Period relations and the Tate conjecture for Hilbert modular surfaces

1987 ◽  
Vol 89 (2) ◽  
pp. 319-345 ◽  
Author(s):  
V. Kumar Murty ◽  
Dinakar Ramakrishnan
Keyword(s):  
Tate Conjecture ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular

scholarly journals Families of abelian surfaces with real multiplication over Hilbert modular surfaces

1982 ◽  
Vol 88 ◽  
pp. 17-53 ◽  
Author(s):  
G. van der Geer ◽  
K. Ueno
Keyword(s):  
Endomorphism Ring ◽  
Symplectic Group ◽  
Abelian Surface ◽  
Quadratic Relation ◽  
Holomorphic Map ◽  
Real Multiplication ◽  
Compact Complex Surfaces ◽  
Hilbert Modular Surfaces ◽  
Hilbert Modular ◽  
Abelian Functions

Around the beginning of this century G. Humbert ([9]) made a detailed study of the properties of compact complex surfaces which can be parametrized by singular abelian functions. A surface parametrized by singular abelian functions is the image under a holomorphic map of a singular abelian surface (i.e. an abelian surface whose endomorphism ring is larger than the ring of rational integers). Humbert showed that the periods of a singular abelian surface satisfy a quadratic relation with integral coefficients and he constructed an invariant D of such a relation with respect to the action of the integral symplectic group on the periods.

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 The Sato-Tate conjecture for Hilbert modular forms

2011 ◽  
Vol 24 (2) ◽  
pp. 411-411 ◽  
Author(s):  
Thomas Barnet-Lamb ◽  
Toby Gee ◽  
David Geraghty
Keyword(s):  
Modular Forms ◽  
Hilbert Modular Forms ◽  
Tate Conjecture ◽  
Hilbert Modular

scholarly journals Tate Cycles on a Product of Two Hilbert Modular Surfaces

2000 ◽  
Vol 80 (1) ◽  
pp. 25-43 ◽  
Author(s):  
V.Kumar Murty ◽  
Dipendra Prasad
Keyword(s):  
Hilbert Modular Surfaces ◽  
Hilbert Modular
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