MATHEMATICS. 321 



MATHEMATICS. 



Morley, Frank., Johns Hopkins University, Baltimore, Maryland. Applica- 

 tion of Cremona groups to the solution of algebraic equations. (For previous 

 reports see Year Books Nos. 9-15.) 



In the series of three articles on "Point Sets and Cremona Groups"^ 

 certain facts were developed which raised a presumption that the 

 self -associated set of 2p+2 points in Sp could be connected with 

 the general theta functions in p variables, or, more specifically, that 

 the absolute invariants of the set of points could be expressed in 

 terms of theta modular functions. If this presumption is correct, 

 then the point set would furnish a geometric background for the 

 general theta function, just as the algebraic curve of genus p has served 

 for the particular Abelian theta function. Thus far the only known 

 cases in point are the hyperelliptic theta functions determined by 

 2p-\-2 points on the rational norm curve in Sp (a particular case of the 

 self -associated set), and the general theta functions for p = 3, which 

 are likewise Abelian. The possibility was strengthened further by 

 the discovery^ of configurations of theta characteristics which were 

 grouped in the same way as the absolute invariants of the point set. 



The investigations of the past year have disclosed that the point 

 set defines by purely projective processes a finite group G which is 

 isomorphic with the modular group H of the odd and even thetas. The 

 extended group, defined in Point Sets II, which is attached to the point 

 set and which for values beyond p = 3 is infinite and discontinuous, con- 

 tains an infinite invariant subgroup whose factor group is G. The 

 identification of this group with H establishes the presumption men- 

 tioned above, though the precise algebraic nature of the connection 

 remains to be investigated. 



The importance of the basis notation, as set forth in an earlier 

 paper,^ for the odd and even thetas is emphasized by the manner in 

 which the group G arises on the projective side, and this notation has 

 been developed further. 



iParts I, II, III, Trans. Amer. Math. Soc, vols. 16, 17, 18 (1915-17). 

 'An Isomorphism between Theta Characteristics and the (2p+2)-Point; Annals of M&the> 

 matics, Ser. II, vol. 17 (1916). 



