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At the end of the Introduction to a recent work of Joël Bellaïche to appear in Inventiones one can read a paragraph that I try to rephrase as follows (see the author's webpage, or the "Online First Articles" section in the aforementioned journal webpage -- subscription required -- for the original statements):

The currently available strategies to produce as many independent extensions in the Selmer group as predicted by the $p$-adic Bloch-Kato conjecture rely on (vast) generalizations of the ideas first appeared in Ribet's gem on the converse to Herbrand (the so-called Ribet's method). Loosely, if one wants to produce elements in ${\rm Sel}(\rho)$ one picks a suitable automorphic form $f$ whose attached Galois representation is reducible, containing $\rho$ and the trivial character as (some of the) constituents, and deforms $f$ into a $p$-adic family whose Galois representation is generically irreducible; then a variant of Ribet's lemma yields extensions as the ones desired that can be shown (when one is lucky enough to be able to rule out the possible ``parasit'' extensions) to actually land in ${\rm Sel}(\rho)$. Further, for the methods to succeed, a good knowledge of the relation between the $p$-adic $L$-functions of $f$ and $\rho$ seems necessary.

What motivates my question is the following tantalizing comment of the author in the above section:

``and there are fundamental reasons that the form $f$ [as above] we work with is critical or $\theta$-critical, that it would take us too far to explain here''

(As explained in the paper, in the case of a classical elliptic cuspidal eigenform $f$, the notions of critical and $\theta$-critical are equivalent, the former meaning that there is some non-classical overconvergent modular form which is a generalized eigenvector for the eigensystem of $f$.)

Hence I would like to openly ask for some explanation on why it is so fundamental that the form $f$ is chosen to be critical or $\theta$-critical.

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May I propose that you just ask the author itself? –  Julien Puydt Nov 6 '11 at 10:58
Dear Julien, I certainly could do that, and with less effort than I needed to pose my question. But I had at least two reasons to ask it here: (1) the answer may be known to other experts; (2) an eventual answer may be useful to other mathematicians. –  monodromy Nov 6 '11 at 12:47
Also: Joël regularly answers question on mathoverflow... I know that in some cases his work can produce a big reducibility locus (and hence a big lower bound on the Selmer group) simply by knowing that he has a critical refinement, so my quick guess is that this is his "fundamental reason" here. –  Rob Harron Nov 6 '11 at 16:10
I am preparing an answer. Since I have a very busy day, it will not be posted before tonight, and probably tomorrow. –  Joël Nov 7 '11 at 14:50
Ok, here it is. –  Joël Nov 14 '11 at 14:33
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1 Answer

up vote 18 down vote accepted

If we start with a Galois representation $\rho_\pi = 1 \oplus \rho \,\oplus$ other terms, with a refinement, and try to apply the method in question, we need at three different steps to assume that the refinement is critical:

  1. To construct by automorphic methods a non-trivial deformation of $\rho_\pi$ with its chosen refinement.

  2. To prove that the deformation constructed in 1. is generically irreducible

  3. And finally, to prove that the non-trivial extensions constructed from this deformation (residually reducible, generically irreducible) by some generalizations of Ribet's lemma have "good reduction" in the sense of Bloch-Kato at places dividing $p$.

Of the three reasons, 3. seems to me the more fondamental. In 2. we can in certain situations do without the criticality hypothesis, but we get much better result using it. In 1., the criticality hypothesis is needed in all cases that have been considered in the literarure so far, but I can imagine cases where it should not be so.

I now want to explain those reasons in more detail, but for this I need to define the terms I use, in particular what is a refinement and what it means for it to be critical (which can be done precisely), and also what is the method I am talking about (which by necessity can only be done with a certain vagueness).

So the method I am considering here, or I should rather say a family of closely related methods, is a proper (and small) subset of the huge world created by Ribet's 1976 paper, used by Chenevier and me in certain of our papers (mainly in our 2004 Annales de l'ENS and our 2009 book at Astérisque), and also by Skinner and Urban (mainly in their 2006 paper at JIMJ and their 2006 announcement at the ICM -- to delimitate the method I should say I don't consider their work on the main conjecture for elliptic curves as part of it). The nethod can be roughly described as follows: the aim is to produce some non-trivial element(s) in the Bloch-Kato Selmer group $H^1_f(\rho)$ of a $p$-adic Galois representation $\rho$ of automorphic origin and of motivic weight $-1$ (this is the most intersting case). The method involves finding, using an hypothesis on $\rho$ (typically a vanishing of its $L$-function) an automorphic representation $\pi$ for a reductive group $G$ over a number field (say $\mathbb{Q}$ to fix ideas), whose attached Galois representation $\rho_\pi$ (semi-simple by construction) is $1 \oplus \rho$ plus other factors deforming $\pi$ $p$-adically in such a way that the corresponding deformation of $\rho_\pi$ is generically irreducible, and use some generalization of Ribet's lemma to construct one or several non-trivial independent extensions of $1$ by $\rho$ which are provably in $H^1_f(\rho)$.

