## The eigencurve at the boundary

Let ${p}$ be a prime (assumed ${>2}$ for simplicity), and fix a character ${\chi:(\mathbf{Z}/p\mathbf{Z})^{\times}\rightarrow\mathbf{Z}_{p}^{\times}}$. Let ${\mathcal{W}\simeq\mathrm{Spf}(\mathbf{Z}_{p}[[T]])^{\mathrm{rig}}}$ denote the ${\chi}$-component of weight space, and let ${w:\mathcal{C}\rightarrow\mathcal{W}}$ denote the ${\chi}$-portion of the eigencurve (of whatever tame level ${N}$). For a point ${\lambda\in\mathcal{W}}$, write ${v_{p}(\lambda)}$ as shorthand for ${v_{p}(T(\lambda))}$. Let ${\mathcal{W}^{+}}$ denote the portion of ${\mathcal{W}}$ cut out by the condition ${|T|>\frac{1}{p}}$, so this is some annulus extending towards the boundary of ${\mathcal{W}}$. There has been some recent remarkable progress on Coleman’s equally remarkable conjecture:

Conjecture (Coleman): There is an ${\mathbf{F}_{p}((T))}$-Banach module ${\overline{M}_{\chi}^{\dagger}}$ of “characteristic ${p}$ overconvergent modular forms” such that ${\overline{F}(X)=\det(1-U_{p}X)|\overline{M}_{\chi}^{\dagger}}$ is an entire power series which coincides with the reduction mod ${p}$ of the Fredholm series ${F(X)=\det(1-U_{p}X)|M_{\mathcal{W}}^{\dagger}}$ of ${U_{p}}$ acting on overconvergent modular forms over ${\mathcal{W}}$.

Furthermore, if ${(n_{i},v_{i})}$ denote the breakpoints of the Newton polygon of ${\overline{F}}$, then for any point ${\lambda\in\mathcal{W}^{+}(\overline{\mathbf{Q}}_{p})}$, the breakpoints of the Newton polygon of the specialization ${F_{\lambda}(X)}$ (which coincides with ${\det(1-U_{p}X)|M_{\lambda}^{\dagger}}$) are exactly the points ${(n_{i},v_{i}\cdot v_{p}(\lambda))}$.

The second part of this conjecture implies (by pure rigid analysis) that over ${\mathcal{W}^{+}}$, ${\mathcal{C}}$ breaks apart into a disjoint union of annuli, each finite and flat over ${\mathcal{W}^{+}}$. This latter phenomenon was observed by Buzzard and Kilford in the case ${p=2}$ and tame level one, prompting Coleman’s conjecture above.

It turns out this conjecture has some other nice structural implications for the eigencurve. These observations are basically trivial, but (judging from some conversations I’ve had recently) don’t seem as well-known as they should be.

Proposition. Let ${\mathcal{C}_{i}}$ be any irreducible component of ${\mathcal{C}}$. Suppose Coleman’s conjecture is true. Then ${\mathcal{C}_{i}}$ contains infinitely many weight two classical points of noncritical slope, and the complement of ${w(\mathcal{C}_{i})}$ in ${\mathcal{W}}$ contains at most finitely many points.

Proof. The curve ${\mathcal{C}_{i}}$ maps finitely and surjectively to its spectral variety, which is some Fredholm hypersuface ${\mathcal{Z}_{i}\subset\mathcal{W}\times\mathbf{A}^{1}}$. Again let ${w}$ denote the projection onto ${\mathcal{W}}$. By e.g. Proposition 4.1.3 in my eigenvariety paper, ${w(\mathcal{Z}_{i})\subset\mathcal{W}}$ is Zariski-open, and by the above conjecture it contains all of ${\mathcal{W}^{+}}$, so the complement of ${w(\mathcal{Z}_{i})}$ is a Zariski-closed analytic subset of any qc open containing ${\mathcal{W}\smallsetminus\mathcal{W}^{+}}$, and is therefore finite. We also see from Prop. 4.1.3 that ${\mathcal{Z}_{i}}$ cannot “avoid the boundary”, and in particular (assuming again the conjecture) ${\mathcal{Z}_{i}\cap w^{-1}(\mathcal{W}^{+})}$ is a nonempty disjoint union of annuli. Let ${\mathcal{A}}$ be any one of these annuli.

For the first part, let ${\psi_{i},i\geq1}$, be a sequence of characters of ${(1+p\mathbf{Z}_{p}^{\times})}$ of exact order ${p^{i}}$, and let ${\lambda_{i}}$ be the point of ${\mathcal{W}(\overline{\mathbf{Q}_{p}})}$ corresponding to ${\chi\cdot\psi_{i}}$. (By my conventions, classical points over ${\lambda_{i}}$ correspond to classical newforms of weight 2 and level ${Np^{i}}$.) Since ${v_{p}(\lambda_{i})=\frac{1}{p^{i-1}(p-1)}}$, these points excur towards the boundary as ${i\rightarrow\infty}$. Let ${z_{i}}$ be any preimage of ${\lambda_{i}}$ in ${\mathcal{A}}$. By the conjecture, there is some ${c\geq0}$ such that the slope of any point ${z\in\mathcal{A}}$ is equal to ${c\cdot v_{p}(w(z))}$. In particular, the slope of ${z_{i}}$ is ${<1}$ for ${i\gg0}$, so Coleman’s classicality theorem kicks in and we conclude: any point in ${\mathcal{C}_{i}}$ lying over ${z_{i}}$ is classical for sufficiently large ${i}$. ${\square}$

Given any irreducible component ${\mathcal{C}_{i}}$ as above, one can define an inertial Weil-Deligne representation for any prime ${\ell \mid N, \, \ell \nmid p}$ (this is the inertial WD rep. associated with the Galois representations carried by ${\mathcal{C}_{i}}$ on some open dense subspace).  If this WD rep is indecomposable for some ${\ell}$, we are in a quaternionic setting, and everything is unconditional.