Beilinson conjectures

Beilinson conjectures
Name	Beilinson conjectures
Field	Number theory, Algebraic geometry, Algebraic K-theory
Proposed by	Alexander Beilinson
Year	1980s

Beilinson conjectures

The Beilinson conjectures propose deep relations among Alexander Beilinson, Algebraic K-theory, Motivic cohomology, Deligne cohomology, and the values of L-functions attached to algebraic varieties over number fields and global fields. They synthesize ideas developed in the work of Bernard Dwork, André Weil, Jean-Pierre Serre, Atle Selberg, and Goro Shimura and influence research by Pierre Deligne, Spencer Bloch, Vladimir Voevodsky, and Stephen Bloch.

Overview

Beilinson formulated a family of conjectural statements linking higher regulators in Algebraic K-theory and Motivic cohomology to leading terms of L-functions for motives over number fields and function fields, drawing on techniques from Hodge theory, Étale cohomology, Arakelov theory, and the study of zeta functions. The program unites earlier predictions such as the Birch–Swinnerton–Dyer conjecture for Elliptic curves, the Tamagawa number conjecture envisioned by Bloch and Kato, and the formulations of Grothendieck and Deligne concerning special values of L-functions. Results by Beilinson, Bloch, Kato, Scholl, Soulé, and Borel have provided the structural framework linking regulator maps to arithmetical invariants.

Statements of the Conjectures

Beilinson conjectures propose that for a pure motive M over a number field or function field, the order of vanishing and the leading nonzero coefficient of the L-function L(M, s) at integer points s = n can be expressed in terms of the rank and determinant of regulator pairings between K-groups K_{2n-1}(X) or Motivic cohomology groups and targets such as Deligne cohomology or Étale cohomology, paralleling predictions from Bloch–Kato conjecture and the Bloch–Kato conjectures for Galois representations. The conjectures include precise formulations of period factors involving comparisons between Betti cohomology, de Rham cohomology, and mixed Hodge structures and incorporate contributions from Tamagawa number factors and local epsilon factors studied by John Tate and Henri Carayol.

Relation to Special Values of L-functions

Beilinson links special values L(M, n) for motives M to regulator determinants constructed from Algebraic K-theory and Motivic cohomology, refining earlier work on Dirichlet L-series by Leopold Kronecker and conjectures on Dedekind zeta functions by Stark and Brumer. The conjectures predict that up to explicit transcendental periods computed from Hodge theory and rational factors arising from Galois cohomology and local fields, the leading coefficient of L(M, s) at s = n equals a determinant of a regulator matrix built from elements of K-groups studied by Quillen and Milnor, generalizing the Class number formula proven by Dirichlet and extended by Dedekind and Hecke.

Motivic Cohomology and Regulators

Central to Beilinson's vision is the identification of higher regulator maps from Algebraic K-theory and Motivic cohomology to realizations such as Deligne cohomology, Syntomic cohomology, and Étale cohomology with coefficients in p-adic Hodge theory objects, building on constructions by Bloch, Quillen, Soulé, and Gillet–Soulé. These regulator maps are expected to carry arithmetic information similar to the classical Abel–Jacobi map for cycles on varieties, and to interact with period isomorphisms developed by Deligne and comparison theorems of Grothendieck and Faltings. The conjectures require precise compatibility between regulator images and canonical lattices coming from integral models and Arakelov theory frameworks advanced by Faltings and Gillet.

Evidence and Known Cases

Evidence includes Bloch and Beilinson computations for K2 of fields and certain curves, Borel's theorems on rational ranks of K-groups for number fields, and proofs of special cases such as the Birch–Swinnerton–Dyer conjecture in analytic rank zero or one for certain Elliptic curves achieved by Kolyvagin, Gross–Zagier theorem by Gross and Zagier, and the work of Kato on modular forms and Iwasawa theory. Results by Scholl for modular motives, Beilinson and Deligne for zeta values of modular curves, and advances in p-adic L-function theory by Perrin-Riou and Mazur provide corroborating instances. Calculations by Goncharov and Levine on polylogarithms illuminate motivic regulators in low weights.

Consequences and Applications

If correct, the conjectures would unify class number formulas, analytic rank predictions, and special value formulae across Elliptic curves, Modular form motives, and higher-dimensional Algebraic varietys, impacting Iwasawa theory, automorphic representations, and the study of Galois representations attached to motives by Langlands correspondences. They would imply precise descriptions of Selmer groups studied by Mazur, refined statements of the Tamagawa number conjecture by Bloch and Kato, and furnish period relations used in computational approaches by researchers such as Cremona and Stein.

Open Problems and Variants

Major open problems include providing a working category of mixed motives as envisioned by Grothendieck and Beilinson and proving the full compatibility of regulator maps with expected period and epsilon factors, as sought in work by Voevodsky, Levine, and Neeman. Variants include equivariant refinements involving Galois group actions and noncommutative extensions inspired by Connes and equivariant Tamagawa number conjectures developed by Burns and Flach. Constructing canonical elements in K-groups, establishing integrality properties predicted by Bloch–Kato conjecture methods, and extending known cases beyond rank-zero or rank-one scenarios remain central challenges pursued by many research programs in arithmetic geometry.

Category:Conjectures in number theory