geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
Beilinson’s conjectures (Beilinson 85) conjecture for arithmetic varieties over number fields
that the realification of the Beilinson regulator exhibits an isomorphism between the relevant algebraic K-theory/motivic cohomology groups and Deligne cohomology (ordinary differential cohomology) groups;
(recalled e.g. as Schneider 88, p. 30, Brylinski-Zucker 91, conjecture 5.20, Deninger-Scholl (3.1.1), Nekovar (6.1 (1))).
induced by this that special values of the (Hasse-Weil-type) L-function are proportional to the Beilinson regulator, in analogy with the class number formula and the Birch and Swinnerton-Dyer conjecture
(recalled e.g. as Schneider 88, p. 31 Brylinski-Zucker 91, conjecture 5.21, Deninger-Scholl (3.1.2), Nekovar (6.1 (2)))).
The Beilinson conjecture for special values of L-functions follows the Birch and Swinnerton-Dyer conjecture and Pierre Deligne‘s conjecture on special value of L-functions.
The original articles are
Alexander BeilinsonHigher regulators and values of L-functions, Journal of Soviet Mathematics 30 (1985), 2036-2070, (mathnet (Russian), DOI)
Alexander Beilinson, Higher regulators of curves, Funct. Anal. Appl. 14 (1980), 116-118, mathnet (Russian).
Alexander Beilinson, Height pairing between algebraic cycles, in K-Theory, Arithmetic and Geometry, Lecture Notes in Mathematics Volume 1289, 1987, pp 1-26, DOI.
Reviews include
Michael Rapoport, Norbert Schappacher, Peter Schneider (eds.), Beilinson's Conjectures on Special Values of L-Functions Perspectives in Mathematics, Volume 4, Academic Press, Inc. 1988 (ISBN:978-0-12-581120-0)
Christophe Soulé, Régulateurs Séminaire Bourbaki, 27 (1984-1985), Exp. No. 644, 17 p. (Numdam)
Peter Schneider, Introduction to the Beilinson conjectures, in Rapoport-Schappacher-Schneider 88 (pdf)
Jan Nekovar, section 3 of Beilinson’s Conjectures (pdf)
Christopher Deninger, Anthony Scholl, The Beilinson conjectures (pdf)
Jean-Luc Brylinski, Steven Zucker, conjecture 5.20,5.21 in An overview of recent advances in Hodge theory, English translation in Several complex variables VI, volume 69 of Encyclopedia of Math. Sciences, pages 39-142, 1990 (original in Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr., 1991, Volume 69, Pages 48–165 (web, pdf (Russian original)))
A noncommutative analogue is considered in