scholarly journals Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic curves defined over $\overline{\mathbb{Q}}$

1998 ◽  
Vol 2 (4) ◽  
pp. 875-920 ◽  
Author(s):  
M. Mulase ◽  
M. Penkava
Keyword(s):  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Quadratic Differentials ◽  
Ribbon Graphs

scholarly journals Tau Function and Moduli of Meromorphic Quadratic Differential

2022 ◽  
Author(s):  
Dmitry Korotkin ◽  
◽  
Peter Zograf ◽  
Keyword(s):  
Moduli Spaces ◽  
Quadratic Differential ◽  
Algebraic Curves ◽  
Picard Group ◽  
Quadratic Differentials ◽  
Tau Function ◽  
Tau Functions

The Bergman tau functions are applied to the study of the Picard group of moduli spaces of quadratic differentials with at most n simple poles on genus g complex algebraic curves. This generalizes our previous results on moduli spaces of holomorphic quadratic differentials.

On the structure of the Whittaker sublocus of the moduli space of algebraic curves

2006 ◽  
Vol 136 (2) ◽  
pp. 337-346
Author(s):  
Ernesto Girondo ◽  
Gabino González-Diez
Keyword(s):  
Differential Equations ◽  
Algebraic Equation ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Hyperelliptic Curves ◽  
Teichmuller Theory ◽  
Compactness Result ◽  
Teichmüller Theory ◽  
Fuchsian Differential Equations

We prove the compactness of the Whittaker sublocus of the moduli space of Riemann surfaces (complex algebraic curves). This is the subset of points representing hyperelliptic curves that satisfy Whittaker's conjecture on the uniformization of hyperelliptic curves via the monodromy of Fuchsian differential equations. In the last part of the paper we devote our attention to the statement made by R. A. Rankin more than 40 years ago, to the effect that the conjecture ‘has not been proved for any algebraic equation containing irremovable arbitrary constants’. We combine our compactness result with other facts about Teichmüller theory to show that, in the most natural interpretations of this statement we can think of, this result is, in fact, impossible.

ON THE CONNECTEDNESS OF THE LOCUS OF REAL ELLIPTIC-HYPERELLIPTIC RIEMANN SURFACES

2009 ◽  
Vol 20 (08) ◽  
pp. 1069-1080 ◽  
Author(s):  
JOSÉ A. BUJALANCE ◽  
ANTONIO F. COSTA ◽  
ANA M. PORTO
Keyword(s):  
Riemann Surface ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Algebraic Curves ◽  
Orbit Space ◽  
Previous Article ◽  
Connected Components ◽  
Hyperelliptic Involution ◽  
Hyperelliptic Surfaces ◽  
Genus One

A Riemann surface X of genus g > 2 is elliptic-hyperelliptic if it admits a conformal involution h such that the orbit space X/〈h〉 has genus one. This elliptic-hyperelliptic involution h is unique for g > 5 [1]. In a previous article [3], we established the non-connectedness of the subspace [Formula: see text] of real elliptic-hyperelliptic algebraic curves in the moduli space [Formula: see text] of Riemann surfaces of genus g, when g is even and > 5. In this paper we improve this result and give a complete answer to the connectedness problem of the space [Formula: see text] of real elliptic-hyperelliptic surfaces of genus > 5: we show that [Formula: see text] is connected if g is odd and has exactly two connected components if g is even; in both cases the closure [Formula: see text] of [Formula: see text] in the compactified moduli space [Formula: see text] is connected.

scholarly journals GENERALISED FERMAT HYPERMAPS AND GALOIS ORBITS

2009 ◽  
Vol 51 (2) ◽  
pp. 289-299 ◽  
Author(s):  
ANTOINE D. COSTE ◽  
GARETH A. JONES ◽  
MANFRED STREIT ◽  
JÜRGEN WOLFART
Keyword(s):  
Galois Group ◽  
Riemann Surfaces ◽  
Prime Power ◽  
Algebraic Curves ◽  
Complete Bipartite Graph ◽  
Number Fields ◽  
Absolute Galois Group ◽  
Graph Theoretic ◽  
Galois Orbits ◽  
Fermat Curves

AbstractWe consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods.

scholarly journals Automorphism Groups of Symmetric and Pseudo-real Riemann Surfaces

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Ewa Tyszkowska
Keyword(s):  
Riemann Surfaces ◽  
Algebraic Curve ◽  
Algebraic Curves ◽  
Automorphism Groups ◽  
Real Riemann Surfaces ◽  
Compact Riemann Surfaces

AbstractThe category of smooth, irreducible, projective, complex algebraic curves is equivalent to the category of compact Riemann surfaces. We study automorphism groups of Riemann surfaces which are equivalent to complex algebraic curves with real moduli. A complex algebraic curve C has real moduli when the corresponding surface $$X_C$$ X C admits an anti-conformal automorphism. If no such an automorphism is an involution (symmetry), then the surface $$X_C$$ X C is called pseudo-real and the curve C is isomorphic to its conjugate, but is not definable over reals. Otherwise, the surface $$X_C$$ X C is called symmetric and the curve C is real.

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