Abelian Functions

2011 ◽  
pp. 347-426
Author(s):  
Eberhard Freitag
Keyword(s):  
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.

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

Linear forms in algebraic points of Abelian functions. II

1976 ◽  
Vol 79 (1) ◽  
pp. 55-70 ◽  
Author(s):  
D. W. Masser
Keyword(s):  
Analytic Function ◽  
Maximum Modulus ◽  
Complex Variables ◽  
Several Complex Variables ◽  
Maximum Modulus Principle ◽  
Linear Forms ◽  
Extrapolation Procedure ◽  
Abelian Functions

In this paper we continue to develop the apparatus needed for the proof of the theorem announced in (11). We retain the notation of (11) together with the assumptions made there about the field of Abelian functions. This section deals with properties of more general functions holomorphic on Cn. When n = 1 the extrapolation procedure in problems of transcendence is essentially the maximum modulus principle together with the act of dividing out zeros of an analytic function. For n > 1, however, this approach is not possible, and some mild theory of several complex variables is required. This was first used in the context of transcendence by Bombieri and Lang in (2) and (12), and we now give a brief account of the basic constructions of their papers.

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