Mazur's Program B

Mazur's Program B
Name	Mazur's Program B
Subject	Number theory
Date	1990s–2000s
Main proponent	Barry Mazur
Related	Iwasawa theory, Galois representations, modular forms

Mazur's Program B

Mazur's Program B is a research agenda proposed by Barry Mazur that reorients parts of number theory around deformation-theoretic analysis of Galois representations, aiming to relate diophantine problems to congruences among automorphic objects. Originating in talks and writings by Mazur during the 1990s and early 2000s, Program B influenced work by researchers connected with Andrew Wiles, Richard Taylor, Jean-Pierre Serre, and others in the study of modular forms, Iwasawa theory, and the Langlands program. The program frames concrete conjectures and strategies that tie the arithmetic of elliptic curves, abelian varieties, and higher-dimensional motives to the structure of deformation rings and Hecke algebras.

Background and Motivation

Program B arose from interactions among the study of elliptic curves over Q, the proof of the Taniyama–Shimura–Weil conjecture as pursued by Andrew Wiles and Richard Taylor, and foundational work of Jean-Pierre Serre on modularity. Mazur synthesized insights from Iwasawa theory of Kenkichi Iwasawa, the deformation theory of Galois representations developed by Barry Mazur himself, and structural properties of Hecke algebras studied by Hida and Mazur to propose a unified program. Influences also include the modularity lifting techniques of Freitag, the reciprocity principles of the Langlands program, and structural results of Grothendieck on étale cohomology.

Statement of Program B

Program B proposes studying arithmetic objects by analyzing universal deformation rings of residual Galois representations and comparing them to algebras generated by correspondingly chosen Hecke operators. It posits that for suitable residual representations arising from elliptic curves, Hilbert modular forms, or automorphic representations, there exist natural maps between universal deformation rings and localized Hecke algebras that are often isomorphisms. The program articulates conjectural compatibilities inspired by the Taylor–Wiles method, expects control of Selmer groups in the style of Mazur and Greenberg, and anticipates applications to questions about rational points on modular curves, Shimura varieties, and deformation-theoretic approaches to the Birch and Swinnerton-Dyer conjecture.

Connections to Galois Representations and Automorphic Forms

Central to Program B are continuous odd two-dimensional Galois representations of Gal(Qbar/Q) with coefficients in finite fields, their lifts to p-adic representations, and the conjectured associations with modular forms and automorphic representations predicted by Langlands correspondences. The program leverages local-global compatibility results exemplified by Fontaine–Mazur conjecture, the Breuil–Mézard conjecture, and the modularity lifting theorems of Wiles, Taylor–Wiles, and later work by Kisin and Calegari. It connects deformation rings to spaces of modular forms constructed by Deligne, Serre, and Shimura, and links congruences between eigenforms to arithmetic phenomena studied by Ribet and Mazur.

Progress and Key Results

Work inspired by Program B includes the Taylor–Wiles modularity lifting breakthrough proving modularity for semistable elliptic curves over Q and later generalizations by Kisin, Calegari, and Geraghty to more general settings. Results verifying R = T theorems (isomorphisms between deformation rings R and Hecke algebras T) have been established in numerous cases by Wiles, Taylor, Diamond, Kisin, and Gee. Advances toward the Breuil–Mézard conjecture and results on patched modules by Emerton, Gee, and Paskunas realize key Program B aspirations. Applications include proofs of modularity lifting for Hilbert modular forms, progress on level-lowering by Ribet, and arithmetic consequences for rational points on modular curves and the structure of Selmer groups explored by Mazur and Greenberg.

Methods and Techniques

Program B employs deformation theory of Galois representations as developed by Barry Mazur, patching methods inspired by Taylor–Wiles, and commutative algebra techniques like Cohen–Macaulay and Gorenstein properties analysis. It uses Hecke algebras acting on completed cohomology as in work of Emerton, local p-adic Hodge theory from Fontaine and Colmez, and patching of modules over power series rings employed by Kisin and Calegari–Geraghty. The program frequently invokes results from the theory of Shimura varieties as developed by Deligne and Kottwitz, rigid analytic geometry contributions from Berkovich, and categorical techniques linked to Vladimir Drinfeld and Pierre Deligne.

Open Problems and Conjectures

Open directions central to Program B include proving broad classes of R = T theorems for higher-dimensional Galois representations and non-self-dual automorphic contexts as envisioned by Langlands, resolving instances of the Breuil–Mézard conjecture and the Fontaine–Mazur conjecture, and extending patching and automorphy lifting to settings treated by Calegari–Geraghty and Scholze. Further goals include establishing modularity over general number fields along lines suggested by Taylor, refining links to the Birch and Swinnerton-Dyer conjecture and Bloch–Kato conjecture explored by Bloch, Kato, and Tate, and developing p-adic local Langlands correspondences in higher rank as pursued by Scholze and Emerton.

Category:Number theory