A symbolic theory of formal modular covariants

Abstract

1. Prologue. Thus far, very little has been published on the general theory of formal modular invariants or covariants. Workers have, on the whole, obtained results for special, more or less isolated, cases; and although some beautiful and important general theorems have been proved, they are more or less unrelated. This is, of course, only natural in any division of knowledge in its formative state. Nevertheless, no worker in the field could fail to be conscious of a certain uniformity common to the special cases that have been studied in detail; though (alas!) this uniformity usually appeared to be broken ruthlessly in the next case studied. This breaking of an apparent law signified, however, merely that we did not know these special cases with a sufficient thoroughness of illuminating detail, or were trying unwittingly to make the laws conform to certain standards, unconsciously preconceived. This latter handicap was laid on us naturally enough by our thorough knowledge of algebraic invariants and the fact that this newer kind of covariants is, in many ways, strikingly like the older, classic covariants, though so tantalisingly different. Their similarity and their difference show themselves in the very beginning of the study: in the definitions, in the simplest examples. Perhaps the differences that first come to mind are those which are inherent in the fields of definition, which, in the case of classic covariants, is the field of reals or ordinary complex numbers and, in the case of modular covariants, is a Galois field, GF [pn] , of order pn . These differences are too obvious to mention in detail, but one who has studied the beautiful proofs given by the old masters of invariant theory has been forced to the conclusion that most of the proofs seemed to use the properties of a field of characteristic zero, not in some accidental manner, but rather in veriest necessity. Growing from the surface differences between the two fields are two very important distinguishing characteristics of the two kinds of covariants. It * Part II was presented to the Society, September 7, 1920; Part III, December 28, 1921; Parts IV and V, December 27, 1922. The reading of the literature in connection with this paper was much facilitated by the purchase of books with a grant made by the American Association for the Advancement of Science and this help is herewith gratefully acknowledged. 286