Almost-universal quadratic forms: an effective solution of a problem of Ramanujan
From the introduction: The object of this paper is to prove several results giving an effective method for deciding whether a positive definite integral quaternary quadratic form is almost universal, that is, whether it represents all large positive integers. In this way we obtain an effective and definitive solution to a problem first addressed and investigated by \textit{S. Ramanujan} 90 years ago [Proc. Camb. Philos. Soc. 19, 11--21 (1917; JFM 46.0240.01)]. The following set \(\Sigma=\{1,2,3,5,6,7,10, 14\}\) plays an important role in our investigations. Observe that the 8 elements of \(\Sigma\) are the smallest possible positive integers representing all 8 different square classes of \(Q\). Theorem 1.1. For any positive definite integral quaternary quadratic form \(f\), the following conditions are equivalent: (i) \(f\) is almost universal. (ii) \(f\) is locally universal and, either (a) \(f\) is \(p\)-isotropic for every prime number \(p\), or (b) \(f\) is 2-anisotropic and represents every element of the set \[ 16\Sigma = \{16,32,48,80,96,112, 160,224\}, \] or (c) \(f\) is equivalent to one of the following four forms \[ x^2 + 2y^2 + 5z^2 + 10t^2,\quad x^2 + 2y^2 + 3z^2 + 5t^2 + 2yz,\quad x^2 + y^2 + 3z^2 + 3t^2,\quad x^2 + 2y^2 + 4z^2 + 7t^2 + 2yz. \] The set \(16\Sigma\) in Theorem 1.1 is optimal in the sense that removing any of its elements would lead to a false statement. Unless stated otherwise, here by an ``integral form'' is always meant a positive definite quadratic form having integer matrix. An integral form \(f\) is called locally universal if \(f\) is \(p\)-universal for every prime \(p\), that is, if \(f\) represents every \(p\)-adic integer over the ring \(\mathbb Z_p\) of \(p\)-adic integers. If \(f\) represents zero nontrivially over \(\mathbb Z_p\), we say that \(f\) is \(p\)-isotropic, otherwise \(f\) is called \(p\)-anisotropic. Effective criteria for \(p\)-universality are known and easy to verify [cf. \textit{G. Pall}, Am. J. Math. 68, 47--58 (1946; Zbl 0060.11003)]. Simple effective criteria for \(p\)-isotropy, involving the discriminant and the Hasse-Minkowski invariant of a form, are also well known. Therefore all conditions characterizing almost universality, stated in Theorem 1.1, are effective. Theorem 1.2. An almost universal integral quadratic form is \(p\)-anisotropic for at most one prime \(p\). The four forms listed in Theorem 1.1 (c) are the only ones (up to equivalence) which are \(p\)-anisotropic for some \(p > 2\). The part of Theorem 1.1 concerning the 2-anisotropic case can be replaced by an even more precise description of almost universal 2-anisotropic forms. First define an invariant \(\beta(f)\) of an integral quaternary 2-anisotropic, 2-universal quadratic form \(f\): \(\beta(f)\) is the smallest integer \(b\geq 1\) such that always, if \(f(x)\equiv 0 \pmod{2b+2}\) then \(x = 2y\) for some \(y\) in \(\mathbb Z^4\). It is shown that \(\beta(f)\) is one of the numbers \(1, 2, 3\) or \(4\), and that any of these numbers is the \(\beta(f)\) for some form \(f\). Theorem 1.3. Given an integral 2-anisotropic quaternary quadratic form \(f\), the following conditions are equivalent: (i) \(f\) is almost universal. (ii) \(f\) is 2-universal and represents every number in the set \(2\beta(f)\Sigma\). The paper is organized as follows. In Section 2, we prove an important technical result saying that every 2-universal and 2-anisotropic integral quaternary quadratic form \(f\) contains a ``subform'' \(g\) which, over \(\mathbb Z^2\), is equivalent to \[ 2{\beta(f)-1}(2x^2 + 2xy + 2y^2 + 4z^2 + 4zt + 4t^2). \] This result is better expressed in the language of lattices which is used in the proofs (cf. Theorem 2.4). In Section 3 we study various properties of the family \(M\). Theorems 2.4 and 3.1 are the main technical ingredients needed in the proof of the results stated above. Section 4 contains the proof of Theorem 1.3, and Section 5 the proofs of Theorems 1.1, 1.2 and 1.5 that provides all necessary information allowing to decide effectively whether a genus contains an almost universal 2-anisotropic form. Section 6 contains Table 4 enumerating all 79 lattices of the family \(M\) and the list of all 65 lattices of the family \(N\) where \(M\) denotes the set of all (up to equivalence over \(\mathbb Z\)) integral quaternary quadratic forms which are even, 2-anisotropic, represent all even positive integers, and have the discriminant of the type \(4(8m + 1)\) for some integer \(m\geq 0\), and \(N\) a subfamily of \(M\) of all forms representing different genera.
scientific article; zbMATH DE number 5532604