On cubic rings and quaternion rings. (English) Zbl 1166.11034

In their book on “The theory of irrationalities of the third degree” [Translations of Mathematical Monographs. 10. American Mathematical Society (1964; Zbl 0133.30202)] B. N. Delone and D. K. Fadeev established a correspondence between three-dimensional \(\mathbb Z\)-orders and binary cubic forms which is essentially one-to one. This remarkable observation was reinspected and generalized recently by W. T. Gan, B. H. Gross, and G. Savin [Duke Math. J. 115, No. 1, 105–169 (2002; Zbl 1165.11315)]. As is well-known, a similar correspondence holds in dimension 4. The authors extend both correspondences to coefficient rings which are either local or PID.


11R16 Cubic and quartic extensions
11E76 Forms of degree higher than two
Full Text: DOI


[1] Bhargava, M., Higher composition laws II: On cubic analogues of Gauss composition, Ann. of Math., 159, 865-886 (2004) · Zbl 1169.11044
[2] Brandt, H., Zur Zahlentheorie der Quaternionen, Jber. Deutsch. Math. Verein., 53, 23-57 (1943) · Zbl 0028.10802
[3] Brzeziński, J., A characterization of Gorenstein orders in quaternion algebras, Math. Scand., 50, 1, 19-24 (1982) · Zbl 0483.16002
[4] Delone, B. N.; Faddeev, D. K., The Theory of Irrationalities of the Third Degree, Transl. Math. Monogr., vol. 10 (1964), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 0133.30202
[5] Gan, W. T.; Gross, B. H.; Savin, G., Fourier coefficients of modular forms on \(G_2\), Duke Math. J., 115, 105-169 (2002) · Zbl 1165.11315
[6] Hermite, C., Sur la théorie des formes quadratiques, J. Reine Angew. Math., 47, 307-342 (1854) · ERAM 047.1273cj
[7] Latimer, C. G., The classes of integral sets in a quaternion algebra, Duke Math. J., 3, 237-247 (1937) · JFM 63.0093.04
[8] Llorente, P., Correspondencia entre formas ternarias enteras y órdenes cuaterniónicos, Rev. R. Acad. Cienc. Exactas Fis. Nat. (Esp.), 94, 3, 397-416 (2000)
[10] Pall, G., On generalized quaternions, Trans. Amer. Math. Soc., 59, 280-332 (1946) · Zbl 0060.11004
[11] Peters, M., Ternäre und quaternäre quadratische Formen und Quaternionenalgebren, Acta Arith., 15, 329-365 (1968/1969) · Zbl 0188.11202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.