On the eigenvariety of Hilbert modular forms at classical parallel weight one points with dihedral projective image

Deo, Shaunak

doi:10.1090/tran/7064

On the eigenvariety of Hilbert modular forms at classical parallel weight one points with dihedral projective image
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Trans. Amer. Math. Soc. 370 (2018), 3885-3912 Request permission

Abstract:

We show that the $p$-adic eigenvariety constructed by Andreatta-Iovita-Pilloni, parameterizing cuspidal Hilbert modular eigenforms defined over a totally real field $F$, is smooth at certain classical parallel weight one points which are regular at every place of $F$ above $p$ and also determine whether the map to the weight space at those points is étale or not. We prove these results assuming the Leopoldt conjecture for certain quadratic extensions of $F$ in some cases, assuming the $p$-adic Schanuel conjecture in some cases, and unconditionally in some cases, using the deformation theory of Galois representations. As a consequence, we also determine whether the cuspidal part of the $1$-dimensional parallel weight eigenvariety, constructed by Kisin-Lai, is smooth or not at those points.

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