As the title says. I’ll freely use the notation of my paper, although unfortunately I can’t figure out how to implement “mathscr” fonts on this blog, so objects denoted in mathscr fonts in the paper are denoted in mathcal fonts below – e.g. is the module of locally analytic distributions of weight , is the weight space of tame level , is the eigenvariety for of tame level , etc.
1. There’s a natural notion of a “cuspidal component” of an eigenvariety. Precisely, given a point with associated eigenpacket of weight , let’s say or is cuspidal if
Here denotes the boundary cohomology of , i.e. the cohomology of the local system induced by on the boundary of the Borel-Serre compactification. Now let’s say an irreducible component of is cuspidal if it contains a Zariski-dense set of cuspidal points. Let denote the union of these irreducible components. It’s natural to expect that is equidimensional of dimension . When , this takes the pleasant form
(Here is the Leopoldt defect of at , so conjecturally.) Hopefully I’ll have more to say about these matters in the near future!
2. In the context of over a totally real or CM field , there’s an evident notion of an “essentially self-dual” or “essentially conjugate self-dual” component of . However, even if is ESD or ECSD, the point should only lie in an ESD/ECSD component of if the character is ESD/ECSD as well. For example, let be a classical holomorphic newform of weight and level prime to , with and the roots of – then typically has six parameters , corresponding to orderings on the set , but only the orderings and are essentially self-dual. By Theorem 5.4.1, the points on associated with and these two orderings really do exist, and they lie on essentially self-dual components, but I have no idea how to say anything about the other four points. In the ‘non-critical’ case these predicted points are easily seen to exist, and one can even check that any irreducible component containing one of them has dimension exactly , but I have no idea how to say anything else qualitative about these components.
3. Here’s some more fun evidence towards Conjecture 1.2.5. Let be a pair of finite slope overconvergent cusp forms. Suppose has weight and very large slope, and that has non-integral weight. The associated Galois representations are trianguline at with unique parameters , and the ordered weights satisfy , , with . The tensor product is trianguline at with two triangulations (cf. section 6.5 of my thesis), say with parameters and , whose -sequences are
Now its easy to see that and as subgroups of , and both ‘s are minimal for the partial ordering on . So Conjecture 1.2.5 predicts eight points on the eigenvariety associated with : one for each .
Theorem. The point exists on the eigenvariety for
When I was trying to formulate Conjecture 1.2.5, the presence of non-simple reflections in this example confused me for a long time. Przemek inadvertently put me out my misery while telling me about his beautiful joint work with John Bergdall, by explaining the relevance of Verma modules, which put me on to Humphreys’s beautiful book on the BGG Category and the realization that the partial ordering is what’s relevant. Indeed, the reader will recognize as essentially the notion of “strong linkage” defined in section 5.1 of Humphreys – silly though it might seem, the key moment for me was the realization that the ‘s in the definition of strong linkage are arbitrary positive roots, not necessarily simple.
4. Is there an example of a singular point on an eigenvariety which meets a unique irreducible component and whose associated Galois representation is irreducible? Already for the eigencurve, this is an open question. If you drop the second condition, the answer is “yes” thanks to some examples of Bellaiche (on eigenvarieties). If you drop the first condition, the answer is “yes” for soft reasons (CM and non-CM components meeting, level-raising a la Newton, etc).
5. Suppose is a global Galois representation which is de Rham and trianguline with a critical refinement (i.e. has a parameter with in the notation of section 6.2). Is the Lie algebra of the Zariski closure of “small” compared to ? This is true in all the examples I know, e.g. critically refinable Eisenstein and CM forms (where is abelian) and tensor products as in Theorem 1.2.8 (where is essentially ). One possible definition of “small”: .