scholarly journals Arithmetic intersection on a Hilbert modular surface and the Faltings height

2013 ◽  
Vol 17 (2) ◽  
pp. 335-382 ◽  
Author(s):  
Tonghai Yang
Keyword(s):  
Modular Surface ◽  
Hilbert Modular Surface ◽  
Hilbert Modular ◽  
Faltings Height

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.

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.

The second main theorem for entire curves into Hilbert modular surfaces

2015 ◽  
Vol 27 (4) ◽  
Author(s):  
Yusaku Tiba
Keyword(s):  
General Type ◽  
Second Main Theorem ◽  
Modular Surface ◽  
Normal Crossing ◽  
Codimension One ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Surface Of General Type ◽  
Simple Normal Crossing ◽  
The Second Main Theorem

AbstractThe main goal of this article is to prove a second main theorem for entire curves into Hilbert modular surfaces. As an application of our main theorem, we obtain a condition ensuring that entire curves in a Hilbert modular surface of general type are contained in the exceptional divisors. We also show a second main theorem for a simple normal crossing divisor which is tangent to a holomorphic distribution of codimension one on a smooth projective algebraic manifold.

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