Bloch group
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| Bloch group | |
|---|---|
| Name | Bloch group |
| Field | Algebraic K-theory; Algebraic geometry |
| Introduced | 1970s |
| Notable people | Spencer Bloch; Don Zagier; Andrei Suslin; Maxim Kontsevich; Alexander Goncharov |
Bloch group is a finitely generated abelian group associated to a field that encodes deep arithmetic and geometric information about values of polylogarithms, regulators, and algebraic K-theory of fields. It was introduced in work of Spencer Bloch motivated by conjectures of Beilinson and insights from computations of special values of L-functions and dilogarithms by Don Zagier and others. The Bloch group connects explicit functional equations of the dilogarithm to algebraic cycles, providing a bridge between algebraic K-theory, motives, and the analytic theory of special functions.
Definition and basic properties
For a field F one defines a pre-Bloch group built from the free abelian group on F\{0,1\} modulo relations coming from the five-term functional equation of the classical dilogarithm; the Bloch group is the quotient of this pre-Bloch group by torsion arising from known trivial relations. Foundational results due to Spencer Bloch, Andrei Suslin, and Don Zagier identify structural properties such as finite generation for number fields and functoriality with respect to field extensions like those studied by Emil Artin and Alexander Grothendieck. Key properties relate to exact sequences connecting Bloch groups to versions of Milnor K-theory and Quillen algebraic K-theory and reflect comparisons with Hodge structures investigated by Pierre Deligne and regulators studied by Beilinson.
Relation to algebraic K-theory
The Bloch group is closely tied to the third algebraic K-theory group K3 of fields via a regulator exact sequence and the Suslin isomorphism for algebraically closed fields; work of Andrei Suslin and Maxim Kontsevich clarifies how the Bloch group maps to K3^{ind} and relates to the Quillen K-theory groups considered by Daniel Quillen. For global fields like Q and number fields such as Q(√-1) or cyclotomic extensions studied by Carl Friedrich Gauss, Bloch groups appear in conjectural descriptions of the motivic cohomology groups of arithmetic schemes put forward by Alexander Beilinson and used in computations of special values of Dedekind zeta functions investigated by Dirichlet and Riemann.
Regulator maps and Bloch–Wigner dilogarithm
Regulator maps from the Bloch group to real cohomology are constructed using the Bloch–Wigner dilogarithm function originally analyzed in classical work by Ludwig Schlömilch and later systematically by Don Zagier. These maps connect elements of the Bloch group to periods and provide explicit formulas for regulators occurring in conjectures of Beilinson and comparisons with the Borel regulator computed in Armand Borel's work on higher regulators for arithmetic groups like SL2(Z). The dilogarithm functional equations used in the Bloch group construction mirror identities studied in the analytic theory of polylogarithms by Leonard Lewin and appear in the study of hyperbolic volume invariants of 3-manifolds investigated by William Thurston and Dylan Thurston.
Computations and examples
Explicit computations of Bloch groups for classical fields such as Q, imaginary quadratic fields like Q(√-3), and cyclotomic fields arising in Pierre de Fermat-related contexts have been carried out by Don Zagier, C. Soulé, and Andrei Suslin. Examples often use elements coming from roots of unity studied by Carl Gustav Jacob Jacobi and cyclotomic units appearing in Ernst Kummer's work, and calculations employ techniques from the cohomology of arithmetic groups developed by Armand Borel and computational methods from the theory of modular forms studied by Jean-Pierre Serre. Numerical verifications comparing regulator values with special values of L-functions have been explored in literature influenced by conjectures of Beilinson and computations by D. Zagier and collaborators.
Connections with motives and polylogarithms
The Bloch group plays a central role in explicit descriptions of mixed Tate motives over number fields, a program advanced by Alexander Goncharov, Maurice Brown, and Maxim Kontsevich. It encodes the extension classes in motivic cohomology governing polylogarithm periods studied by Leonard Lewin and appears in the motivic Lie coalgebra formalism developed by Alexander Goncharov. These connections tie the Bloch group to conjectural frameworks linking special values of L-functions to motivic regulators as predicted by Beilinson and elaborated in the context of mixed motives by Pierre Deligne and Alexander Beilinson.
Variants and generalizations
Generalizations of the Bloch group include higher Bloch groups connected to higher polylogarithms and higher algebraic K-theory investigated by Spencer Bloch, Alexander Goncharov, and Martin Snaith; equivariant and p-adic variants studied by Kazuya Kato and John Coates appear in Iwasawa-theoretic contexts related to Kenkichi Iwasawa. Analogs for function fields over finite fields and for schemes have been developed in work of Suslin and Charles Weibel, and categorical formulations linking to triangulated categories of motives were pursued by Vladimir Voevodsky and Maxim Kontsevich.