Colloquium publications.

INVARIANTS AND NUMBER THEORY.

55

to ao8al8. Hence C- P1a2. Similarly,
L4/L3 - a + a32Q32 - a3PQ31 + a3L3P-1 (mod p),
where the Q's are defined by (16) and are congruent to*
Q3 = Q(L3/L2)-1 + L2P2-, Q32 = (L3/L2) -1+ QP,
with Q as above. Hence for integral values of the a's,
Q31- (1 - P1)Pla2 - 0, Q32 1- P1 (1 - a2p-) = 1-P2
L4/L3 - P2a3a
13. Modular Covariants.-Extending the usual definition of a
covariant of an algebraic form f to the case in which the group is
the set of all linear transformations with integral coefficients
taken modulo p, we obtain the concepts modular covariants or
formal modular covariants according as the coefficients of f
are integers taken modulo p or are indeterminates. The contrast
is the same as in ~ 5. The universal covariants obtained in ~ 2
and ~ 4 do not involve the coefficients of f and hence are formal
covariants.
I have recently provedt that all rational integral modular
covariants of any system of modular forms are rational integral
functions of a finite number of these covariants. In the same paper
I proved that a fundamental system of modular covariants of the
binary quadratic form (25) modulo 3 is given by the form f itself,
its discriminant A, the universal covariants L and Q, together witht
q = (a + c)(b2 + ac - 1), f4 = ax4 + b3y + bxy3+ cy4,
(63) C1 = (a2b - b3)x2 + 2(b2 + ac) (c - a)xy + (b3 - bc2)y2,
02 = (A + a2)x2 - 2b(a + c)xy + (A + c2)y2.
Here f4 is a formal covariant, which is congruent to f for integral
* Transactions of the American Mathematical Society, vol. 12 (1911), p. 77.
t Transactions of the American Mathematical Society, vol. 14 (1913), pp.
299-310. The extension to cogredient sets of variables has since been made
by Professor F. B. Wiley, and will be published in his Chicago dissertation.
t No one of the eight is a rational integral function of the remaining seven
even in the case of integral coefficients a, b, c taken modulo 3.