PROCEEDINGS 
OF THE 
NATIONAL ACADEMY OF SCIENCES 
Volume 1 JANUARY 15, 1915 Number 1 
RECENT PROGRESS IN THE THEORIES OF MODULAR AND 
FORMAL INVARIANTS AND IN MODULAR GEOMETRY 
By L. E. Dickson 
DEPARTMENT OF MATHEMATICS. UNIVERSITY OF CHICAGO 
Presented to the Academy, November 7, 1914 
1. Contrast between algebraic and modular invariants. By way of 
introduction we recall the argument, made in certain texts on invariants, 
to prove that a linear form I = ax by has no invariant. For, the 
vanishing of an invariant / {a, b) of / would imply a property of those 
forms / for which 7 = 0, not possessed by the forms I for which 7 4= 0. 
But all forms I are equivalent since each can be transformed into x. 
This argument is erroneous since the identically vanishing form / (with 
a = b = 0) cannot be transformed into x. Nor is the conclusion cor- 
rect. The function 7 {a, b), defined in the sense of Dirichlet to be unity 
if / = 0 and zero if / is not identically zero, is evidently an invariant of /, 
since it has the same value for all equivalent forms /. 
In the number-theoretic case in which the coefficients of / and of the 
linear transformation are integers taken modulo p (a prime) , the bizarre 
Dirichlet function 7 (a, b) employed in the algebraic case is no longer 
necessary, since it may now be replaced by the polynomial invariant 
/=(a^-i--l) (5^-1-1), 
with the value unity if a = Z) = 0 (mod p) and the value zero if a and b 
are not both congruent to zero. Hence 7 is an invariant of / modvilo p. 
It is called a modular invariant of I. 
2. Formal invariants and their construction. Let the coefficients a and 
6 of / be independent variables as in the theory of algebraic invariants. 
But let the coefficients of the linear transformations be integers taken 
modulo p as in the theory of modular invariants. Invariants arising in 
this composite case are called formal invariants and were first intro- 
1 
