Tsuchihashi's cusp singularities and automorphisms of Hilbert modular variety cusps

1986 ◽  
Vol 274 (4) ◽  
pp. 691-698 ◽  
Author(s):  
G. K. Sankaran
Keyword(s):  
Modular Variety ◽  
Hilbert Modular ◽  
Hilbert Modular Variety ◽  
Cusp Singularities

scholarly journals On Ihara’s lemma for Hilbert modular varieties

2009 ◽  
Vol 145 (5) ◽  
pp. 1114-1146 ◽  
Author(s):  
Mladen Dimitrov
Keyword(s):  
Selmer Group ◽  
Absolute Galois Group ◽  
Modular Variety ◽  
Low Weight ◽  
Hilbert Modular ◽  
Finite Set ◽  
Hilbert Modular Varieties ◽  
Totally Real ◽  
Modular Varieties ◽  
Hilbert Modular Variety

AbstractLet ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.

Effective resolution of cusps on Hilbert modular varieties

1986 ◽  
Vol 99 (1) ◽  
pp. 51-57
Author(s):  
G. K. Sankaran
Keyword(s):  
Modular Variety ◽  
Hilbert Modular ◽  
Hilbert Modular Varieties ◽  
Modular Varieties ◽  
Hilbert Modular Variety

In this paper, we use the Shintani decomposition, known to number theorists, to describe an effective method of finding a resolution of the cusps of a Hilbert modular variety, in any dimension.

Special Values of Class Group L-Functions for CM Fields

2010 ◽  
Vol 62 (1) ◽  
pp. 157-181
Author(s):  
Riad Masri
Keyword(s):  
Modular Function ◽  
Special Values ◽  
Relative Class ◽  
Hilbert Modular ◽  
Cm Points ◽  
Relative Class Number ◽  
Rational Multiple ◽  
Totally Real ◽  
Elliptic Units ◽  
Hilbert Modular Variety

Abstract. Let H be the Hilbert class field of a CM number field K with maximal totally real subfield F of degree n over ℚ. We evaluate the second term in the Taylor expansion at s = 0 of the Galoisequivariant L-function ΘS∞(s) associated to the unramified abelian characters of Gal(H/K). This is an identity in the group ring C[Gal(H/K)] expressing Θ (n)S∞ (0) as essentially a linear combination of logarithms of special values ﹛Ψ (zσ)﹜, where Ψ: ℍn →ℝ is a Hilbert modular function for a congruence subgroup of SL2 (OF) and ﹛zσ : σ ∈ Gal(H/K)﹜ are CM points on a universal Hilbert modular variety. We apply this result to express the relative class number hH/hK as a rational multiple of the determinant of an (hK − 1) × (hK − 1) matrix of logarithms of ratios of special values Ψ (zσ), thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for Ψ (zσ) in terms of exponentials of special values of L-functions.

scholarly journals Cusps of Hilbert modular varieties

2008 ◽  
Vol 144 (3) ◽  
pp. 749-759 ◽  
Author(s):  
D. B. McREYNOLDS
Keyword(s):  
Cross Section ◽  
Cross Sections ◽  
Modular Surface ◽  
Hilbert Modular Surface ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Necessary And Sufficient ◽  
Hilbert Modular Varieties ◽  
Modular Varieties ◽  
Hilbert Modular Variety

AbstractMotivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifoldMto be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeo morphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3–manifolds that cannot arise as a cusp cross-section of a 1–cusped nonsingular Hilbert modular surface.

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