Smoothness of definite unitary eigenvarieties at critical points

TL;DR

This paper establishes an upper bound for the tangent space dimension at classical points of eigenvarieties for definite unitary groups, showing smoothness at critical points when the bound is tight and linking local rings to Galois deformation rings.

Contribution

It provides a new upper bound for tangent space dimensions at classical points of eigenvarieties, especially at critical points, and identifies conditions for smoothness and Galois deformation ring structures.

Findings

Bound matches eigenvariety dimension at certain points

Eigenvarieties are smooth when the bound is minimized

Local rings are universal Galois deformation rings at these points

Abstract

We compute an upper bound for the dimension of the tangent spaces at classical points of certain eigenvarieties associated with definite unitary groups, especially including the so-called critically refined cases. Our bound is given in terms of "critical types" and when our bound is minimized it matches the dimension of the eigenvariety. In those cases, which we explicitly determine, the eigenvariety is necessarily smooth and our proof also shows that the completed local ring on the eigenvariety is naturally a certain universal Galois deformation ring.