Bloch–Kato

Bloch–Kato
Name	Bloch–Kato
Field	Algebraic number theory
Notable conjecture	Bloch–Kato conjecture
Related people	Spencer Bloch; Kazuya Kato; Barry Mazur; Pierre Deligne; Jean-Pierre Serre

Contents

Bloch–Kato.

Definition and Overview

Bloch–Kato denotes a foundational nexus in algebraic number theory, connecting conjectures about Galois cohomology, motivic cohomology, K-theory, Hodge theory and p-adic Hodge theory through the work of Spencer Bloch and Kazuya Kato; key players include John Tate, Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne and Barry Mazur who influenced the surrounding landscape. The Bloch–Kato statements relate special values of L-functions, regulators in K-theory, and Selmer groups studied by Andrew Wiles, Richard Taylor, Vladimir Voevodsky and Aise Johan de Jong and connect to results by Gerd Faltings, Ken Ribet and Alexander Beilinson. Central themes invoke the work of Iwasawa, Kazuya Kato, Spencer Bloch, Kazuya Kato (again), Jean-Marc Fontaine, Jannsen, and later contributions by Terry Tao-adjacent collaborators, while influencing programs around Langlands program, Birch and Swinnerton-Dyer conjecture, Beilinson conjectures and Bloch–Kato conjecture-related theorems.

Origins trace to conjectural frameworks proposed by Spencer Bloch and Kazuya Kato building on insights of John Tate, André Weil, Alexander Grothendieck and Jean-Pierre Serre. Early antecedents include Milnor K-theory by John Milnor, reciprocity laws of Emil Artin, and the cohomological formalism advanced by Henri Cartan, Samuel Eilenberg and Henri Poincaré with later structural input from Pierre Deligne and Grothendieck. Major contributors who advanced proofs and techniques include Vladimir Voevodsky for motivic homotopy, Marcus Rost for norm varieties, Matilde Marcolli for arithmetic applications, Kazuya Kato for p-adic methods, Barry Mazur for deformation theory, Andrew Wiles and Richard Taylor for modularity links, and Jan Nekovář for Selmer complexes. Subsequent progress involved teams around Florian Pop, Christopher Skinner, Wintenberger, Tetsushi Ito, Weil group researchers, and institutes like Institute for Advanced Study, Princeton University, École Normale Supérieure and Clay Mathematics Institute.

The Bloch–Kato conjecture predicts precise isomorphisms between graded pieces of Milnor K-theory mod n and Galois cohomology groups, generalizing the Milnor conjecture proved by Vladimir Voevodsky, and connecting with the Tate conjecture and Beilinson conjectures; protagonists in the statement include Spencer Bloch, Kazuya Kato, and later proofs relied on work by Voevodsky, Marcus Rost, Fabien Morel and Charles Weibel. The theorem version arose after proofs of the mod-2 Milnor conjecture by Voevodsky and the full norm residue isomorphism by collaboration among Rost, Voevodsky, Weibel and others, which tied into the Bloch–Kato conjecture via techniques of motivic cohomology, cycle class maps, and Gersten complexes. These developments interacted with the Langlands program through modularity results of Andrew Wiles and Richard Taylor and with Iwasawa theory advanced by Kenkichi Iwasawa and Ruochuan Liu.

The framework uses Galois cohomology, Milnor K-theory, motivic cohomology, etale cohomology, p-adic Hodge theory, and regulator maps; foundational tools were developed by Alexander Grothendieck, Jean-Pierre Serre, Pierre Deligne, Jean-Marc Fontaine and Barry Mazur. Central constructions include norm varieties introduced by Marcus Rost, cycle complexes by Spencer Bloch, and motivic complexes advanced by Vladimir Voevodsky and Fabien Morel. The conjectures and theorems also invoke the theory of Selmer groups shaped by John Coates, Robert Greenberg, Jan Nekovář, and deformation theory influenced by Barry Mazur and Richard Taylor. Analytic inputs derive from L-functions studied by Goro Shimura, André Weil, and Atle Selberg, while categorical and homotopical methods trace to Daniel Quillen, Quillen, Grothendieck and Bousfield developments in algebraic K-theory.

Major milestones include the proof of the Milnor conjecture by Vladimir Voevodsky, the Rost–Voevodsky proof of the Bloch–Kato norm residue isomorphism using Rost's norm varieties and Voevodsky's motivic methods, and subsequent refinements by Weibel and collaborators. Techniques draw on motivic homotopy theory of Voevodsky and Fabien Morel, cycle-theoretic ideas from Spencer Bloch, and cohomological machinery developed by Jean-Pierre Serre and Pierre Deligne. Related results include work on the Beilinson conjectures by Alexander Beilinson, the Birch and Swinnerton-Dyer conjecture investigations by John Coates and Andrew Wiles, and applications in Iwasawa theory pursued by Kenkichi Iwasawa and Ralph Greenberg. Ongoing progress involves researchers such as Matthew Morrow, Kazuya Kato (again), Hélène Esnault, Aise Johan de Jong and collaborations at MSRI, IHES and Mathematical Sciences Research Institute.

Consequences touch L-functions, special value conjectures of Beilinson and Bloch, explicit reciprocity laws of Emil Artin-type generalized by Kazuya Kato, and implications for the Birch and Swinnerton-Dyer conjecture and Iwasawa theory studied by Andrew Wiles, Mazur and Kenkichi Iwasawa. The Bloch–Kato framework informs work in the Langlands program as pursued by Robert Langlands, Michael Harris, Richard Taylor and Pierre Deligne and influences arithmetic geometry at institutions like Harvard University, Princeton University, Universität Bonn and École Normale Supérieure. It also underpins explicit computations in arithmetic of elliptic curves overseen by John Cremona, insights into motives developed by Yves André and Uwe Jannsen, and algorithmic approaches connected to Bhargava-style enumerations and computational projects at L-functions and Modular Forms Database institutions.