Local Conditions for Global Representations of Quadratic Forms

We show that the theorem of Ellenberg and Venkatesh on representation of integral quadratic forms by integral positive definite quadratic forms is valid under weaker conditions on the represented form. In the article [5] Ellenberg and Venkatesh prove that for any integral positive definite quadratic form f in n variables there is a constant C(f) such that integral quadratic forms g of square free discriminant in m ≤ n − 5 variables with minimum μ(g) > C(f) are represented by f if and only if they are represented by f locally everywhere, i.e., over R and over all p-adic integers. If one fixes an odd prime p not dividing the discriminant of f one can find a constant C(f, p) such that representability is even guaranteed for g of rank m ≤ n − 3 with μ(g) > C(f, p), provided the discriminant of g is further restricted to be prime to p. It is mentioned in [5] that I have suggested to replace the condition of square free discriminant on g by a weaker condition. This suggestion is worked out here. 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. We will work throughout in the language of lattices as described e.g. in [12, 15] (with the exception of Theorem 11). We fix a totally real number field F with ring of integers o and a totally positive definite quadratic space (V,Q) over F of dimension n ≥ 3; the quadratic form Q may be written as Q(x) = 〈x, x〉 with a scalar product 〈 , 〉 on V . By OV (F ) we denote the group of isometries of V with respect to Q (the orthogonal group of the quadratic space (V,Q)), by OV (A) its adelization, by SOV (F ) resp. SOV (A) their subgroups of elements of determinant 1. For a lattice Λ on V we denote its automorphism group (or unit group) {σ ∈ OV (F ) | σ(Λ) = Λ} by OΛ(o), similarly for the local or adelic analogues. The minimum of Λ is μ(Λ) := min{N Q(Q(x)) | x ∈ Λ,x 6= 0}. For the question which lattices have large minimum it does not matter whether we chose this definition or min{Tr Q(Q(x)) | x ∈ Λ,x 6= 0} instead, see the remark in [7, p.139]. MSC 2000: Primary 11E12, Secondary 11E04, key words: representation of quadratic forms, integral quadratic forms.