Timeline for Iwasawa main conjectures vs Bloch-Kato conjectures
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17 events
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| Sep 11, 2015 at 8:01 | history | edited | Olivier | CC BY-SA 3.0 |
Corrected Latex
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| Dec 16, 2010 at 14:54 | comment | added | Joël | Good luck for the anticycl. ETNC (and don't forget it's very common than younger mathematicians outrun well-established specialists in the field). To answer your question about BK: yes I believe it is accessible, in two sense: (1) it is accessible by the means of algebraic number theory (and automorphic forms) while I believe Perrin-Riou's conjecture, which contains as a very very special case Leopoldt's is not (my original meaning), and (2) I believe that the lower bound on Selmer in BK in any rank for automorphic Gal. rep. is doable(Chenevier and I did some cases in any rank years ago) | |
| Dec 14, 2010 at 23:58 | comment | added | Olivier | Still, thanks to your question and the ensuing discussion, I just realized that the full ETNC in the anticyclotomic direction for an automorphic form attached to a definite quaternion algebra should be at the very limit of what is accessible nowadays. Perhaps I should look into it (but, if I am right, I don't see how I could outrun Darmon anyway). | |
| Dec 14, 2010 at 23:55 | comment | added | Olivier | You are right about this entangling business. But the morale of this story (to me) is that as far as I know, the boot-strapping process I alluded too above (which could be summed up as "prove the ETNC in some situations where it is relatively easy then show that it is enough to prove the IMC then show that this proves the ETNC in harder cases as well") has never shed any light on the part we missed in the first place. By the way, you consider Bloch-Kato accessible? In high ranks as well? This surprises me a little. And why are we writing in English? | |
| Dec 14, 2010 at 23:37 | comment | added | Joël | Now I know why you say that, and this is because of Perrin-Riou's point of view, where indeed the IMF comes last, in a form where Mazur and Wiles, let alone Iwasawa, would find hard to recognize the link to what they have done. I like Perrin-Riou a lot, and I have immense respects for her mathematical achievements, which have provided us with a clear picture of what is expected, but to my sense her point of view has the shortcoming of entangling the accessible-very-hard conjectures (e.g. the IMF or Bloch-Kato) with the inaccessible ($p$-adic Beilinson, etc.). | |
| Dec 14, 2010 at 23:26 | comment | added | Joël | I am happy to have this discussion. Olivier, I strongly disagree with you when you say "in the grand web of what we expect, the IMC comes very last". The history shows the exact contrary. For the Kubota-Leopoldt's zeta function the IMC (the original one) has been known for more than twenty-five years. Yet we don't know today if $\zeta_p(1-k)$ vanishes for $k$ even negative, nor if $H^1(G_Q,\Q_p(k))=0$. For the the Zeta function of a totally real field, the IMC is known sine 1990, but the Leopoldt's conjecture is still open, maybe for a long long time. Etc. | |
| Dec 14, 2010 at 22:13 | comment | added | Olivier | One very last thing, and then I shut up for good, though I have immense respect for Jay's mathematical achievements (and a great fondness towards the character), I don't think he would be too offended if I point out that relating p-adic L-functions with Iwasawa modules was done in considerable generality by B.Perrin-Riou when he was still in high-school. The sentence "in the sense of Greenberg, or more generally Pottharst" makes me cringe a little. | |
| Dec 14, 2010 at 22:03 | comment | added | Olivier | I would, of course, be extremely interested to be proven wrong. Also, I start to have some sympathies for BCrd style of MO comment. Who needs vowels, when the price to pay is 4 comments in a row? | |
| Dec 14, 2010 at 22:02 | comment | added | Olivier | In the real world, this crucial input is given to us (at some good classical points) by automorphic considerations which are outside of what we have discussed up to now so we can boot-strap the process to formulate an IMC (and even sometimes prove it, again by the same boot-strapping process). However, this should not obscure the fact that in the grand web of what we expect, the IMC comes very last. It would be marvellous if this boot-strapping process could give us a missing piece, but I don't see that this happened ever outside of the generic case (i.e minimal order of vanishing). | |
| Dec 14, 2010 at 21:52 | comment | added | Olivier | Yet I also think that your point of view on this is slightly misguided in the following sense: the modern (and correct) way to look at the IMC is to see it as a generalization (or interpolation, if you wish) of the ETNC at each classical point and each finite layer. In order to formulate the ETNC in this generality, you need as a pre-requisite something like Bloch-Kato's conjecture on the order of vanishing (or something like the non-vanishing of the $p$-adic regulator or something like the semi-simplicity of some complexes). | |
| Dec 14, 2010 at 21:49 | comment | added | Olivier | Well then, as I wrote already, I think the answer to the question "what does the main conjecture say about the dimension of a Selmer group, or the vanishing of the p-adic L-function?" is "not much". | |
| Dec 14, 2010 at 21:27 | comment | added | Joël | In other words, your answer (E) satisfies condition (1), but not (2). It is not a consequence of the main conjecture alone, but it is the consequence of the main conjecture, plus another conjecture, namely the non-annulation of a $p$-adic regulator. In other words again, (E) is nice and interesting, but it is not an answer to the question "what does the main conjecture says about the dimension of a Selmer group, or the vanishing of the $p$-adic L-function?" It is an answer to the more vague question "what do we expect to be true". Yet I believe that the orignal question has a simple answer. | |
| Dec 14, 2010 at 21:22 | comment | added | Joël | To explain this, let me consider the same example as above: the representation $\Q_p(k)$ of $G_Q$ for $k$ negative even. In this case, the relevant $p$-adic $L$-function is Kubota-Leopoldt's $\zeta_p$, and your answer becomes: "Grenberg conjectures that there is an equivalence (E) : $H^1_f(Q,Q_p(k)) = 0$ iff $\zeta_p(1-k) \neq 0$"; since the LHS is known, Greenberg conjectures the RHS. Yet the RHS is not known. Hence the equivalence (E) is not known either. Thus (E) can not possibly be a consequence of the main conjecture, because the main conjecture is known in this context by Mazur-Wiles. | |
| Dec 14, 2010 at 21:07 | comment | added | Joël | It is important here to be careful about the difference between well-believed conjectures and unconditional results. What I am looking for is a result that satisfies two conditions: 1) it relates the vanishing (or not) of a $p$-adic L-function of a Galois representation $\rho$ (defined however you like) at an integer $n$, with the dimension of a Selmer group of $\rho^\ast(1-n)$ plus corrective local terms. 2) it is an unconditional consequence of the main conjecture (in the sense of Greenberg, or more generally Pottharst) relating the p-adic L-function of (1) with an Iwasawa Selmer module. | |
| Dec 14, 2010 at 20:54 | comment | added | Joël | Thanks very much to you Olivier for your long answer. Point taken about the fact that the dimension we're looking for may be not the dimension of a subspace of a full global H^1, but rather may hace correcting local terms as well. Yet, your answer is not exactly what I was looking for, but that's my fault as I realized (after all this discussion) that I didn't ask my question clearly enough. Let me explain what I am really looking for (this is also an answer to Rob H's recent comment under my question): | |
| Dec 14, 2010 at 17:12 | history | edited | Olivier | CC BY-SA 2.5 |
Corrected typos
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| Dec 14, 2010 at 17:07 | history | answered | Olivier | CC BY-SA 2.5 |