On the connexion of algebraic functions with automorphic functions

1898 ◽  
Vol 63 (389-400) ◽  
pp. 267-268 ◽  
Keyword(s):  
Riemann Surface ◽  
Existence Theorem ◽  
Algebraic Equation ◽  
Rational Functions ◽  
Algebraic Relation ◽  
Periodic Functions ◽  
Algebraic Functions ◽  
Automorphic Functions

If u and z are variables connected by an algebraic equation, they are, in general, multiform functions of each other; the multiformity can be represented by a Riemann surface, to each point of which corresponds a pair of values of u and z . Poincaré and Klein have proved that a variable t exists, of which u and z are uniform automorphic functions; the existence-theorem, however, does not connect t analytically with u and z . When the genus ( genre, Geschlecht ) of the algebraic relation is zero oi unity, t can be found by known methods; the automorphic functions required are rational functions, and doubly periodic functions, in the two case respectively.

scholarly journals I. On the connexion of algebraic functions with automorphic functions

1899 ◽  
Vol 192 ◽  
pp. 1-32 ◽  
Keyword(s):  
Algebraic Curve ◽  
Natural Extension ◽  
Elliptic Functions ◽  
Rational Functions ◽  
Single Variable ◽  
Algebraic Functions ◽  
Automorphic Functions

It is well known that if f ( u, z ) = 0 . . . . . . . . . (1) is the equation of an algebraic curve of genus ( genre, Geschlecht ) zero, then u and z can be expressed as rational functions of a single variable t . If, however, the genus of the curve (1) is unity, u and z can be expressed as uniform elliptic functions of a variable t . The natural extension of these results was effected in 1881 by the discovery of automorphic functions; whatever be the genus of the curve (1), u and z can be expressed as uniform automorphic functions of a new variable.

Continuity properties of compact right topological groups

1979 ◽  
Vol 86 (3) ◽  
pp. 427-435 ◽  
Author(s):  
Paul Milnes
Keyword(s):  
Almost Periodic ◽  
Periodic Functions ◽  
Topological Groups ◽  
Almost Automorphic ◽  
Almost Automorphic Function ◽  
Almost Automorphic Functions ◽  
Continuity Properties ◽  
Enveloping Semigroups ◽  
The One ◽  
Automorphic Functions

AbstractCompact right topological groups appear naturally in topological dynamics. Some continuity properties of the one arising as an enveloping semigroup from the distal function are considered here (and, by way of comparison, the enveloping semigroups arising from two almost automorphic functions are discussed). The continuity properties are established either explicitly or by citing a theorem which is proved here and gives some characterizations of almost periodic functions. One characterization is proved using the result (essentially due to W. A. Veech) that a distal, almost automorphic function is almost periodic. A proof of this last result is also given.

A Symmetric Proof of the Riemann-Roch Theorem, and a New Form of the Unit Theorem

1952 ◽  
Vol 4 ◽  
pp. 136-148
Author(s):  
S. Beatty ◽  
N. D. Lane
Keyword(s):  
Algebraic Equation ◽  
Rational Functions ◽  
Independent Variable ◽  
New Form ◽  
Roch Theorem ◽  
Image Position ◽  
Complex Coefficients

Let F(z, u) denote1where F1(z),… , Fn(z) are rational functions of z with complex coefficients. We shall speak of F (z, u) = 0 as the fundamental algebraic equation and shall adopt z as the independent variable and u as the dependent, except in § 4, where we use x and y instead of them, and where it is understood that x and y are connected birationally with z and u.

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.

scholarly journals Automorphic forms on nondiscrete Möbius groups

1996 ◽  
Vol 38 (2) ◽  
pp. 249-253
Author(s):  
A. F. Beardon
Keyword(s):  
Riemann Surface ◽  
Discrete Group ◽  
Meromorphic Functions ◽  
Automorphic Forms ◽  
Quotient Space ◽  
General Situation ◽  
Möbius Group ◽  
Upper Half Plane ◽  
Möbius Groups ◽  
Automorphic Functions

If Г is a discrete Möbius group acting on the upper half-plane ℋ of the complex plane, the quotient space ℋ/Г is a Riemann surface ℛ and the automorphic functions on Г correspond to meromorphic functions on ℛ. If Г is a nondiscrete Möbius group acting on ℋ, then ℋ/Г is no longer a Riemann surface, and it is obvious that in this case there are no nonconstant automorphic functions on Г. The situation for automorphic forms is quite different. Automorphic forms of integral dimension for a discrete group Г correspond to meromorphic differentials on ℛ, but even if Г is nondiscrete it may still support nontrivial automorphic forms. The problem of classifying those nondiscrete Möbius groups which act on ℋ and which support nonconstant automorphic forms of arbitrary real dimension was raised and solved (rather indirectly) in [2] where, roughly speaking, function-theoretic methods are used to analyse all possible polynomial automorphic forms of integral dimension, and the results obtained then used to analyse the more general situation.

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