The Hilbert modular surface for the ideal $$(\sqrt 5 )$$ and the Horrocks-Mumford bundle

1988 ◽  
Vol 198 (1) ◽  
pp. 95-116 ◽  
Author(s):  
K. Hulek ◽  
H. Lange
Keyword(s):  
Modular Surface ◽  
Hilbert Modular Surface ◽  
Hilbert Modular ◽  
The Ideal

scholarly journals TWO MODULI SPACES OF ABELIAN FOURFOLDS WITH AN AUTOMORPHISM OF ORDER FIVE

2012 ◽  
Vol 23 (10) ◽  
pp. 1250108 ◽  
Author(s):  
BERT VAN GEEMEN ◽  
MATTHIAS SCHÜTT
Keyword(s):  
Complete Intersection ◽  
Moduli Spaces ◽  
Compact Surface ◽  
Canonical Model ◽  
Modular Surface ◽  
Shimura Curve ◽  
Hilbert Modular Surface ◽  
Hilbert Modular

We find explicit projective models of a compact Shimura curve and of a (non-compact) surface which are the moduli spaces of principally polarized abelian fourfolds with an automorphism of order five. The surface has a 24-nodal canonical model in P4 which is the complete intersection of two S5-invariant cubics. It is dominated by a Hilbert modular surface and we give a modular interpretation for this. We also determine the L-series of these varieties as well as those of several modular covers of the Shimura curve.

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 Real multiplication through explicit correspondences

2016 ◽  
Vol 19 (A) ◽  
pp. 29-42 ◽  
Author(s):  
Abhinav Kumar ◽  
Ronen E. Mukamel
Keyword(s):  
Riemann Surfaces ◽  
Algebraic Models ◽  
Modular Surface ◽  
Real Multiplication ◽  
Universal Family ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Genus 2 ◽  
Genus 2 Curves ◽  
Do So

We compute equations for real multiplication on the divisor classes of genus-2 curves via algebraic correspondences. We do so by implementing van Wamelen’s method for computing equations for endomorphisms of Jacobians on examples drawn from the algebraic models for Hilbert modular surfaces computed by Elkies and Kumar. We also compute a correspondence over the universal family for the Hilbert modular surface of discriminant $5$ and use our equations to prove a conjecture of A. Wright on dynamics over the moduli space of Riemann surfaces.

scholarly journals Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1206 ◽  
Author(s):  
Alex Brandts ◽  
Tali Pinsky ◽  
Lior Silberman
Keyword(s):  
Class Group ◽  
Quadratic Field ◽  
Finite Collection ◽  
Ideal Class Group ◽  
Modular Surface ◽  
Closed Curves ◽  
Hyperbolic Three Manifolds ◽  
Unit Tangent Bundle ◽  
Real Quadratic ◽  
The Ideal

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) ∖ PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics.

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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