Extensions of representations of integral quadratic forms

Abstract

Let N and M be quadratic Z-lattices, and K be a sublattice of N. A representation σ : K → M is said to be extensible to N if there exists a representation ρ : N → M such that ρ| K = σ . We prove in this paper a local-global principle for extensibility of representation, which is a generalization of the main theorems on representations by positive definite Z-lattices by Hsia, Kitaoka and Kneser (J. Reine Angew. Math. 301:132-141, 1978) and Jöchner and Kitaoka (J. Number Theory 48:88-101, 1994). Applications to almost n-universal lattices and systems of quadratic equations with linear conditions are discussed.

Local conditions for global representations of quadratic forms

“…Proof. This is proven in the same way as Theorem IV' in [1, p.95] and Theorem 2.1 in [2], namely by constructing a representation ρ of M ∩ (F R) ⊥ into Λ ∩ (F R) ⊥ satisfying suitable congruence conditions at the primes dividing the discriminant of one of Λ, R and pasting it together with the given σ, the congruence conditions being chosen so that ρ ⊥ σ actually maps M into Λ. Notice for this that the condition of bounded imprimitivity implies that the index of M ∩ (F R) ⊥ in the orthogonal projection π(M ) of M onto (F R) ⊥ is bounded independently of M (for an integral primitive sublattice this index is bounded by the index of R in its dual lattice R # ).…”

“…Combining our version of the result of [5] with results of Kitaoka we also obtain some new cases in which with a suitable fixed prime q the only condition on g (apart from µ(g) > C(f, q) and representability of g by f locally everywhere) is bounded divisibility of the discriminant of g by q. Moreover, results on extensions of representations as given in [1,2] can be obtained with new dimension bounds. We take the occasion to reformulate some of the proofs of [5] in a way that is closer to other work on the subject.…”

“…Then there exists a constant C := C(Λ, R, j, w, c) such that if M ⊇ R is an o-lattice of rank m ≤ n − 3 and : (i) for each place v of F there is a representation τ v : M v → Λ v with τ v | Rv = σ v with imprimitivity bounded by c and with isotropic orthogonal complement in Λ at the place w, Proof. This is proven in the same way as Theorem IV in [1, p. 95] and Theorem 2.1 in [2], namely by constructing a representation of M ∩(F R) ⊥ into Λ ∩ (F R) ⊥ satisfying suitable congruence conditions at the primes dividing the discriminant of one of Λ, R and glueing it to the given σ, the congruence conditions being chosen so that ⊥ σ actually maps M into Λ.…”

“…It appears that at least the present method is not able to give such a result: If we consider an infinite sequence of lattices M i ⊆ Λ whose minimum is bounded, there must be infinite subsequences having a nonzero intersection, since there are only finitely many vectors of given length in Λ. Propositions 5 and 6 can therefore not be applied to such a situation. One can however easily adapt the method to the case of extensions of representations of some fixed lattice as treated in [1,2], see Corollary 9 below. v) At least presently there seems to be no way to obtain an effective bound on the constant C of the Theorem.…”

“…Notice (for comparison with Theorem 2.1 of [2]) that the boundedness of this index also implies that the quotient of the minimum of M ∩ (F R) ⊥ and the minimum of π(M ) is bounded by some constant independent of M , so it is relevant only for the size of the constant C which of these minima one considers.…”