78 Mr Richmond, On Automorphic Functions 



On Automorphic Functions and the general theory of Algebraic 

 Curves. By H. W. Richmond, M.A., King's College. 



[Received 19 February 1903.] 



The propositions enunciated in this short abstract are con- 

 cerned with the theory of algebraic curves in its broadest aspect, 

 which embraces curves of any genus (or deficiency) and of any 

 order, belonging 1 to space of any number of dimensions : they are 

 immediate consequences of the researches of Humbert upon the 

 application of Poincare's automorphic functions to plane curves. 

 These researches put us in possession of the parametric equations 

 of the most general algebraic curve conceivable, i.e. equations in 

 which the homogeneous coordinates of points of the curve are 

 expressed by ratios of uniform functions of a parameter ; — rational 

 functions if the genus p = 0, elliptic o--functions if p = 1, holo- 

 morphic thetafuchsian functions if p > 1. 



Curves in which the coordinates of points are expressed by 

 functions founded upon the same group of substitutions are called 

 Curves of the Group. All rational curves (p — 0) are considered 

 as belonging to a group, as are also all elliptic curves (p = 1) 

 whose points have their coordinates expressed by elliptic functions 

 with the same periods. All propositions here deal with curves 

 which belong to a group. 



Prop. I. Between any two curves of a group a rational point- 

 to-point correspondence may be established, corresponding points 

 having equal parameters, and the coordinates of any point of 

 either curve being expressible rationally in terms of those of the 

 corresponding point of the other curve. 



Prop. II. A curve which belongs to space of d dimensions 

 is defined by the ratios of d+1 coordinates (x Q , x l9 x 2 ~, ... x d ) 

 expressed as uniform functions of a parameter, among which no 

 linear syzygy can hold. Any three of these (x , x 1} x 2 ) are connected 

 by a homogeneous relation of a degree normally equal to and 

 never greater than the order of the curve ; the ratio of any 

 other coordinate x r to x may as a rule be expressed in rational 

 terms of x 1 jx and x 2 /x . 



1 A curve ' belongs to ' space of d dimensions when it lies wholly in space of 

 d dimensions and does not lie wholly in any space of fewer than d dimensions. 



