The ring of Hilbert modular forms for real quadratic fields of small discriminant

1987 ◽  
pp. 501-536
Author(s):  
Friedrich Hirzebruch
Keyword(s):  
Modular Forms ◽  
Quadratic Fields ◽  
Hilbert Modular Forms ◽  
Real Quadratic Fields ◽  
Hilbert Modular ◽  
Real Quadratic

scholarly journals Ramification of the Eigencurve at Classical RM Points

2019 ◽  
Vol 72 (1) ◽  
pp. 57-88
Author(s):  
Adel Betina
Keyword(s):  
Modular Forms ◽  
Quadratic Field ◽  
Galois Representations ◽  
Theta Series ◽  
Base Change ◽  
Hilbert Modular Forms ◽  
Deformation Problem ◽  
Ramification Index ◽  
Hilbert Modular ◽  
Real Quadratic

AbstractJ. Bellaïche and M. Dimitrov showed that the $p$-adic eigencurve is smooth but not étale over the weight space at $p$-regular theta series attached to a character of a real quadratic field $F$ in which $p$ splits. In this paper we prove the existence of an isomorphism between the subring fixed by the Atkin–Lehner involution of the completed local ring of the eigencurve at these points and a universal ring representing a pseudo-deformation problem. Additionally, we give a precise criterion for which the ramification index is exactly 2. We finish this paper by proving the smoothness of the nearly ordinary and ordinary Hecke algebras for Hilbert modular forms over $F$ at the overconvergent cuspidal Eisenstein points, being the base change lift for $\text{GL}(2)_{/F}$ of these theta series. Our approach uses deformations and pseudo-deformations of reducible Galois representations.

RESTRICTIONS OF HILBERT MODULAR FORMS

2009 ◽  
Vol 05 (01) ◽  
pp. 67-80
Author(s):  
NAJIB OULED AZAIEZ
Keyword(s):  
Modular Forms ◽  
Modular Group ◽  
Quadratic Field ◽  
Hilbert Modular Forms ◽  
Real Quadratic Field ◽  
Modular Surface ◽  
Modular Curve ◽  
Hilbert Modular Group ◽  
Hilbert Modular ◽  
Real Quadratic

Let Γ ⊂ PSL (2, ℝ) be a discrete and finite covolume subgroup. We suppose that the modular curve [Formula: see text] is "embedded" in a Hilbert modular surface [Formula: see text], where ΓK is the Hilbert modular group associated to a real quadratic field K. We define a sequence of restrictions (ρn)n∈ℕ of Hilbert modular forms for ΓK to modular forms for Γ. We denote by Mk1, k2(ΓK) the space of Hilbert modular forms of weight (k1, k2) for ΓK. We prove that ∑n∈ℕ ∑k1, k2∈ℕ ρn(Mk1, k2(ΓK)) is a subalgebra closed under Rankin–Cohen brackets of the algebra ⊕k∈ℕ Mk(Γ) of modular forms for Γ.

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