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.

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.

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.

scholarly journals Minimal weights of Hilbert modular forms in characteristic

2017 ◽  
Vol 153 (9) ◽  
pp. 1769-1778 ◽  
Author(s):  
Fred Diamond ◽  
Payman L Kassaei
Keyword(s):  
Modular Forms ◽  
Real Field ◽  
Hilbert Modular Forms ◽  
Hilbert Modular ◽  
Hilbert Modular Varieties ◽  
Totally Real ◽  
Modular Varieties ◽  
Totally Real Field

We consider mod $p$ Hilbert modular forms associated to a totally real field of degree $d$ in which $p$ is unramified. We prove that every such form arises by multiplication by partial Hasse invariants from one whose weight (a $d$-tuple of integers) lies in a certain cone contained in the set of non-negative weights, answering a question of Andreatta and Goren. The proof is based on properties of the Goren–Oort stratification on mod $p$ Hilbert modular varieties established by Goren and Oort, and Tian and Xiao.

scholarly journals ALGEBRAIC CYCLES AND TATE CLASSES ON HILBERT MODULAR VARIETIES

2014 ◽  
Vol 10 (01) ◽  
pp. 161-176
Author(s):  
JAYCE R. GETZ ◽  
HEEKYOUNG HAHN
Keyword(s):  
Discrete Series ◽  
Algebraic Cycles ◽  
Automorphic Representation ◽  
Abelian Extensions ◽  
Real Place ◽  
Hilbert Modular ◽  
Rank One ◽  
Hilbert Modular Varieties ◽  
Totally Real ◽  
Modular Varieties

Let E/ℚ be a totally real number field that is Galois over ℚ, and let π be a cuspidal, nondihedral automorphic representation of GL 2(𝔸E) that is in the lowest weight discrete series at every real place of E. The representation π cuts out a "motive" M ét (π∞) from the ℓ-adic middle degree intersection cohomology of an appropriate Hilbert modular variety. If ℓ is sufficiently large in a sense that depends on π we compute the dimension of the space of Tate classes in M ét (π∞). Moreover if the space of Tate classes on this motive over all finite abelian extensions k/E is at most of rank one as a Hecke module, we prove that the space of Tate classes in M ét (π∞) is spanned by algebraic cycles.

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