Eigenvarieties and functoriality

Question

In Langlands' review of Hida's book "$p$-adic automorphic forms on Shimura varieties", he discusses a nexus of 4 areas of modern number theory: automorphic representations, motives, spaces of $p$-adic Galois representations, and a fourth less well-defined area which he describes in some cases as spaces of representations of $p$-adic Hecke rings. These two final objects should correspond to each other in some exact way through the $p$-adic Langlands conjectures 'in families'.

My understanding is that the fourth area is now underway via the theory of eigenvarieties.

• Is it indeed the case that eigenvarieties (are conjectured to) play this role in the Langlands set-up?

Langlands states many conjectural relationships between this fourth area and the other 3, and I would like to find out to what extent these relationships have been satisfied by the theory of eigenvarieties. The first two questions concern the relationship between automorphic representations and eigenvarieties, and the third concerns that between $p$-adic Galois representations and eigenvarieties.

1. Are there known examples/a general theory for functoriality of eigenvarieties not simply arising from known cases of functoriality for automorphic representations, and would functoriality at the level of eigenvarieties get us any closer to the global conjectures?
2. An automorphic representation $\pi$ on an adelic group $G(\mathbb{A}_F)$ has attached to it a pair $(\{A(\pi_{\mathfrak{p}})\}, {}^{\lambda}H_{\pi})$, consisting of the set of Frobenius-Hecke conjugacy classes $\{A(\pi_{\mathfrak{p}})\}$ contained in some subgroup $^{\lambda}H_{\pi}\subset {}^L G$ of the L-group, which is a reductive algebraic group over $\mathbb{C}$. This pair should be equal to the Frobenius-Hecke conjugacy classes of the corresponding motive, $M$, along with the reductive algebraic group $ ^{\mu}H_M$, attached by the Tannakian theory. In the world of Galois $p$-adic representations we get Frobenius conjugacy classes in $p$-adic groups. For the points on the eigenvariety over a prime $\mathfrak{q}$, do we attach conjugacy classes in some reductive algebraic group over $F_{\mathfrak{q}}$, in comparison with those attached to the corresponding $\mathfrak{q}$-adic Galois representation? If this is the case, are we conjecturing some rationality results on $ ^{\lambda} H_{\pi}$, or just on the classes $\{A(\pi_{\mathfrak{p}})\}$, so that they can be considered in the $p$-adic world? I suppose this is a generalisation on the results of Shimura on the finite dimension of the Hecke field...
3. Where do we stand on the relation between eigenvarieties and deformation spaces for $p$-adic Galois representations? I suppose these are in the area of the $ \mathcal{R}=\mathbb{T}$ theorems. Now that we have constructed some (all?) eigenvarieties, do we know that their dimensions of these varieties match with the Galois side? Can we construct the $p$-adic Langlands correspondence in families even for $GL_1$?

Apologies for the broad question, I am not very familiar with the area. Thank you in advance!

Answer 1

You have asked a lot of questions at once, and it is impossible to give more than a hint at a small subset of these questions.

I think the general theme here is: the existence of eigenvarieties doesn't "create information from nowhere" about the core questions of global Langlands (functoriality and reciprocity); but it allows you to "move information from place to place" -- allowing you to propagate instances of functoriality/reciprocity from some restricted class of automorphic forms to a bigger class.

(1) All of the cases of functoriality of eigenvarieties I'm aware of take some kind of "classical" functoriality as an input (proved either via classical automorphic methods, e.g. comparison of trace formulas, converse theorems, etc, or by fancier methods coming from modularity lifting etc). You can sometimes get new, purely p-adic functoriality results out of the combination of classical functoriality + eigenvariety deformation; e.g. there are some (rather fragmentary) results by Kilford giving a p-adic Jacquet--Langlands correspondence for weight 1 modular forms, which is a purely p-adic thing not seen on the classical side. But it's maybe not terribly satisfying because you don't actually get much information about the resulting maps.

(2) What you're describing -- the collection of Frobenius conj classes -- is a sort of "skeleton" on which you can build $\ell$-adic Galois reps for all $\ell$. However, you wouldn't expect the $\ell$-adic guys to deform over eigenvarieties for all $\ell$. As you correctly guessed, it is only the p-adic Galois reps which deform over the p-adic eigenvariety -- the $\ell$-adic reps for $\ell \ne p$ don't deform well. The cases where we can construct eigenvarieties are indeed a subset (probably strict!) of the cases where we know some kind of finite generation of Hecke eigenvalue fields a la Shimura, so it makes sense to consider these objects p-adically.

(3) Eigenvarieties actually play a rather crucial role in lots of work on the "reciprocity" side of global Langlands (going back + forth between Galois and automorphic objects). The general theme here is that one constructs Galois reps attached to automorphic forms of "sufficiently regular weight" using etale coh. of Shimura varieties, and then uses deformation over an eigenvariety to fill in the "bad" weights where the etale coh argument breaks down (e.g. non-regular limits of discrete series, like weight 1 modular forms for GL2). See e.g. Chenevier's article in the Paris Book Project.

However, it's important here to be aware that the eigenvariety can't see all automorphic forms, it only detects those $\pi$ for which $\pi_{\mathfrak{p}}$ for $\mathfrak{p} \mid p$ are "finite-slope", i.e. subquotients of principal series (and you need to make a choice of one among the Weyl-group orbit of characters from which $\pi_{\mathfrak{p}}$ is induced, which we call a "p-refinement" or "p-stabilisation"). This needs to be reflected on the Galois side; so an eigenvariety should match up with a slightly modified deformation space for p-adic Galois reps -- parametrising Galois reps with some additional "triangulation" data. However, we're rather a long way from proving this for general reductive groups; for $GL_2 / \mathbf{Q}$ there is an almost complete (but not 100% complete) picture but things get very hard very quickly once you step away from that. (For GL1, all representations are vacuously finite-slope, so that case is a little misleadingly easy.)

(Footnote: I think we're a long way from constructing "all eigenvarieties" just yet! We can pick up those $\pi$ which have non-vanishing $(\mathfrak{g}, K)$-cohomology, or those which have non-vanishing $(\mathfrak{p}, K)$-cohomology if $G$ admits a Shimura variety; but that leaves out lots of interesting automorphic reps which we have no way of seeing at present.)