### A characterization of tame Hilbert-symbol equivalence

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We investigate the almost regular positive definite integral quaternary quadratic forms. In particular, we show that every such form is $p$-anisotropic for at most one prime number $p$. Moreover, for a prime $p$ there is an almost regular $p$-anisotropic quaternary quadratic form if and only if $p\le 37$. We also study the genera containing some almost regular $p$-anisotropic quaternary form. We show several finiteness results concerning the families of these genera and give effective criteria for almost regularity....

We improve the known upper bound of the dimension $n$ of an indecomposable unimodular lattice whose shadow has the third largest possible length, $n-16$.