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

@article{Betina2018OnTH, title={On the Hilbert eigenvariety at exotic and CM classical weight 1 points}, author={Adel Betina and Shaunak V. Deo and Francesc Fit'e}, journal={arXiv: Number Theory}, year={2018} }

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… Expand

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#### References

SHOWING 1-10 OF 42 REFERENCES

On the failure of Gorensteinness at weight 1 Eisenstein points of the eigencurve

- Mathematics
- 2018

We prove that the cuspidal $p$-adic eigencurve $C_{cusp}$ is etale over the weight space at any classical weight $1$ Eisenstein point $f$. Further we show that $C_{cusp}$ meets transversely at $f$… Expand

Geometry of the eigencurve at CM points and trivial zeros of Katz p-adic L-functions

- Mathematics
- 2019

The primary goal of this paper is to study the geometry of the p-adic eigencurve at a point f corresponding to a weight one theta series irregular at p. We show that f belongs to exactly three or… Expand

STARK POINTS AND $p$-ADIC ITERATED INTEGRALS ATTACHED TO MODULAR FORMS OF WEIGHT ONE

- Mathematics
- Forum of Mathematics, Pi
- 2015

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let ${\it\varrho}_{\flat }$ and ${\it\varrho}_{\sharp }$ be odd two-dimensional Artin representations for which ${\it\varrho}_{\flat }\otimes… Expand

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

- Mathematics
- 2017

We show that the p-adic eigenvariety constructed by AndreattaIovita-Pilloni, parameterizing cuspidal Hilbert modular eigenforms defined over a totally real field F , is smooth at certain classical… Expand

On Nearly Ordinary Hecke Algebras for $GL(2)$ over Totally Real Fields

- Mathematics
- 1989

Since this work is a continuation of our previous paper [8], we shall suppose the familiarity on the reader's part with the result and the notation in [8]. Especially we fix a rational prime p and a… Expand

Iwasawa Theory for Artin Representations, I

- Mathematics
- 2018

This article is the first of a pair of articles dealing with the Iwasawa theory of modular forms of weight 1 and, more generally, of Artin representations satisfying certain conditions. The main… Expand

Les Vari\'et\'es de Hecke-Hilbert aux points classiques de poids $1$

- Mathematics
- 2016

We show that the Eigenvariety attached to Hilbert modular forms over a totally real field $F$ is smooth at the points corresponding to certain classical weight one theta series and we give a precise… Expand

On Galois representations associated to Hilbert modular forms.

- Mathematics
- 1997

In this paper, we prove that, to any Hilbert cuspidal eigenform, one may attach a compatible system of Galois representations. This result extends the analogous results of Deligne and Deligne–Serre… Expand

Overconvergent Hilbert modular forms

- Mathematics
- 2005

<abstract abstract-type="TeX"><p>We generalize the construction of the eigencurve by Coleman-Mazur to the setting of totally real fields, and show that a finite slope Hilbert modular eigenform can be… Expand

Anti-cyclotomic Katz $p$-adic $L$-functions and congruence modules

- Mathematics
- 1993

— The purpose of this paper is to prove the divisibility of the characteristic power series of the congruence module of a Hida /?-adic family of theta series coming from a CM-field (with fixed… Expand