Reference Request: Beilinson-Bloch conjecture in terms of Beilinson regulator isomorphism

Question

I'm looking for a reference that provides a concise statement of the Beilinson-Bloch conjecture, specifically formulated in terms of an isomorphism under the Beilinson regulator map.

More precisely, I'm interested in a statement that relates:

1. The algebraic K-theory of a smooth projective scheme X over a field $F$.
2. The Galois cohomology of the étale cohomology of X with, or without cyclotomic character coefficients.

Ideally, the reference would include:

• A precise formulation of the Beilinson regulator map in this context.
• The conjectured isomorphism between (rationalized) K-groups and the relevant Galois cohomology groups. Something like: $r_{p,n}: K_{2n+1-j}(X)_{\mathbb{Q}_p} \to H^1(F, H^{2n}_{\text{ét}}(\overline{X}, \mathbb{Q}_p(j)))$ is an isomorphism. (Apologies if my indices are incorrect).
• Any connection to special values of L-functions, if applicable in this formulation.

I'm aware of Beilinson's original 1985 paper and Bloch's 1986 work, but I'm hoping for a more recent treatment that might synthesize these ideas into a single, clear statement.

Any pointers to survey articles, textbooks, or recent research papers that provide such a formulation would be greatly appreciated. Thank you!

Answer 1

Some conjectures – like this one and also the BSD conjecture – are hard to find in precise form in a single place, because the community's understanding of statement of the conjecture changed over time: additional statements got "folded in" as researchers came to see them as related.

A large chunk of the conjecture in the form you want – relating K-theory and etale cohomology – is given as Conjecture 5.3(i) of the Bloch–Kato paper in the Grothendieck Festschrift:

Bloch, Spencer; Kato, Kazuya, L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. I, Prog. Math. 86, 333-400 (1990). ZBL0768.14001.

Answer 2

The article "Autour des Conjectures de Bloch et Kato" by Fontaine and Perrin-Riou, which can be found on p.599-706 of Volume I of [Motives], has a lot of what you want in a way that adapts to Galois representations that are not a full étale cohomology group of a variety (but are conjecturally subquotients of such).

Regarding $L$-functions, Fontaine--Perrin-Riou conjecture in III.4.2.2 that for a $p$-adic Galois representation $V$ cut out from the étale cohomology of variety via algebraic cycles that: $$ \dim_{\mathbb{Q}_p} H^1_f(G_K;V) = \operatorname{ord}_{s=0} L(V,s). $$

In III.4.1.5 of the same paper, they have a conjecture predicting that $\dim_{\mathbb{Q}_p} H^1_f(G_K;V)$ is the same as the dimension of a corresponding $\operatorname{Ext}$ group in a category of "motivic structures" (note the definition of $L$-admissible goes back ultimately to III.3.2.1). They do not define this category, but I have a joint paper coming out soon that (among other more important things) defines such a category precisely and shows that in the situation of Bloch--Kato, the $\operatorname{Ext}$ group receives a map from the corresponding algebraic $K$-theory group.

[Motives]: Jannsen, Uwe (ed.); Kleiman, Steven (ed.); Serre, Jean-Pierre (ed.), Motives. Proceedings of the summer research conference on motives, held at the University of Washington, Seattle, WA, USA, July 20-August 2, 1991, Proceedings of Symposia in Pure Mathematics. 55, Pt. 1. Providence, RI: American Mathematical Society (AMS). xiv, (1994). ZBL0788.00053.