Modular Invariant Theory Research Articles

A fundamental system of formal covariants modulo 2 of the binary cubic

whose coefficients are arbitrary variables, be transformed by the group A of all linear substitutions on xi, x2, whose coefficients are least positive residues modulo p, a prime number, there is brought into existence an infinitude of rational integral functions of ao, *-* *, am, x1, X2, which are invariants under the group. Whether this infinite system possesses the property of finiteness, in general, is an unsolved problem, but in this paper I show that, when the modulus is 2, the system of covariants of a cubief3 is finite and that the fundamental set consists of twenty quantics. This system of covariants, five of which are pure invariants, is derived in explicit form. The methods of generation and proof of the completeness of the fundamental set are developed from the point of view emphasized in a paper on the formal modular invariant theory, by the present writer, in volume 17 of these T r a n s a c t i o n s . t These methods presuppose a knowledge of a fundamental system of formal seminvariants of the given ground form; but this seminvariant system has been given previouslyt for f3 and the modulus 2, by Dickson.

The formal modular invariant theory of binary quantics

is a binary quantic of order m whose coefficients are variables. We study the functions of the a's and x's which are invariantive under the operation of transforming fm by G. Sections 1, 2, 3 deal with methods of constructing such functions, especial attention being given to the case p = 2. In ? 5 fundamental systems of first degree concomitants of the quartic and quintic are derived. Section 4 is devoted to a proof that if the systems of concomitants modulo 2 of the forms of orders 1, 2, 3 and the simultaneous systems obtained by combining forms of these three orders are all finite, then the system of fm is likewise finite. The hitherto undemonstrated theorem on the finiteness of the formal concomitants modulo 2 is thus reduced to a comparatively simple problem. In ? 6 there is proved a theorem on the reducibility of any covariant, modulo 2, of the binary cubic in terms of a set of fourteen invariants and covariants.