  We conjecture two generalisations of Elkies' theorem on unimodular quadratic
forms to non-unimodular forms. We give some evidence for these conjectures
including a result for determinant 3. These conjectures, when combined with
results of Froyshov and of Ozsvath and Szabo, would give a simple test of
whether a rational homology 3-sphere may bound a negative-definite
four-manifold. We verify some predictions using Donaldson's theorem. Based on
this we compute the four-ball genus of some Montesinos knots.
