On the Hilbert eigenvariety at exotic and CM classical weight 1 points
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Mathematische Zeitschrift
On the Hilbert eigenvariety at exotic and CM classical weight 1 points Adel Betina1 · Shaunak V. Deo2 · Francesc Fité3 Received: 9 January 2019 / Accepted: 21 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 E corresponding to an ordinary p-stabilization of f . We show that if the p-adic Schanuel conjecture is true, then E is smooth at x if f has CM. If we additionally assume that F/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. Keywords Parallel weight one Hilbert modular forms · Deformation of Galois representations · Eigenvariety Mathematics Subject Classification 11F80 (Primary) · 11F41 · 11R37
1 Introduction The main goal of this paper is to study the geometry of the eigenvariety of Hilbert modular forms at classical points of parallel weight one. Before proceeding further, we will first fix some notations and describe the objects of interest. Let p be a prime number, F ⊆ Q be a
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Shaunak V. Deo [email protected] Adel Betina [email protected] Francesc Fité [email protected]
1
Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Wien, Austria
2
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
3
Department of Mathematics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
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totally real number field of degree d over Q with ring of integers O F , and n be an ideal of O F coprime to p. We will denote by E the p-adic Hilbert eigenvariety of tame level n constructed by Andreatta, Iovita and Pilloni in [1], parameterizing systems of Hecke eigenvalues of overconvergent cuspidal Hilbert modular eigenforms over F of tame level n, having weights of same parity and finite slope. Recall that there exists a locally finite morphism w = (k, v) : E → W called the weight map, where W is the rigid space over Q p representing morphisms × Z× p × (O F ⊗ Z p ) → Gm . Recall also that locally on E and W , the morphism w is finite, open and surjective, though it is not necessarily flat. Thus, the p-adic eigenvariety E is equidimensional of dimension d + 1. Let f be a classical cuspidal Hilbert modular eigenform over F of tame level n having weights of same parity. A p-stabilization of f with finite slope (when it exists) defines a point x on E . A well-known result of Hida ([11] if F = Q, [12] in general) asserts that w is étale at p-ordinary eigenforms of cohomological weights (i.e. all of its weights have the same parity and are at leas
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