Continuing an amazingly interesting chain of answers about motivic cohomology, I thought I should learn about the Beilinson conjectures, referred there.

I have found some references, and they seem to present the conjectures from different sides, e.g. there's the statement about vanishing but then there are also connections to motivic polylogarithms. 

What I miss from these articles in a general picture that would allow us to start somewhere natural. So,


  how would you describe an introduction into Beilinson conjectures in motivic homotopy?


Sorry for such a loaded question — I really don't know how to make it fit MathOverflow format better. One could theoreticlly post lost of specific questions on the topic, but to ask the right questions in this case you might need to know more than I do. Also, I know there are some technical developments, e.g. the language of derived stacks, and my hope would be that somebody could make a connection to these conjectures using some  clear and suitable language.

Let me talk about Beilinson's conjectures by beginning with $\zeta$-functions of number fields and $K$-theory. Space is limited, but let me see if I can tell a coherent story.

The Dedekind zeta function and the Dirichlet regulator

Suppose $F$ a number field, with
$$[F:\mathbf{Q}]=n=r_1+2r_2,$$
where $r_1$ is the number of real embeddings, and $r_2$ is the number of complex embeddings. Write $\mathcal{O}$ for the ring of integers of $F$.

Here's the power series for the Dedekind zeta function:
$$\zeta_F(s)=\sum|(\mathcal{O}/I)|^{-s},$$
where the sum is taken over nonzero ideals $I$ of $\mathcal{O}$.

Here are a few key analytical facts about this power series:


This power series converges absolutely for $\Re(s)>1$.
The function $\zeta_F(s)$ can be analytically continued to a meromorphic function on $\mathbf{C}$ with a simple pole at $s=1$.
There is the Euler product expansion:
$$\zeta_F(s)=\prod_{0\neq p\in\mathrm{Spec}(\mathcal{O}_F)}\frac{1}{1-|(\mathcal{O}_F/p)|^{-s}}.$$
The Dedekind zeta function satisfies a functional equation relating $\zeta_F(1-s)$ and $\zeta_F(s).$
If $m$ is a positive integer, $\zeta_F(s)$ has a (possible) zero at $s=1-m$ of order
$$d_m=\begin{cases}r_1+r_2-1&\textrm{if }m=1;\\
r_1+r_2&\textrm{if }m>1\textrm{ is odd};\\
r_2&\textrm{if }m>1\textrm{ is even},
\end{cases}$$
and its special value at $s=1-m$ is
$$\zeta_F^{\star}(1-m)=\lim_{s\to 1-m}(s+m-1)^{-d_m}\zeta_F(s),$$
the first nonzero coefficient of the Taylor expansion around $1-m$.


Our interest is in these special values of $\zeta_F(s)$ at $s=1-m$. At the end of the 19th century, Dirichlet discovered an arithmetic interpretation of the special value $\zeta_F^{\star}(0)$. Recall that the Dirichlet regulator map is the logarithmic embedding
$$\rho_F^D:\mathcal{O}_F^{\times}/\mu_F\to\mathbf{R}^{r_1+r_2-1},$$
where $\mu_F$ is the group of roots of unity of $F$. The covolume of the image lattice is the the Dirichlet regulator $R^D_F$. With this, we have the

Dirichlet Analytic Class Number Formula. The order of vanishing of $\zeta_F(s)$ at $s=0$ is $\operatorname{rank}_\mathbf{Z}\mathcal{O}_F^\times$, and the special value of $\zeta_F(s)$ at $s=0$ is given by the formula
$$\zeta_F^{\star}(0)=-\frac{|\mathrm{Pic}(\mathcal{O}_F)|}{|\mu_F|}R^D_F.$$

Now, using what we know about the lower $K$-theory, we have:
$$K_0(\mathcal{O})\cong\mathbf{Z}\oplus\mathrm{Pic}(\mathcal{O})$$
and
$$K_1(\mathcal{O}_F)\cong\mathcal{O}_F^{\times}.$$

So the Dirichlet Analytic Class Number Formula reads:
$$\zeta_F^{\star}(0)=-\frac{|{}^{\tau}K_0(\mathcal{O})|}{|{}^{\tau}K_1(\mathcal{O})|}R^D_F,$$
where ${}^{\tau}A$ denotes the torsion subgroup of the abelian group $A$.

The Borel regulator and the Lichtenbaum conjectures

Let us keep the notations from the previous section.

Theorem [Borel]. If $m>0$ is even, then $K_m(\mathcal{O})$ is finite.

In the early 1970s, A. Borel constructed the Borel regulator maps, using the structure of the homology of $SL_n(\mathcal{O})$. These are homomorphisms
$$\rho_{F,m}^B:K_{2m-1}(\mathcal{O})\to\mathbf{R}^{d_m},$$
one for every integer $m>0$, generalizing the Dirichlet regulator (which is the Borel regulator when $m=1$). Borel showed that for any integer $m>0$ the kernel of $\rho_{F,m}^B$ is finite, and that the induced map
$$\rho_{F,m}^B\otimes\mathbf{R}:K_{2m-1}(\mathcal{O})\otimes\mathbf{R}\to\mathbf{R}^{d_m}$$
is an isomorphism. That is, the rank of $K_{2m-1}(\mathcal{O})$ is equal to the order of vanishing $d_m$ of the Dedekind zeta function $\zeta_F(s)$ at $s=1-m$. Hence the image of $\rho_{F,m}^B$ is a lattice in $\mathbf{R}^{d_m}$; its covolume is called the Borel regulator $R_{F,m}^B$.

