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?

19more comments