Let be a prime (assumed for simplicity), and fix a character . Let denote the -component of weight space, and let denote the -portion of the eigencurve (of whatever tame level ). For a point , write as shorthand for . Let denote the portion of cut out by the condition , so this is some annulus extending towards the boundary of . There has been some recent remarkable progress on Coleman’s equally remarkable conjecture:
Conjecture (Coleman): There is an -Banach module of “characteristic overconvergent modular forms” such that is an entire power series which coincides with the reduction mod of the Fredholm series of acting on overconvergent modular forms over .
Furthermore, if denote the breakpoints of the Newton polygon of , then for any point , the breakpoints of the Newton polygon of the specialization (which coincides with ) are exactly the points .
The second part of this conjecture implies (by pure rigid analysis) that over , breaks apart into a disjoint union of annuli, each finite and flat over . This latter phenomenon was observed by Buzzard and Kilford in the case 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 be any irreducible component of . Suppose Coleman’s conjecture is true. Then contains infinitely many weight two classical points of noncritical slope, and the complement of in contains at most finitely many points.
Proof. The curve maps finitely and surjectively to its spectral variety, which is some Fredholm hypersuface . Again let denote the projection onto . By e.g. Proposition 4.1.3 in my eigenvariety paper, is Zariski-open, and by the above conjecture it contains all of , so the complement of is a Zariski-closed analytic subset of any qc open containing , and is therefore finite. We also see from Prop. 4.1.3 that cannot “avoid the boundary”, and in particular (assuming again the conjecture) is a nonempty disjoint union of annuli. Let be any one of these annuli.
For the first part, let , be a sequence of characters of of exact order , and let be the point of corresponding to . (By my conventions, classical points over correspond to classical newforms of weight 2 and level .) Since , these points excur towards the boundary as . Let be any preimage of in . By the conjecture, there is some such that the slope of any point is equal to . In particular, the slope of is for , so Coleman’s classicality theorem kicks in and we conclude: any point in lying over is classical for sufficiently large .
Given any irreducible component as above, one can define an inertial Weil-Deligne representation for any prime (this is the inertial WD rep. associated with the Galois representations carried by on some open dense subspace). If this WD rep is indecomposable for some , we are in a quaternionic setting, and everything is unconditional.