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Quotients of Fermat curves and a Hecke character. (English) Zbl 1076.11039

The authors consider a family of quotients of Fermat curves, and show that some members in the family have simple Jacobians with complex multiplication (CM) by a non-cyclotomic field. In an earlier paper [Math. Comput. 74, 499–518 (2005; Zbl 1049.14014)], the second and third authors gave an algorithm for constructing CM Picard curves starting from a given CM field, but they were unable to give a rigorous proof that the curves obtained via this algorithm have complex multiplication by the given CM field. The family considered here contains some of the examples from this previous paper, and thus provides such a rigorous proof in some cases. The majority of this article concentrates on one such example, the curve \(y^3=x(x^7+1)\). The authors compute the local zeta functions of this curve using two different methods - Jacobi sums and the general theory of complex multiplication – by using facts about the CM field \({\mathbb Q}(\zeta_3,\zeta_7 + \zeta_7^{-1})\).

MSC:

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14G15 Finite ground fields in algebraic geometry
11G15 Complex multiplication and moduli of abelian varieties
11T24 Other character sums and Gauss sums
14H40 Jacobians, Prym varieties

Citations:

Zbl 1049.14014
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Full Text: DOI

References:

[1] Berndt, B. C.; Evans, R. J.; Williams, K. S., Gauss and Jacobi Sums (1997), Wiley-Interscience Publication: Wiley-Interscience Publication New York
[2] Davenport, H.; Hasse, H., Die Nullstellen der Kongruenzzetafunktionen in gewissen zyklischen Fällen, J. Reine Angew. Math., 172, 151-182 (1934) · JFM 60.0913.01
[3] Holzapfel, R. P., The Ball and Some Hilbert Problems (1995), Birkhäuser: Birkhäuser Basel
[4] R.P. Holzapfel, F. Nicolae, Arithmetic on a family of Picard curves, Proceedings of the Sixth International Conference on Finite Fields with Applications 2001, Springer, Berlin, 2003, pp. 187-208.; R.P. Holzapfel, F. Nicolae, Arithmetic on a family of Picard curves, Proceedings of the Sixth International Conference on Finite Fields with Applications 2001, Springer, Berlin, 2003, pp. 187-208. · Zbl 1101.14043
[5] K. Koike, A. Weng, Construction of CM-Picard curves with application to cryptography, preprint, 2003.; K. Koike, A. Weng, Construction of CM-Picard curves with application to cryptography, preprint, 2003.
[6] Lang, S., Complex Multiplication (1983), Springer: Springer Berlin · Zbl 0536.14029
[7] Lemmermeyer, F., Reciprocity Laws (2000), Springer: Springer Berlin · Zbl 0949.11002
[8] Weng, A., A class of hyperelliptic CM-curves of genus three, J. Ramanujan Math. Soc., 16, 4, 339-372 (2001) · Zbl 1066.11028
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