Voi,. 7, 1921 
MATHEMATICS: A. B. COBLE 
245- 
GEOMETRIC ASPECTS OF THE ABELIAN MODULAR 
FUNCTIONS OF GENUS FOUR (I) 
By Arthur B. Coble 1 
Department of Mathematics, University of Illinois 
Communicated by B. H. Moore, June 21, 1921 
1. Introduction. — The plane curve of genus 4 has a canonical series 
g| and is mapped from the plane by the canonical adjoin ts into the normal 
curve of genus 4, a space sextic which is the complete intersection of a 
quadric and a cubic surface. If we denote a point of this quadric by the 
parameters t, r of the cross generators through it the equation of this 
sextic is F = (ar) 3 (at) 3 — 0. For geometric purposes we may define 
a modular function to be any rational or irrational invariant of the form 
F, bi-cubic in the digredient binary variables r, t; for transcendental 
purposes it is desirable to restrict this definition by requiring further 
that this invariant, regarded as a function of the normalized periods co# 
of the abelian integrals attached to the curve, be uniform. 
There seems to be an unusually rich variety of geometric entities which 
center about this normal curve. Some of these have received independent 
investigation. It is the purpose of this series of abstracts to indicate a 
number of new relations among these various entities and to connect each 
with the normal sextic F. The methods employed are in the main geo- 
metric. Direct algebraic attack on problems which contain nine irremov- 
able constants, or moduli, is difficult. However much information is 
gained by a free use of algebraic forms containing sets of variables drawn 
from different domains. Both finite and infinite discontinuous groups 
are utilized at various times. 
2. The Figure of Two Space Cubic Curves. — White 2 has introduced for 
other purposes the interpretation of the form F = 0 as the incidence con- 
dition of the point r of the space cubic curve Ci(t) and the plane t of 
the space cubic Ci(0- There is dually an incidence condition of plane 
t of Ci(t) and point t of C 2 (X)> expressed by a form F= (At) 3 (At) 3 = 0. 
We call the sextics of genus 4 determined by F = 0 and F = 0 reciprocal. 
Each is the same covariant of degree three of the other. 
3. A. Set of Four Mutually Related Rational Plane Sextics. — On each 
of the cubic curves G(t), C 2 (0 regarded as a point locus there is a net of 
point quadrics Qi, Q 2 , respectively; on each regarded as a plane locus there 
is a net of quadric envelopes, Qi, Q 2 , respectively. The net Qi cuts the 
curve C 2 (£) in 00 2 sets of six points which lie in an lt(t)- An if on a binary 
domain may be visualized as the line sections of a protectively definite 
(but not localized) rational plane sextic S 2 (t) . Thus the four nets determine 
the four rational plane sextics of the array 
