Local conditions for global representations of quadratic forms

Abstract

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.Comment: 9 pages, v2: corrected typos, added a corollary using results of Kitaoka. v3: added results about representations with congruence conditions and about extensions of representations. v4: minor corrections. Paper to appear in Acta Arithmetic

Averages of Fourier coefficients of Siegel modular forms and representation of binary quadratic forms by quadratic forms in four variables

“…where T is another (half-) integral symmetric matrix of size n ≤ m an (integral) representation of T by S. It is well known that the local global principle of Minkowski and Hasse is not valid for integral representations, but if m is large enough compared to n one can prove that a positive definite T which is represented by S over all Z p and is large enough in a suitable sense is indeed represented by S over the rational integers Z, at least under some mild additional conditions. The bound on the size of m necessary for this has recently been pushed down to m ≥ n + 3, again under suitable additional conditions, in [10], see also [27] for an attempt to optimize those additional conditions. The case m = n + 2 brings some limitations due to the existence of the so called spinor exceptions (see [16,14]), taking these into account a result of the desired type could be reached in [6] for n = 1, m = 3, i. e. for representations of sufficiently large numbers by ternary forms.…”

On the representation of quadratic forms by quadratic forms

“…Recent work of Ellenberg and Venkatesh [7] used ergodic theory to show that the condition on n can be greatly improved to n ≥ m + 5, under the additional assumption that the discriminant of B is square-free. This latter condition has been refined by Schulze-Pillot [15]. The approaches above do not give any quantitative information about integer solutions to (1).…”