Abstract
Let S be a bielliptic surface over a finite field, and let the elliptic curve B be the image of the Albanese mapping S → B. In this case, the zeta function of the surface is equal to the zeta function of the direct product ℙ1 × B. A classification of the possible zeta functions of bielliptic surfaces is also presented in the paper.
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References
M. A. Tsfasman, “Nombre de points des surfaces sur un corps fini,” in Arithmetic, Geometry, and Coding Theory, Luminy, 1993 (Walter de Gruyter, Berlin, 1996), pp. 209–224.
S. Rybakov, “Zeta functions of conic bundles and Del Pezzo surfaces of degree 4 over finite fields,” Mosc. Math. J. 5(4), 919–926 (2005).
E. Freitag and R. Kiehl, Étale Cohomology and the Weil Conjecture, in Ergebnisse der Mathematik und ihrer Grenzgebiete (3), With an historical introduction by J. A. Dieudonné (Springer-Verlag, Berlin, 1988), Vol. 13.
A. Weil, Variétes abéliennes et courbes algébraiques (Hermann & Cie., Paris, 1948).
W. Waterhouse, “Abelian varieties over finite fields,” Ann. Sci. École Norm. Sup. (4) 2, 521–560 (1969).
W. Waterhouse and J. Milne, “Abelian varieties over finite fields,” in Proc. Sympos. Pure Math., State Univ. New York, Stony Brook, NY, 1969 (Amer.Math. Soc., Providence, RI, 1971), Vol. XX, pp. 53–64.
M. A. Tsfasman, “Group of points of an elliptic curve over a finite field,” in Theory of Numbers and Its Applications (Tbilisi, 1985), pp. 286–287.
H.-G. Rück, “A note on elliptic curves over finite fields,” Math. Comp. 49(179), 301–304 (1987).
J. F. Voloch, “A note on elliptic curves over finite fields,” Bull. Soc. Math. France 116(4), 455–458 (1988).
R. Schoof, “Nonsingular plane cubic curves over finite fields,” J. Combin. Theory. Ser. A 46(2), 183–211 (1987).
E. Bombieri and D. Mumford, “Enriques’ classification of surfaces in char. p. II,” in Complex Analysis and Algebraic Geometry (Iwanami Shoten, Tokyo, 1977), pp. 23–42.
L. Bădescu, Algebraic Surfaces, in Universitext (Springer-Verlag, New York, 2001).
Yu. I. Manin, Cubic Forms. Algebra, Geometry, Arithmetic (Nauka, Moscow, 1972; North-Holland Publishing Co., Amsterdam-New York, 1986).
K. Matsuki, Introduction to the Mori Program, in Universitext (Springer-Verlag, New York, 2002).
J.-P. Serre, Galois Cohomology (Springer-Verlag, Berlin-New York, 1965 [in French]; Mir, Moscow, 1968 [in Russian]; Springer-Verlag, Berlin, 1997 [in English]).
R. Hartshorne, Algebraic Geometry (Springer-Verlag, New York-Heidelberg, 1977; Mir, Moscow, 1981).
D. Mumford, Abelian Varieties (Bombay; Oxford University Press, London, 1970; Mir, Moscow, 1971).
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Original Russian Text © S. Yu. Rybakov, 2008, published in Matematicheskie Zametki, 2008, Vol. 83, No. 2, pp. 273–285.
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Rybakov, S.Y. Zeta functions of bielliptic surfaces over finite fields. Math Notes 83, 246–256 (2008). https://doi.org/10.1134/S0001434608010264
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DOI: https://doi.org/10.1134/S0001434608010264