Arithmetic behaviour of the sums of three squares. (English) Zbl 0623.10035

Arenas, A.; Bayer, P.

J. Number Theory 27, 273-284 (1987).

The purpose of this paper is to calculate, for a positive integer \(n\) which has a representation as a sum of three integral squares, the maximum number of summands which can be prime to \(n\). The theorem is that this number is 3 or 2 according to whether \(n\) is coprime to 10, and provided \(n\) is large enough (depending on the radical of \(n\)). The proof uses the evaluation of the number of representations of \(n\) by ternary quadratic forms of special types. This in turn depends on classical results of C. L. Siegel [Ann. Math. (2) 36, 527–606 (1935; Zbl 0012.19703)]. Other elements in the proof are Shimura’s correspondence and the theory of theta-functions given by R. Schulze-Pillot [Invent. Math. 75, 283–299 (1984; Zbl 0533.10021)].
As a consequence, applying directly a theorem of N. Vila [Arch. Math. 44, 424–437 (1985; Zbl 0562.12011)], the authors conclude that if \(n\) is congruent to 3 modulo 8 and sufficiently large (as above) then every central extension of the alternating group \(A_n\) can be realized as a Galois group over \(\mathbb Q\).

Reviewer: James L. Hafner (San José)

MSC:

11P05	Waring’s problem and variants
11R32	Galois theory
11E12	Quadratic forms over global rings and fields
11F11	Holomorphic modular forms of integral weight
11E45	Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)

Keywords:

sum of three integral squares; maximum number of summands; number of representations; ternary quadratic forms; Shimura’s correspondence; theta-functions; central extension; alternating group; Galois group

References:

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