MATHEMATICS: 0. E. GLENN 
107 
COVARIANTS OF BINARY MODULAR GROUPS 
By Oliver Edmunds Glenn 
Defartment of Mathematics, University of Pennsylvania 
Communicated by E. H. Moore, January 30, 1919 
To transform the general case of a binary quantic 
fm = (^^0, fli, . . . , a^lxi, 
by substituting for its variables 
where the coefficients of the transformation are least positive residues of a 
prime number p, is to generate an infinitude of binary forms associated with 
ffn, which are invariants of the linear modular group G of order (p- — p) 
X (/>2 — 1), on the variables 
Much difficulty has been encountered by those who have investigated these 
invariants, both as to methods of generating complete systems and as to proofs 
of their finiteness. This paper is in the form of a summary of the present 
writer's research on this problem. It contains an outline of a method of con- 
struction of the formal modular covariants which has a measure of generality, 
— and which has proved to be definitive, — at least for particular moduli. 
Secondly we give, in explicit form, the finite modular systems of the cubic 
(mod 2) and of the quadratic (mod 3). Phases of the formal modular 
theory not mentioned in this paper are treated in articles by the present 
writer, referred to at the end of this paper, and in Dickson's Madison Col- 
loquium lectures. 
1. Modular convolution. — If 5(^0, di, • • •) is any modular seminvariant 
of which satisfies a certain pair of conditions,^° and ao\ ai, . . . are the 
coefficients of the transformed of f„^ by means of 
xi = Xi -\- tX2y xi = X'l'^ {t any residue mod p), 
then, S' being the conjugate to 5 under the substitution (aoa2)(ai), 
y (ao^ a/, . . .) ^ S{a,, . . .)t^~' + S,t^-' + . . + (mod p), (1) 
written with homogeneous variables x\, X2, instead of /, is a formal covariant 
modulo p. When this principle is applied in the case of the seminvariant 
leading coefficient of the covariant K^'. 
where is a number of the form 
v= {sp- v'){p- 1) = a{p- 1), 
