## Almost-universal quadratic forms: an effective solution of a problem of Ramanujan.(English)Zbl 1242.11028

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 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. 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.

### MSC:

 1.1e+21 General ternary and quaternary quadratic forms; forms of more than two variables 1.1e+13 Quadratic forms over global rings and fields

### Citations:

JFM 46.0240.01; Zbl 0060.11003
Full Text:

### References:

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