by Barry Mazur, Ken Ribet, and researchers at Institute for Advanced Study; these topics connect to non-Archimedean potential theory explored by Robert Rumely.
Applications in Number Theory and Arithmetic Geometry
p-adic analytic methods are central to modern results in arithmetic geometry, including proofs and refinements related to the Taniyama–Shimura conjecture and modularity theorems advanced by Andrew Wiles and Richard Taylor. Iwasawa theory, Euler systems, and the study of Selmer groups use tools from Kenkichi Iwasawa, Barry Mazur, and Kazuya Kato. p-adic Hodge theory, developed by Jean-Michel Fontaine and extended by Gerd Faltings and Pierre Deligne, relates de Rham, crystalline, and étale cohomologies central to proofs by Andrew Wiles and subsequent work by Richard Taylor and Ken Ribet.