Colloquium publications.

14

THE MADISON COLLOQUIUM.

For example, using (19), we see that a complete set of linearly
independent modular invariants of the quadratic form qm
modulo p (p > 2) is given by
(23) Io, A, Ar2 (r = 1,..., m- 1), Dk (k = 1, p-1).
12. Fundamental Systems of Modular Invariants.-While, by
(22), the characteristic invariants Io, ~ ~, In1 form a fundamental
system of modular invariants of a system S of modular forms, it
is usually much easier to find another fundamental system. In
fact, certain invariants are usually known in advance, e. g., the
invariants of the corresponding system of algebraic forms. We
shall prove the following fundamental theorem:
If the modular invariants A, B,.., L completely characterize
the classes, they form a fundamental system of modular invariants.
For example, Io, - -, n-i evidently completely characterize
the classes and were seen to form a fundamental system.
Let c1, * * *, c, be the coefficients of the forms in the system S.
Let each ci take the values 0, 1, *., p - 1. For the resulting
ps sets of values of the c's, let the rational integral functions
A, B, * *, L of cl, * *, c, take the distinct sets of values
Ai, B.,., Li (i = 0,,n- 1).
Thus there are n classes of systems S and by hypothesis the ith
class is uniquely defined by the values A, ~* *, Li of our invariants.
A rational integral invariant 4(cl, * *., c,) takes the same value
for all systems of forms in the ith class, so that this value may
be designated by 0i. Now the polynomial
n-i
P(A, B,..., L) - f {1 - (A - A)- }... { 1 - (L - Li)p-1}
i=0
is congruent to ~i when A = Ai,.., L - Li (mod p). Hence
0(cl,. *, Cs) - P(A, B,.., L) (mod p)
for all sets of integral values of cl, - *, cs. In view of Fermat's
theorem, we may assume that each exponent in f(c1,,, s)
is less than p. If we replace A,.., L by their expressions in