On the orders and genera of ternary quadratic forms

Abstract

The object of this Paper is to supply demonstrations of the undemonstrated results, relating to Ternary Quadratic Forms, which are contained in an important memoir of Eisenstein’s ( “Neue Theoreme der höheren Arithmetik,” Crelle’s Journal, vol. xxxv. p. 117),— and, at the same time, to extend those results to the cases not considered by him in that Memoir. The following are the principal points in which the theory of Eisenstein has been thus further developed:— 1. In Eisenstein’s Memoir forms of an even discriminant only are considered. Such forms, and their contravariants, are always properly primitive; they have particular generic characters with respect to uneven primes dividing the discriminant, but have no supplementary characters ( i. e . characters with respect to 4 or 8). The case of forms of an even discriminant more complicated. Besides the properly primitive order, there may exist, in this case, an improperly primitive order, in which the forms themselves are improperly primitive, and their contravariants properly primitive,—or, again, an improperly primitive order, in which the forms themselves are properly primitive, and their contravariants improperly primitive. Further, forms of an even discriminant may have characters with respect to 4 or 8; and a complete enumeration of these supplementary characters requires a careful distinction of cases. To facilitate this enumeration, a Table is given in the Paper for finding the supplemental characters of any proposed form.