On the Hilbert eigenvariety at exotic and CM classical weight 1 points

TL;DR

This paper investigates the smoothness and local structure of the Hilbert eigenvariety at certain weight 1 points, linking geometric properties to Galois deformation rings and assuming the p-adic Schanuel Conjecture.

Contribution

It establishes conditions under which the eigenvariety is smooth or étale at CM and exotic points, connecting these properties to Galois deformation theory.

Findings

Eigenvariety is smooth at CM points assuming p-adic Schanuel Conjecture.

The weight map is étale at certain points with CM or exotic projective image.

Local rings are isomorphic to universal nearly ordinary Galois deformation rings.

Abstract

Let $F$ be a totally real number field and let $f$ be a classical cuspidal $p$-regular Hilbert modular eigenform over $F$ of parallel weight $1$. Let $x$ be the point on the $p$-adic Hilbert eigenvariety $\mathcal E$ corresponding to an ordinary $p$-stabilization of $f$. We show that if the $p$-adic Schanuel Conjecture is true, then $\mathcal E$ is smooth at $x$ if $f$ has CM. If we additionally assume that $F/\mathbb Q$ is Galois, we show that the weight map is \'etale at $x$ if $f$ has either CM or exotic projective image (which is the case for almost all cuspidal Hilbert modular eigenforms of parallel weight $1$). We prove these results by showing that the completed local ring of the eigenvariety at $x$ is isomorphic to a universal nearly ordinary Galois deformation ring.