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Fermat Jacobians of Prime Degree over Finite Fields

Published online by Cambridge University Press:  20 November 2018

Josep González
Josep González*
Affiliation:
Escola Universitària Politècnica de Vilanova i la Geltrú Departament de Matemàtica Aplicada i Telemàtica Av. Victor Balaguer s/n Vilanova i la Geltrú 08800 Spain, e-mail: [email protected]
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Abstract

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We study the splitting of Fermat Jacobians of prime degree $\ell $ over an algebraic closure of a finite field of characteristic $p$ not equal to $\ell $. We prove that their decomposition is determined by the residue degree of $p$ in the cyclotomic field of the $\ell $-th roots of unity. We provide a numerical criterion that allows to compute the absolutely simple subvarieties and their multiplicity in the Fermat Jacobian.

Keywords

11G2014H40
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 1 , 01 March 1999 , pp. 78 - 86
Copyright
Copyright © Canadian Mathematical Society 1999

References

[Da-Ha 35] Davenport, H. and Hasse, H., Die Nullstellen der Kongruenz-zetafunktionen in gewissen zyklischen Fällen. J. Reine Angew.Math. 172 (1935), 151–182.Google Scholar
[Go 97] González, J., Hasse-Witt matrices for the Fermat curves of prime degree. Tôhoku Math. J. 49 (1997), 149–163.Google Scholar
[Go 98] González, J., On the p-rank of an abelian variety and its endomorphism algebra. Publ.Mat. 42, to appear.Google Scholar
[Ko-Ro 78] Koblitz, N. and Rohrlich, D., Simple factors in the Jacobian of a Fermat curve. Canad. J. Math. 6 (1978), 1183–1205.Google Scholar
[Sch 84] Chmidt, C. G., Arithmetik Abelscher Varietäten mit komplexer Multiplikation. Lecture Notes in Math. 1082, Springer, 1984.Google Scholar
[Se 58] Serre, J. P., Sur la topologie des variétés algébriques en caractéristique p. Symp. Int. Top. Alg. México (ed. J. P. Serre), OEUVRES I, Springer, 1958, 24–53.Google Scholar
[Shi-Ka 79] Shioda, T. and Katsura, T., On Fermat Varieties. Tôhoku Math. J. 31 (1979), 97–115.Google Scholar
[St 79] Stichtenoth, H., Die Hasse-Witt-Invariante eines Kongruenzfunktionenkörpers. Arch. Math. 33 (1979), 357–360.Google Scholar
[Ta 66] Tate, J., Endomorphisms of abelian varieties over finite fields. Invent.Math. 2 (1966), 134–144.Google Scholar
[Ta 68] Tate, J., Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T.Honda). Sém. Bourbaki (1968/69), 95–110.Google Scholar
[Wa 69] Waterhouse, W. C., Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. 2 (1969), 521–560.Google Scholar
[We 49] Weil, A., Number of solutions of equations in finite fields. Bull. Amer.Math. Soc. 55, Collected Papers I, Springer, 1949, 497–508.Google Scholar
[Yu 80] Yui, N., On the Jacobian Variety of the Fermat curve. J. Algebra 65 (1980), 1–45.Google Scholar