Iwasawa main conjectures vs Bloch-Kato conjectures

Question

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the maximal extension of $K$ unramified outside $S$, $\rho: G_K \rightarrow Gl_d({\mathbb Q}_p)$ a geometric irreducible representation of $G_K$. For $n$ any integer, $\rho(n)$ is the Tate twist of $\rho$, that is $\rho$ tensor the cyclotomic character to the power $n$.

The Bloch-Kato Selmer group of $\rho$, denoted $H^1_f(G,\rho)$ is defined as an explicit subspace of $H^1(G,\rho)$ (continuous cohomology): $$H^1_f(G,\rho) = \ker \left(H^1(G,\rho) \rightarrow \prod_{v \in S_K-S_p} H^1(I_v,\rho) \times \prod_{v \in S_p} H^1(D_v, \rho \otimes B_{crys})\right),$$ where $D_v$, $I_v$ are respectively a decomposition subgroup and an inertia subgroup at $v$ of $G$, and the $\rightarrow$ is the product of the restriction maps.

The first statement of the Bloch-Kato conjecture is (for all $n \in \mathbb{Z}$):

CONJECTURE: $\dim H^1_f(G_K,\rho(n)) - \dim H^0(G_K,\rho(n)) = \text{ord}_{s=1-n} L(\rho^\ast,s).$

Here $L(\rho,s)$ is the complex $L$-function (we assume it has a meromorphic continuation over $\mathbb{C}$)

There are other statements concerning the principal values of the L-function at $1-n$, that I do not consider here. Note that this conjecture is obviously invariant by Tate twists. Also, the $H^0$ term is $0$ except if $\rho(n)$ is the trivial representation.

Now I come to my question: It is clear that the Iwasawa main conjectures (by which I mean not only Iwasawa's original conjecture on the Kubota-Leopoldt $\zeta$-function, but its modern generalizations) belongs to the same circle of idea. But what exactly is the relation?

To make my question more precise, let us consider to fix ideas Greenberg's form of the main conjecture, as stated for examples in his paper in Motives. A condition on $\rho$, called the Panchiskin condition, is needed to formulate the conjecture. Then a Selmer group is defined as a module over the Iwasawa algebra $\Lambda$, and this module is conjectured to be co-finite and related to the $p$-adic $L$-function of $\rho$. Unfortunately, Iwasawa-theorist tend to use a different language than Bloch-Kato-theorists: they work with modules like $\mathbb{Q}_p/\mathbb{Z}_p$ instead of $\mathbb{Z}_p$ or $\mathbb{Q}_p$ and properties like co-finite instead of finite (perhpaps they are comathematicians). After one takes cohomology, families, etc, the translation between the two languages becomes far from transparent. Yet, I know that the Iwasawa main conjectures have consequences that can be stated in a way very similar to the Bloch-Kato's conjecture.

Can you state such a consequence of Iwasawa's main conjecture in a language closer to Bloch-Kato, precisely : relating (probably in a weaker sense that in BK) the dimension of a suitablle Selmer groups defined as a subspace of $H^1(G,\rho(n))$ cut by local conditions with the order of vanishing of the p-adic L-function of $\rho^\ast$ (assuming it exists) at some points ($1-n$?). Or is such a thing written somewhere?

I apologize that my question is at the same time technical and elementary. Yet an answer would help me a lot, and possibly may help other people who want to get a global picture of this kind of conjectures, and of the progresses made so far. For example, my question contains as a special case:

What does the Iwasawa main conjecture for ordinary elliptic curces implies for the BSD conjecture?

Answer 1

If I understand your question properly, then I think much is known. Let me sum up what I understand about this picture.

First a short answer to your question. Contrary to what you ask for, it is not expected that the dimension of a subspace of $H^{1}$ cut by local conditions should express the order of vanishing of the $p$-adic $L$-function.

Let us start with Bloch-Kato conjecture. This conjecture can be interpreted as a description of cohomological invariants of motives using special values of the $L$-function (many people think of it in the converse way, as description of special values of the $L$-function in terms of Galois invariants). The first question to ask is "which cohomological invariants are we trying to describe?" and the most reasonable answer is "the complex $C$ of motivic cohomology with compact support" (not known to exist in general). Then the order of vanishing of the $L$-function gives the Euler characteristic of $C\otimes_{\mathbb Q}\mathbb R$ whereas the $p$-adic valuation of the principal term of the $L$-function (divided by the period defined in Bloch-Kato) is a $\mathbb Z_{p}$-basis of the determinant of $C\otimes_{\mathbb Q}\mathbb Q_{p}$ (more precisely, of the inverse of the determinant). Even though you knew all this already, I found it necessary to recall it in order to state what forms the IMC takes in this context.

