Some miscellaneous remarks on eigenvarieties

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. ${\mathcal{D}_{\lambda}}$ is the module of locally analytic distributions of weight ${\lambda}$, ${\mathcal{W}_{K^{p}}}$ is the weight space of tame level ${K^{p}}$, ${\mathcal{X}_{\mathbf{G},K^{p}}}$ is the eigenvariety for ${\mathbf{G}}$ of tame level ${K^{p}}$, etc.

1. There’s a natural notion of a “cuspidal component” of an eigenvariety. Precisely, given a point ${x\in\mathcal{X}_{\mathbf{G},K^{p}}(\overline{\mathbf{Q}_{p}})}$ with associated eigenpacket ${\phi_{x}}$ of weight ${\lambda_{x}}$, let’s say ${x}$ or ${\phi_{x}}$ is cuspidal if

$\displaystyle H_{\partial}^{\ast}(Y(K^{p}I)^{\mathrm{BS}},\mathcal{D}_{\lambda_{x}})_{\ker\phi_{x}}=0.$

Here ${H_{\partial}^{\ast}(Y(K_{f})^{\mathrm{BS}},M)}$ denotes the boundary cohomology of ${M}$, i.e. the cohomology of the local system induced by ${M}$ on the boundary ${Y(K_{f})^{\mathrm{BS}}\smallsetminus Y(K_{f})}$ of the Borel-Serre compactification. Now let’s say an irreducible component of ${\mathcal{X}_{\mathbf{G},K^{p}}}$ is cuspidal if it contains a Zariski-dense set of cuspidal points. Let ${\mathcal{X}_{\mathbf{G},K^{p}}^{\mathrm{cusp}}}$ denote the union of these irreducible components. It’s natural to expect that ${\mathcal{X}_{\mathbf{G},K^{p}}^{\mathrm{cusp}}}$ is equidimensional of dimension ${\mathrm{dim}\mathcal{W}_{K^{p}}-l(\mathbf{G})}$. When ${\mathbf{G}=\mathrm{Res}_{F/\mathbf{Q}}\mathrm{GL}_{n}}$, this takes the pleasant form

$\displaystyle \mathrm{dim}\mathcal{W}_{K^{p}}-l(\mathbf{G})=1+\delta_{F,p}+\left\lfloor \frac{n}{2}\right\rfloor r_{1}+nr_{2}.$

(Here ${\delta_{F,p}}$ is the Leopoldt defect of ${F}$ at ${p}$, so ${\delta_{F,p}=0}$ conjecturally.) Hopefully I’ll have more to say about these matters in the near future!

2. In the context of ${\mathrm{GL}_{n}}$ over a totally real or CM field ${F}$, there’s an evident notion of an “essentially self-dual” or “essentially conjugate self-dual” component of ${\mathcal{X}_{\mathbf{G},K^{p}}}$. However, even if ${\rho_{x}}$ is ESD or ECSD, the point ${x}$ should only lie in an ESD/ECSD component of ${\mathcal{X}}$ if the character ${\delta_{x}}$ is ESD/ECSD as well. For example, let ${f}$ be a classical holomorphic newform of weight ${k}$ and level prime to ${p}$, with ${\alpha}$ and ${\beta}$ the roots of ${X^{2}-a_{f}(p)X+p^{k-1}\varepsilon(p)}$ – then ${\mathrm{sym}^{2}\rho_{f}}$ typically has six parameters ${\delta}$, corresponding to orderings on the set ${\{\alpha^{2},\beta^{2},\alpha\beta\}}$, but only the orderings ${(\alpha^{2},\alpha\beta,\beta^{2})}$ and ${(\beta^{2},\alpha\beta,\alpha^{2})}$ are essentially self-dual. By Theorem 5.4.1, the points on ${\mathcal{X}_{\mathrm{GL}_{3}}}$ associated with ${\mathrm{sym}^{2}\rho_{f}}$ 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 ${2}$, 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 ${f,g}$ be a pair of finite slope overconvergent cusp forms. Suppose ${f}$ has weight ${2}$ and very large slope, and that ${g}$ has non-integral weight. The associated Galois representations are trianguline at ${p}$ with unique parameters ${\delta_{f},\delta_{g}}$, and the ordered weights satisfy ${k_{1}(\delta_{f})=-1,k_{2}(\delta_{f})=0}$, ${k_{1}(\delta_{g})=0}$, ${k_{2}(\delta_{g})=a}$ with ${a\notin\mathbf{Z}}$. The tensor product ${\rho=\rho_{f}\otimes\rho_{g}}$ is trianguline at ${p}$ with two triangulations (cf. section 6.5 of my thesis), say with parameters ${\delta_{1}}$ and ${\delta_{2}}$, whose ${k_{i}}$-sequences are

$\displaystyle (-1,0,a-1,a)$

and

$\displaystyle (-1,a-1,0,a).$

Now its easy to see that ${W(\delta_{1})=\left\{ 1,(12),(34),(12)(34)\right\} }$ and ${W(\delta_{2})=\left\{ 1,(13),(24),(13)(24)\right\} }$ as subgroups of ${S_{4}}$, and both ${\delta_{i}}$‘s are minimal for the partial ordering ${\preceq}$ on ${W(\delta_{i})\cdot\delta_{i}}$. So Conjecture 1.2.5 predicts eight points on the ${\mathrm{GL}_{4}}$ eigenvariety associated with ${\rho}$: one for each ${\eta\in W(\delta_{i})\cdot\delta_{i}}$.

Theorem. The point ${x=x(\rho,\eta)}$ exists on the ${\mathrm{GL}_{4}}$ eigenvariety for

$\displaystyle \eta\in\{\delta_{1},(12)(34)\cdot\delta_{1},\delta_{2},(13)(24)\cdot\delta_{2}\}.$

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 ${\mathcal{O}}$ and the realization that the partial ordering ${\preceq}$ is what’s relevant. Indeed, the reader will recognize ${\preceq}$ 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 ${\alpha_{i}}$‘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 ${U(3)}$ 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 ${\rho:G_{F}\rightarrow\mathrm{GL}_{n}(\overline{\mathbf{Q}_{p}})}$ is a global Galois representation which is de Rham and trianguline with a critical refinement (i.e. ${\rho}$ has a parameter with ${\{\delta\}\subsetneq\mathcal{T}[\delta]}$ in the notation of section 6.2). Is the Lie algebra ${\mathfrak{r}}$ of the Zariski closure of ${\rho(G_{F})}$ “small” compared to ${\mathfrak{gl}_{n}}$? This is true in all the examples I know, e.g. critically refinable Eisenstein and CM forms (where ${\mathfrak{r}}$ is abelian) and tensor products as in Theorem 1.2.8 (where ${\mathfrak{r}}$ is essentially ${\mathfrak{go}_{4}}$). One possible definition of “small”: ${\mathfrak{\mathrm{dim}}\mathfrak{r}^{\mathrm{der}}<\mathrm{dim}\mathfrak{sl}_{n}}$.