One-class genera of maximal integral quadratic forms

Abstract

Suppose Q is a definite quadratic form on a vector space V over some totally real field K = Q. Then the maximal integral ZK -lattices in (V, Q) are locally isometric everywhere and hence form a single genus. We enumerate all orthogonal spaces (V, Q) of dimension at least 3, where the corresponding genus of maximal integral lattices consists of a single isometry class. It turns out, there are 471 such genera. Moreover, the dimension of V and the degree of K are bounded by 6 and 5 respectively. This classification also yields all maximal quaternion orders of type number one.

Euclidean quadratic forms and ADC forms II: integral forms

“…If so, it should be possible to resolve further cases of Conjecture 1: it is a result of Pfeuffer [Pf79] that there are only finitely many class number one totally positive forms as we range over all rings of integers of totally real number fields. In fact, M. Kirschmer has just given an enumeration of the maximal such forms [Ki14]. Thus the complete classification of positive Euclidean forms over rings of integers of totally real number fields may be within reach.…”