Let us assume to fix ideas that $\rho$ is crystalline at $p$, and $\rho_\pi$ as well (in practice the extra factors are crystalline, so this is a consequence of the assumption on $\rho$). Let $D$ be the filtered $\phi$-module attached the the restriction of $\rho_\pi$ to the decomposition group at $p$, and $\phi$ the crystalline Frobenius acting on it. For simplicity, extend the field of ceofficients so that all eigenvalues of $\phi$ are in it, and assume these eigenvalues are distinct. Then a {\it refinement} of $\rho_\pi$ is an ordering of the eigenvalues of $\phi$ on $D$, or equivalently a, $\phi$-stable complete flag in $D$. So there are $d!$ refinments, if $d$ is the dimension of $\rho_\pi$. We say that a refinement is {\it non-critical} if this flag is in generic position w.r.t. the Hodge filtration on $D$. It is an exercise that for every representation there is at least one non-critical refinement (and in general, many). For a representation which is decomposable like $\rho_\pi$ there is also always one (and actually many) critical refinement.

There is also a notion of refinement on the automorphic side: Let say that $G(\mathbb{Q}_p)=GL_d(\mathbb{Q}_p)$ for simplicity, to avoid mentioning the $L$-group; for $\pi$ an automorphic form such that $\pi_p$ is unramified, an {\it automorphic refinement} is a character of the Atkin-Lehner algebra $A_p$ (for $G=Gl_2$, this is just the lagebra generated by the Atkin=Lehenr operator $U_p$) appearing in $\pi_p^{I_p}$, where $I_p$ is the Iwahori subgroup. There is an injective map from the set of automorphic refinements of $\pi$ to the set of refinements of $\rho_\pi$. It is surjective when $\pi_p$ is a full unramified principal series (that is, a represresentation induced from a Borel of an unramified character of a maximal torus which happens to be irreducible), which is the generic situation.

Now I can explain why the criticality of the refinement is needed in step 1. It turns out that if we want a Galois representation $\rho_\pi$ containing $1 \oplus \rho$ which is attached to an automorphic representatin $\pi$, then in the current state of knowledge, this representation also has to contain the cyclotomic character $\chi$. This is because we only know how to attach Galois representations to groups like unitary or symplectic groups, and the Galois representation attached to those groups are (conjugate) self-dual up tp a twist by $\chi$ -- so if they contain $1$, they must also contain $\chi$. (Actualy even if we could work for $G=GL_d$, if we wanted our $\pi$ to appear in the discrete spectrum, Arthur's conjectures would also force $\rho_\pi$ to contain $\chi$. Only for Eisenstein series for $GL_d$ would it perhabs be possible to avoid the $\chi$). Now the very presence of a $1$ and a $\chi$ in $\rho_\pi$ forces the unramified representation $\pi_p$ to be a proper subquotient of the unramified principal series where it belongs (cf. the classification of unramified principal series). In other words, in our case, not all refinements of $\rho_\pi$ comes form an automorphic refinement of $\pi$. A closer analysis shows that all the refinements of $\rho_\pi$ that comes from $\pi$ are critical. Since the theory of deformations of automorphic representations (a.k.a eigenvarieties) can only deform representations with a given automorphic refinement, this forces our refinement of $\rho_\pi$ to be critical in order to carry out step 1.

For step 2. it is sometimes possible to prove by a tedious case analysis using the classification of automorphic representation for $G$ when it is known, that the deformation of $\rho_\pi$ we constructed in 1. is generically irreducible (this is what I did in my thesis, and that Skinner-Urban did in their JIMJ paper), but there is a better argument, purely Galois-theoretical, which proves that if we start with a Galois representation $\rho_\pi$ with a suitable refinement, and deform it (in a way compatible with the refinement, a notion I have not explained), then the deformation obtained has very strong irreducibility properties, even after restriction to $G_{\mathbb{Q}_p}$. Again, the suitable condition to impose on the refinement implies it to be critical. This is the method used in my paper with Chenevier at Annales de l'ENS and in our book, as well as by Skinner-Urban in their ICM's paper.

After step 2, the details vary a lot and can be very messy, but let me explain the common idea: a generalization of Ribet's lemma ensures the existence of either a non-trivial extension of $1$ by the cychotomic character $\chi$, or a non-trivial extension of $1$ by $\rho$. We need to know that those extensions have good reduction at $p$: this will ensure that the first one do not exist, and tha the secons is really in $H^1_f(\rho)$. This is done by applying a lemma of Kisin, whose result depend of the refinement we started with. Ensuring by this method that the extension of $1$ by $\chi$ has good reduction always requires the refinement to be critical (this is also true in most case for the extension of $1$ by $\rho$, though when the Hodge-Tate weights of $\rho$ are well positionned we can avoid the use of Kisin's lemma and the criticality hypothesis for this particular point). So the criticality assumption is necessary for step 3.

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Dear Joël, Thanks very much for posting this. Best wishes, Matt –  Emerton Nov 14 '11 at 21:03
I second Emerton in his comment. Thanks, Joël. Your paper is about constructing the $p$-adic L-functions attached to critically refined modular forms. I would guess that in the general framework that you have marsterfully described, the need of having a good understanding of the $p$-adic L-functions attached to critically refined automorphic forms appears in step 3. Is this right? And back to the setting of your paper, do the new cases of your construction have (expected?) arithmetic applications to, for example, the classical Iwasawa's Main Conjecture (in the form of a more simplified proof)? –  monodromy Nov 15 '11 at 4:12
Thanks Joël, this is very informative. –  user1125 Nov 15 '11 at 6:41
Thanks for this beautiful answer! –  Arijit Feb 6 '13 at 18:33
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