Determinant Formulae in Abelian Functions for a General Trigonal Curve of Degree Five

2012 ◽  
Vol 11 (2) ◽  
pp. 547-574 ◽  
Author(s):  
Yoshihiro Onishi
Keyword(s):  
Trigonal Curve ◽  
Abelian Functions

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 The Alexander module of a trigonal curve

2014 ◽  
Vol 30 (1) ◽  
pp. 25-64
Author(s):  
Alex Degtyarev
Keyword(s):  
Trigonal Curve

Introduction to the Classical Theory of Abelian Functions

2006 ◽  
Author(s):  
A. Markushevich
Keyword(s):  
Classical Theory ◽  
Abelian Functions

On the tritangent planes of a quadri-cubic space curve

1935 ◽  
Vol 4 (3) ◽  
pp. 159-169
Author(s):  
H. W. Richmond
Keyword(s):  
Plane Curve ◽  
Space Curve ◽  
Quadric Surface ◽  
Geometrical Aspect ◽  
Quartic Curve ◽  
Abelian Functions

With any twisted curve of order six is associated a system of planes, usually finite in number, which touch the curve at three distinct points. The curve with its system of tritangent planes possesses properties which recall the properties of a plane quartic curve and its system of bitangent lines; and this is specially true of the sextic which is the intersection of a cubic and a quadric surface. But whereas the properties of the plane curve were discovered by geometrical methods, such methods have only recently been applied with success to the space-curve; the earliest properties were obtained by Clebsch from his Theory of Abelian Functions. In the absence of any one place to which reference can conveniently be made, an account of these properties in their geometrical aspect will be useful.

scholarly journals VIII. On Abel’s Theorem and Abelian functions

1883 ◽  
Vol 174 ◽  
pp. 323-368
Keyword(s):  
Integral Function ◽  
Upper Limits ◽  
Abelian Functions ◽  
And Function

The present paper is divided into two sections. The object of Section I. is to obtain an expression for an integral more general than, but intimately connected with, that occurring in Abel’s theorem. The latter, as enunciated by Professor Rowe in his memoir in the Phil. Trans., 1881, is as follows:—If χ ( x, y ) = 0 be a rational algebraical equation between x and y , then an expression can always be found for Σ∫U dx / f ( x ) ∂χ / ∂y where f ( x ) is a function of x only, U a rational algebraical integral function of x and y , and the upper limits of the series of integrals are the roots of the eliminant with regard to y of χ ( x, y ) = 0 and function θ ( x, y ).

scholarly journals ABELIAN FUNCTIONS FOR TRIGONAL CURVES OF DEGREE FOUR AND DETERMINANTAL FORMULAE IN PURELY TRIGONAL CASE

2009 ◽  
Vol 20 (04) ◽  
pp. 427-441 ◽  
Author(s):  
YOSHIHIRO ÔNISHI
Keyword(s):  
General Theory ◽  
Trigonal Curves ◽  
Abelian Functions ◽  
Curves Of Genus Three

We give the Frobenius–Stickelberger-type and Kiepert-type determinantal formulae for purely trigonal curves of genus three. We explain also general theory of Abelian functions for any trigonal curves of genus three.

scholarly journals Deriving Bases for Abelian Functions Matthew England

2012 ◽  
Vol 11 (2) ◽  
pp. 617-654 ◽  
Author(s):  
Matthew England
Keyword(s):  
Abelian Functions
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