Iwasawa main conjectures vs Bloch-Kato conjectures - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T22:15:56Z http://mathoverflow.net/feeds/question/49269 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/49269/iwasawa-main-conjectures-vs-bloch-kato-conjectures Iwasawa main conjectures vs Bloch-Kato conjectures Joël 2010-12-13T15:49:04Z 2010-12-14T17:12:22Z <p>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 $n$.</p> <p>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.</p> <p>The first statement of the Bloch-Kato conjecture is (for all $n \in \mathbb{Z}$):</p> <p>CONJECTURE: $\dim H^1_f(G_K,\rho(n)) - \dim H^0(G_K,\rho(n)) = \text{ord}_{s=1-n} L(\rho^\ast,s).$</p> <p>Here $L(\rho,s)$ is the complex $L$-function (we assume it has a meromorphic continuation over $\mathbb{C}$)</p> <p>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.</p> <p>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?</p> <p>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.</p> <blockquote> <p>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?</p> </blockquote> <p>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: </p> <blockquote> <p>What does the Iwasawa main conjecture for ordinary elliptic curces implies for the BSD conjecture?</p> </blockquote> http://mathoverflow.net/questions/49269/iwasawa-main-conjectures-vs-bloch-kato-conjectures/49407#49407 Answer by Olivier for Iwasawa main conjectures vs Bloch-Kato conjectures Olivier 2010-12-14T17:07:21Z 2010-12-14T17:12:22Z <p>Dear Joël,</p> <p>If I understand your question properly, then I think much is known. Let me sum up what I understand about this picture.</p> <p>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. </p> <p>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.</p> <p>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 <code>$D^{\phi=p^{-1}}$</code>. 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 $\ellâ‰ p$ of ramifications of $V$ and with the Bloch-Kato condition $\textit{at the level of complexe}$ at $p$.</p> <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 $p$).</p> <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 <code>$H^{0}(G_{\mathbb Q_{p}},V^{*}(1)/F^{+}V^{*}(1))$</code> 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).</p> <p>Hope this helped somehow.</p> <p>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 perhaps, 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.</p> <p>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. </p> <p>Again, hope this helped (but doubt it somehow).</p>