Assume now that our $p$-adic Galois representation $V$ comes from a pure motive and is crystalline at $p$ (I realize that you don't want to make such a strong assumption, but I think all I will say will continue to hold, at least conjecturally). As pointed out in comments already, and as you know, the IMC will say something about the interpolation of the Bloch-Kato conjecture in a $\mathbb Z_{p}$-extension (or more generally in a universal deformation space). I will discuss here only the case of the cyclotomic $\mathbb Z_{p}$-extension. Inside $D_{cris}(V)$ sits $D^{\phi=p^{-1}}$. Let $e$ denotes the dimension of this space over $\mathbb Q_{p}$. Then the cohomological object described by the special values of the (putative) $p$-adic $L$-function is the Selmer complex $S$ of $V$ with the unramified conditions at places $\ell≠p$ of ramifications of $V$ and with the Bloch-Kato condition at the level of complex at $p$.

Based on Bloch-Kato, we should thus expect the Euler characteristic of $S$ evaluated at a character (this is to say of $S\otimes_{\Lambda}\mathbb Z_{p}[\chi]$) to be the order of vanishing of the $p$-adic $L$-function and the $p$-adic $L$-function to give a basis of $\det_{\Lambda} S$. Alas, things are not so easy, because of the infamous trivial zeroes phenomena. So what you can show (possibly assuming plausible conjectures or restricting yourself to rank at most 2 along the way, I'll make an effort to state something really precise if you need to) is that, under Bloch-Kato, the Euler characteristic of $S\otimes_{\Lambda}\mathbb Z_{p}[\chi]$ is equal to the order of vanishing of the usual $L$-function twisted by $\chi$ (as expected) plus $e$ (this is the contribution of the trivial zeroes) $\textit{provided}$ the $\mathcal L$-invariant does not vanish (this is, or should be, equivalent to the semi-simplicity of the complex giving the local condition at $p$).

All this having been said, perhaps you want a concrete answer for a concrete representation. In that case, nothing is simpler than a brave old ordinary representation. For ordinary representation, the local condition at $p$ for the Selmer complex $S$ is simply $R\Gamma(G_{\mathbb Q_{p}},V)$. Hence, the order of vanishing of the $p$-adic $L$-function at a given $\chi$ should simply be the order of vanishing of the $L$-function plus the dimension of $H^{0}(G_{\mathbb Q_{p}},V^{*}(1)/F^{+}V^{*}(1))$ plus or minus simple terms (like the zeroes or poles of the Gamma factors). This reflects the fact that in the generic case, the order of vanishing of the $p$-adic $L$-function should be the dimension of the first cohomology of $S$ (which is not a subspace of $H^{1}$, hence my word of warning at the beginning).

Hope this helped somehow.

Now, let us move on to your second question. I think that if you knew only the IMC, then you couldn't say much about the order of vanishing part of Bloch-Kato. However, if you knew the IMC as well as non-degeneracy of the $p$-adic height pairing (required to formulate the Equivariant Tamagawa Number Conjecture) as well as the Equivariant Tamagawa Number Conjecture for each layer of the cyclotomic extension and/or the vanishing of the $\mu$-invariant, then the order part of Bloch-Kato would follow. Here is how I would try to prove this. First, I would define $S$ (no problem here,as we are in the ordinary case). Then I would construct a canonical trivialization of this complex at each finite layer using the non-degeneracy of the height pairing. Then I would use the ETNC (or I would deduce the ETNC from the IMC using the vanishing of the $\mu$-invariant) to show that the image of the determinant of $S$ at a finite layer under my canonical trivialization is really the value of the principal term of the analytic $L$-function (perhaps times the $\mathcal L$-invariant, but I would know this to be non-zero by semi-simplicity of my complexes). In this way, I would manufacture a complex $L$-function which would agree with the ordinary $L$-function at many (not necessarily classical) points (this would presumably require the IMC and ETNC not only for the cyclotomic extension but for the Hida family containing $E$) and would thus be equal to it. Now, I would know the order of vanishing of my algebraic complex $L$-function at a classical point, so I would know the order of vanishing of the complex $L$-function as well so (finally!) I could check Bloch-Kato.

So, yeah, if you knew the ETNC for the full Hida family and/or the vanishing of the $\mu$-invariant plus the non-degeneracy of the $p$-height pairing, you can, I think, collect the order part of Bloch-Kato as a bonus. Perhaps a moment of sober reflexion is in order now.

Again, hope this helped (but doubt it somehow).