Can we recover an integral quadratic form by representing all its subforms?

Abstract

Let o be the ring of integers of a number field. If f is a quadratic form over o and g is another quadratic form over o which represents all proper subforms of f, does g represent f? We show that if g is indefinite, then g indeed represents f. However, when f is positive definite and indecomposable, then there exists a g which represents all proper subforms of f but not f itself. Along the way we give a new characterization of positive definite decomposable quadratic forms over o and a number-field generalization of the finiteness theorem of representations of quadratic forms by quadratic forms over Z[13, Theorem 3.3] which asserts that given any infinite set S of classes of positive definite integral quadratic forms over o of a fixed rank, there exists a finite subset S0 of S with the property that a positive definite quadratic form over o represents all classes in S if and only if it represents all classes in S0.