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 étale 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.
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Acknowledgements
Betina was supported by the EPSRC Grant EP/R006563/1. Betina was partially supported by START-Prize Y966 of the Austrian Science Fund (FWF). This work was done when Deo was a postdoc at the Mathematics Research Unit of University of Luxembourg. Fité was funded by the Excellence Program María de Maeztu MDM-2014-0445. Fité was partially supported by MTM2015-63829-P. Fité was partially supported by the Simons Foundation Grant 550033. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (Grant agreement no. 682152). We thank the anonymous referee for many useful suggestions which helped us in improving the exposition.
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Betina, A., Deo, S.V. & Fité, F. On the Hilbert eigenvariety at exotic and CM classical weight 1 points. Math. Z. 298, 1077–1096 (2021). https://doi.org/10.1007/s00209-020-02626-1
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DOI: https://doi.org/10.1007/s00209-020-02626-1