Borel showed that the special value of $\zeta_F(s)$ at $s=1-m$ is a rational multiple of the Borel regulator $R_{F,m}^B$, viz.:
$$\zeta_F^{\star}(1-m)=Q_{F,m}R_{F,m}^B.$$
Lichtenbaum was led to give the following conjecture in around 1971, which gives a conjectural description of $Q_{F,m}$.

Conjecture [Lichtenbaum]. For any integer $m>0$, one has
$$|\zeta_F^{\star}(1-m)|"="\frac{|{}^{\tau}K_{2m-2}(\mathcal{O})|}{|{}^{\tau}K_{2m-1}(\mathcal{O})|}R_{F,m}^B.$$
(Here the notation $"="$ indicates that one has equality up to a power of $2$.)

Beilinson's conjectures

Suppose now that $X$ is a smooth proper variety of dimension $n$ over $F$; for simplicity, let's assume that $X$ has good reduction at all primes. The question we might ask is, what could be an analogue for the Lichtenbaum conjectures that might provide us with an interpretation of the special values of $L$-functions of $X$? It turns out that since number fields have motivic cohomological dimension $1$, special values of their $\zeta$-functions can be formulated using only $K$-theory, but life is not so easy if we have higher-dimensional varieties; for this, we must use the weight filtration on $K$-theory in detail; this leads us to motivic cohomology.

Write $\overline{X}:=X\otimes_F\overline{F}$. Now for every nonzero prime $p\in\mathrm{Spec}(\mathcal{O})$, we may choose a prime $q\in\mathrm{Spec}(\overline{\mathcal{O}})$ lying over $p$, and we can contemplate the decomposition subgroup $D_{q}\subset G_F$ and the inertia subgroup $I_{q}\subset D_{q}$.

Now if $\ell$ is a prime over which $p$ does not lie and $0\leq i\leq 2n$, then the inverse $\phi_{q}^{-1}$ of the arithmetic Frobenius $\phi_{q}\in D_{q}/I_{q}$ acts on the $I_{q}$-invariant subspace $H_{\ell}^i(\overline{X})^{I_{q}}$ of the $\ell$-adic cohomology $H_{\ell}^i(\overline{X})$. We can contemplate the characteristic polynomial of this action:
$$P_{p}(i,x):=\det(1-x\phi_{q}^{-1}).$$
One sees that $P_{p}(i,x)$ does not depend on the particular choice of $q$, and it is a consequence of Deligne's proof of the Weil conjectures that the polynomial $P_{p}(i,x)$ has integer coefficients that are independent of $\ell$. (If there are primes of bad reduction, this is expected by a conjecture of Serre.)

This permits us to define the local $L$-factor at the corresponding finite place $\nu(p)$:
$$L_{\nu(p)}(X,i,s):=\frac{1}{P_{p}(i,p^{-s})}$$
We can also define local $L$-factors at infinite places as well. For the sake of brevity, let me skip over this for now. (I can fill in the details later if you like.)

With these local $L$-factors, we define the $L$-function of $X$ via the Euler product expansion
$$L(X,i,s):=\prod_{0\neq p\in\mathrm{Spec}(\mathcal{O})}L_{\nu(p)}(X,i,s);$$
this product converges absolutely for $\Re(s)\gg 0$. We also define the $L$-function at the infinite prime
$$L_{\infty}(X,i,s):=\prod_{\nu|\infty}L_{\nu}(X,i,s)$$
and the full $L$-function
$$\Lambda(X,i,s)=L_{\infty}(X,i,s)L(X,i,s).$$

Here are the expected analytical properties of the $L$-function of $X$.


The Euler product converges absolutely for $\Re(s)>\frac{i}{2}+1$.
$L(X,i,s)$ admits a meromorphic continuation to the complex plane, and the only possible pole occurs at $s=\frac{i}{2}+1$ for $i$ even.
$L\left(X,i,\frac{i}{2}+1\right)\neq 0$.
There is a functional equation relating $\Lambda(X,i,s)$ and $\Lambda(X,i,i+1-s).$


Beilinson constructs the Beilinson regulator $\rho$ from the part $H^{i+1}_{\mu}(\mathcal{X},\mathbf{Q}(r))$ of rational motivic cohomology of $X$ coming from a smooth and proper model $\mathcal{X}$ of $X$ (conjectured to be an invariant of the choice of $\mathcal{X}$) to Deligne-Beilinson cohomology $D^{i+1}(X,\mathbf{R}(r))$. This has already been discussed here. It's nice to know that we now have a precise relationship between the Beilinson regulator and the Borel regulator. (They agree up to exactly the fudge factor power of $2$ that appears in the statement of the Lichtenbaum conjecture above.)

Let's now assume $r<\frac{i}{2}$.

Conjecture [Beilinson]. The Beilinson regulator $\rho$ induces an isomorphism
$$H^{i+1}_{\mu}(\mathcal{X},\mathbf{Q}(r))\otimes\mathbf{R}\cong D^{i+1}(X,\mathbf{R}(r)),$$
and if $c_X(r)\in\mathbf{R}^{\times}/\mathbf{Q}^{\times}$ is the isomorphism above calculated in rational bases, then
$$L^{\star}(X,i,r)\equiv c_X(r)\mod\mathbf{Q}^{\times}.$$
