A modular group and Riemann surfaces of genus 2

1975 ◽  
Vol 142 (3) ◽  
pp. 205-219 ◽  
Author(s):  
Linda Keen
Keyword(s):  
Riemann Surfaces ◽  
Modular Group ◽  
Genus 2

Small eigenvalues on Riemann surfaces of genus 2

1991 ◽  
Vol 106 (1) ◽  
pp. 121-138 ◽  
Author(s):  
Paul Schmutz
Keyword(s):  
Riemann Surfaces ◽  
Genus 2 ◽  
Small Eigenvalues

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.

Theta Constants, Riemann Surfaces and the Modular Group

2001 ◽  
Author(s):  
Hershel Farkas ◽  
Irwin Kra
Keyword(s):  
Riemann Surfaces ◽  
Modular Group

scholarly journals Poincaré's theorem for the modular group of real Riemann surfaces

2009 ◽  
Vol 27 (5) ◽  
pp. 680-690 ◽  
Author(s):  
Antonio F. Costa ◽  
Sergey M. Natanzon
Keyword(s):  
Riemann Surfaces ◽  
Modular Group ◽  
Real Riemann Surfaces

scholarly journals GENUS 2 SEMI-REGULAR COVERINGS WITH LIFTING SYMMETRIES

2008 ◽  
Vol 50 (3) ◽  
pp. 379-394 ◽  
Author(s):  
YOLANDA FUERTES ◽  
ALEXANDER MEDNYKH
Keyword(s):  
Riemann Surfaces ◽  
Compact Riemann Surface ◽  
Riemann Sphere ◽  
Target Surface ◽  
Algebraic Equations ◽  
Main Ingredient ◽  
Joint Paper ◽  
Genus 2 ◽  
Compact Riemann Surfaces ◽  
Uniform Covering

AbstractIn this paper, we obtain algebraic equations for all genus 2 compact Riemann surfaces that admit a semi-regular (or uniform) covering of the Riemann sphere with more than two lifting symmetries. By a lifting symmetry, we mean an automorphism of the target surface which can be lifted to the covering. We restrict ourselves to the genus 2 surfaces in order to make computations easier and to make possible to find their algebraic equations as well. At the same time, the main ingredient (Main Proposition) depends neither on the genus, nor on the order of the group of lifting symmetries. Because of this, the paper can be thought as a generalisation for the non-normal case to the question of lifting automorphisms of a compact Riemann surface to a normal covering, treated, for instance, by E. Bujalance and M. Conder in a joint paper, or by P. Turbek solely.

Moduli for Special Riemann Surfaces of Genus 2

1969 ◽  
Vol 144 ◽  
pp. 95 ◽  
Author(s):  
John Schiller
Keyword(s):  
Riemann Surfaces ◽  
Genus 2

scholarly journals Perfect Non-Extremal Riemann Surfaces

2000 ◽  
Vol 43 (1) ◽  
pp. 115-125 ◽  
Author(s):  
Paul Schmutz Schaller
Keyword(s):  
Riemann Surface ◽  
Quadratic Forms ◽  
Riemann Surfaces ◽  
Closed Geodesic ◽  
Modular Group ◽  
Infinite Family ◽  
Positive Definite ◽  
Normal Subgroups

AbstractAn infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group PSL(2, ℤ) of level 6. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function syst which associates to every Riemann surface M the length of a systole, which is a shortest closed geodesic of M.

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