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Rationality of the zeta-function of a set of equations over a finite field

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Book cover p-adic Numbers, p-adic Analysis, and Zeta-Functions

Part of the book series: Graduate Texts in Mathematics ((GTM,volume 58))

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Abstract

If F is a field, let \(\mathbb{A}_{F}^{n}\) denote “n-dimensional affine space over F,” i.e., the set of ordered n-tuples (x 1, …, x n ) of elements x i of F. Let f(X 1,…, X n ) ∈F[X 1…, X n ] be a polynomial in the n variables X 1 ,…, X n .

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Bibliography

  1. A. Weil, Number of solutions of equations in finite fields, Bull. Amer. Math. Soc., 55 (1949), 497–508.

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  2. J.-P. Serre, Rationalité des fonctions ζ des variétés algébriques (d’après Bernard Dwork), Séminaire Bourbaki, No. 198, February 1960.

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  3. B. Dwork, On the rationality of the zeta function of an algebraic variety, Amer. J. Math., 82 (1960), 631–648.

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  4. P. Monsky, p-adic Analysis and Zeta Functions, Lectures at Kyoto University, Kinokuniya Book Store, Tokyo, or Brandeis Univ. Math. Dept., 1970.

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  5. N. Katz, Une formule de congruence pour la fonction ζ, S.G.A. 7 II (Lecture Notes in Math. 340), Springer-Verlag, 1973.

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  6. B. Dwork, On the zeta function of a hypersurface, Publ. Math. I.H.E.S., 12 (1962), 5–68.

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  7. B. Dwork, On the zeta function of a hypersurface II, Ann. of Math., 80 (1964), 227–299.

    Article  MathSciNet  Google Scholar 

  8. B. Dwork, A deformation theory for the zeta function of a hypersurface, Proc. Int. Cong. Math. 1962 Stockholm, 247–259.

    Google Scholar 

  9. N. Katz, Travaux de Dwork, Séminaire Bourbaki, No. 409 (Lecture Notes in Math. 317), Springer-Verlag, 1973.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, Cambridge, Massachusetts, 02138, USA

    Neal Koblitz

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  1. Neal Koblitz
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© 1977 Springer-Verlag, New York Inc.

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Koblitz, N. (1977). Rationality of the zeta-function of a set of equations over a finite field. In: p-adic Numbers, p-adic Analysis, and Zeta-Functions. Graduate Texts in Mathematics, vol 58. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0047-2_5

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  • DOI: https://doi.org/10.1007/978-1-4684-0047-2_5

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4684-0049-6

  • Online ISBN: 978-1-4684-0047-2

  • eBook Packages: Springer Book Archive

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