scholarly journals Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?

2020 ◽  
Author(s):  
Edward Frenkel ◽  
Keyword(s):  
Algebraic Curves ◽  
Analytic Theory ◽  
Automorphic Functions

On the Algebraic Curves Uniformized by Arithmetical Automorphic Functions

1967 ◽  
Vol 86 (3) ◽  
pp. 449 ◽  
Author(s):  
Koji Doi ◽  
Hidehisa Naganuma
Keyword(s):  
Algebraic Curves ◽  
Automorphic Functions

scholarly journals On The Uniformisation of Algebraic Curves of Genus 3

1930 ◽  
Vol 2 (2) ◽  
pp. 102-107 ◽  
Author(s):  
M. Mursi
Keyword(s):  
Algebraic Equation ◽  
Algebraic Curves ◽  
Image Position ◽  
Automorphic Functions

An algebraic equationdetermines, in general, s as a many valued function of z. If s and z can be expressed as one valued functions of a third variable t, then t is called the uniformising variable. As Poincaré showed, s and z are automorphic functions of t.

IV.—The Differential Equations Associated with the Uniformization of Certain Algebraic Curves

1958 ◽  
Vol 65 (1) ◽  
pp. 35-62 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Rational Function ◽  
Algebraic Equation ◽  
Algebraic Curves ◽  
Algebraic Equations ◽  
Branch Points ◽  
Unknown Parameters ◽  
Image Position ◽  
Automorphic Functions

SynopsisEvery algebraic equation can be uniformized by automorphic functions belonging to a certain group of bilinear transformations. In certain cases, such as for hyperelliptic equations, this group is a subgroup of the monodromic group of a differential equation of the formwhere R(z) is a rational function which, in general, contains unknown parameters as coefficients. A conjecture of E. T. Whittaker regarding the values of these parameters for the hyperelliptic case is proved for a wide variety of algebraic equations whose branch points possess certain symmetric properties, and is extended to equations of higher type. In several cases, the uniformizing functions belong to subgroups of the groups of the Riemann-Schwarz triangle functions.

GEOMETRIC THEORY OF MULTIDIMENSIONAL INTERPOLATION

2020 ◽  
Vol 2020 (1) ◽  
pp. 9-16
Author(s):  
Evgeniy Konopatskiy
Keyword(s):  
Response Function ◽  
Algebraic Curves ◽  
Geometric Theory ◽  
Analytical Description ◽  
Geometric Interpretation ◽  
Affine Geometry ◽  
Mathematical Apparatus ◽  
Multidimensional Interpolation ◽  
Pass Through

The paper presents a geometric theory of multidimensional interpolation based on invariants of affine geometry. The analytical description of geometric interpolants is performed within the framework of the mathematical apparatus BN-calculation using algebraic curves that pass through preset points. A geometric interpretation of the interaction of parameters, factors, and the response function is presented, which makes it possible to generalize the geometric theory of multidimensional interpolation in the direction of increasing the dimension of space. The conceptual principles of forming the tree of the geometric interpolant model as a geometric basis for modeling multi-factor processes and phenomena are described.

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