Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-17T13:34:48.875Z Has data issue: false hasContentIssue false
  • English
  • Français

The distribution of certain special values of the cubic Legendre symbol

Published online by Cambridge University Press:  18 May 2009

S. J. Patterson
S. J. Patterson
Affiliation:
Mathematisches Institut, Bunsenstr. 3–5 D-3400 Göttingen, W. Germany
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let ω be a primitive cube root of unity. We define the cubic residue symbol (Legendre symbol) on ℤ[ω] as follows. Let πεℤ[ω] be a prime, (3, π)=1. For α ε ℤ[ω] such that (α, π)=1 we let be that third root of unity so that

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 27 , October 1985 , pp. 165 - 184
Copyright
Copyright © Glasgow Mathematical Journal Trust 1985

References

REFERENCES

1.Bass, H., Milnor, J. and Serre, J-P, Solution of the congruence subgroup problem for SLn(n≥3) and Sp2n(n≥2). Publ. Math. I.H.E.S. 33 (1967), 59–137.CrossRefGoogle Scholar
2.Elstrodt, J., Grunewald, E. and Mennicke, J., Discontinuous groups on three-dimensional hyperbolic space; analytic theory and number-theory applications, work in progress.Google Scholar
3.Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (O.U.P., 1960).Google Scholar
4.Heath-Brown, D. R. and Patterson, S. J., On the distribution of cubic Gauss sums at prime arguments, J. Reine Angew. Math. 310 (1979), 111–130.Google Scholar
5.Kazhdan, D. A. and Patterson, S. J., Metaplectic forms, Publ. Math. I.H.E.S., 59 (1984), 35–142.CrossRefGoogle Scholar
6.Lang, S., Algebraic number theory (Addison-Wesley, 1970).Google Scholar
7.Matthews, C. R., Gauss sums and elliptic functions, II. The quartic case, Invent. Math. 54 (1979), 23–52.Google Scholar
8.Neuenhöffer, H., Über die analytische Fortsetzung von Poincaréreihen, S.B. Heidelberger Akad. Wiss. Math.-Nat. Kl. (1973), 33–90.Google Scholar
9.Ogg, A., Modular forms and Dirichlet series (Benjamin, 1969).Google Scholar
10.Patterson, S. J., A cubic analogue of the theta series, J. Reine Angew. Math. 296 (1977), 125–161, 217–220.Google Scholar
11.Patterson, S. J., The distribution of general Gauss sums at prime arguments, in Halberstam, H. and Hooley, C., ed., Recent progress in analytic number theory, Vol. 2 (Academic Press, 1981).Google Scholar
12.Rademacher, H., Topics in analytic number theory (Springer, 1973).Google Scholar
13.Serre, J-P., A functorial property of power residue symbols, Publ. Math. I.H.E.S. 44 (1974), 241–244.Google Scholar
14.Titchmarsh, E. C., The theory of the Riemann zeta-function (O.U.P., 1951).Google Scholar