A characterisation of the $\mathbf{n\langle1\rangle \oplus\langle3\rangle}$ form and applications to rational homology spheres

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We conjecture two generalisations of Elkies' theorem on unimodular quadratic forms to non-unimodular forms. We give some evidence for these conjec- tures 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.