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

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)
Full Text: DOI
[1] \scA. Arenas, An arithmetic problem on the sums of three squares. Acta Arith., \bf51, No. 3, in press.
[2] \scP. Bayer and E. Nart, Zeta functions and genus of quadratic forms, to appear. · Zbl 0702.11022
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[4] Loo Keng, H., ()
[5] Schulze-Pillot, R., Theta-reihen positiv definiter quadratischer formen, Invent. math., 75, 283-299, (1984) · Zbl 0533.10021
[6] Shimura, G., On modular forms of half integral weight, Ann. of math., 97, 440-481, (1973) · Zbl 0266.10022
[7] Siegel, C.L.; Siegel, C.L., Über die analytische theorie der quadratischen formen, (), 36, 527-606, (1935) · JFM 61.0140.01
[8] Vila, N., On central extensions of An as a Galois group over \bfq, Arch. math., 44, 424-437, (1985) · Zbl 0562.12011
[9] \scN. Vila, On stem extensions of Sn as Galois group over number fields. J. Algebra, in press. · Zbl 0662.12011
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