Today I wanted to mention a theorem of Chenever about components of the Eigencurve. Let \(\mathcal{W}\) denote weight space (which is basically a union of discs), and let
\(\pi: \mathcal{E} \rightarrow \mathcal{W}\)
be the Coleman-Mazur eigencurve together with its natural map to \(\mathcal{W}.\) It will do well to also consider the versions of the eigencurve corresponding to quaternion algebras \(D/\mathbf{Q}\) as well.
Theorem: [Chenevier] Suppose that
- \(\mathcal{E}\) has “no holes” (that is, a family of finite slope forms over the punctured disc extends over the missing point),
- The “halo” of \(\mathcal{E}\) is given by a union of finite flat components whose slope tends to zero as \(x \in \mathcal{W}\) tends to the boundary of the disc.
Then every non-ordinary component of \(\mathcal{E}\) has infinite degree.
In particular, since both of these theorems are now known in many cases (properness by Hansheng Diao and Ruochuan Liu, and haloness by Ruochuan Liu, Daqing Wan, and Liang Xiao, at least in the definite quaternion algebra case), the conclusion is also known.
The proof is basically the following. Given a component \(C\) of finite degree, the first assumption implies that it actually is proper and finite. One may then consider the norm of \(U_p\) on \(C\) to the Iwasawa algebra to obtain a bounded (hence Iwasawa) function \(F = \mathrm{Norm}(U_p).\) This function cannot have any zeros (again by properness), and hence, by the Weierstrass preparation theorem, it is a power of \(p\) times a unit. But that implies that \(F\) has constant valuation near the boundary, which contradicts the fact that the slopes are tending to zero (except in the ordinary case).
Naturally one may ask whether \(\mathcal{E}\) has only finitely many components, although this seems somewhat harder to prove.