# Irreducible components of the eigencurve of finite degree are finite over the weight space

## TL;DR

This paper proves that finite degree irreducible components of the p-adic eigencurve are actually finite over the weight space, supporting the conjecture that only ordinary components have finite degree.

## Contribution

It establishes that any finite degree irreducible component of the eigencurve must be finite over the weight space, confirming a key aspect of the conjecture.

## Key findings

- Finite degree components are finite over the weight space.
- Supports the conjecture that only ordinary components are finite.
- Provides a link between geometric properties and the conjecture.

## Abstract

Let p be a rational prime and N a positive integer which is prime to p. Let W be the p-adic weight space for GL_{2,Q}. Let C_N be the p-adic Coleman-Mazur eigencurve of tame level N. In this paper, we prove that any irreducible component of C_N which is of finite degree over W is in fact finite over W. Combined with an argument of Chenevier and a conjecture of Coleman-Mazur-Buzzard-Kilford (which has been proven in special cases, and for general quaternionic eigencurves) this shows that the only finite degree components of the eigencurve are the ordinary components.