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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 toan ordinary $p$-stabilization of $f$. We show that if the $p$-adic SchanuelConjecture is true, then $\mathcal E$ is smooth at $x$ if $f$ has CM. If weadditionally assume that $F/\mathbb Q$ is Galois, we show that the weight mapis \'etale at $x$ if $f$ has either CM or exotic projective image (which is thecase for almost all cuspidal Hilbert modular eigenforms of parallel weight$1$). We prove these results by showing that the completed local ring of theeigenvariety at $x$ is isomorphic to a universal nearly ordinary Galoisdeformation ring.

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