×

zbMATH — the first resource for mathematics

Abelian surfaces and Jacobian varieties over finite fields. (English) Zbl 0742.14037
Let \(C\) be a projective, smooth curve of genus 2 over the finite field \(\mathbb{F}_ q\); \(q=p^ n\). The zeta function of \(C\) is a rational function of the form \((1+a_ 1T+a_ 2T^ 2+a_ 1qT^ 3+q^ 2T^ 4)/(1-T)(1- qT)\); where the Riemann hypothesis restricts the integers \(\{a_ 1,a_ 2\}\) to \(| a_ 1|\leq 4q^{1/2}\) and \(| a_ 2|\leq 6q\). The author presents a partial solution to the problem of finding all \(\{a_ 1,a_ 2\}\) which can occur via the use of the theory of abelian varieties over finite fields. (The paper also contains a nice, short summary of this theory.).
Reviewer: D.Goss (Columbus)

MSC:
14K15 Arithmetic ground fields for abelian varieties
14G15 Finite ground fields in algebraic geometry
14H40 Jacobians, Prym varieties
PDF BibTeX XML Cite
Full Text: Numdam EuDML
References:
[1] Hayashida, T. - Nishi, M. : Existence of curves of genus two on a product of two elliptic curves , J. Math. Soc. Japan, Vol. 17 (1965), 1-16. · Zbl 0132.41701 · doi:10.2969/jmsj/01710001
[2] Lang, S. : Algebraic groups over finite fields , Amer. J. of Math. 78 (1956), 555-563. · Zbl 0073.37901 · doi:10.2307/2372673
[3] Manin, Y.I. : The Theory of commutative formal groups over fields of finite characteristic , Russian Math. Surv., Vol. 18 (1963), 1-83. · Zbl 0128.15603 · doi:10.1070/rm1963v018n06ABEH001142
[4] Mumford, D. : Abelian Varieties , Second Edition, Oxford Univ. Press, Oxford (1974). · Zbl 0326.14012
[5] Oda T. : The first de Rham cohomology group and Dieudonné modules , Ann. scient. Éc. Norm. Sup., (4), t. 2 (1969), 63-135. · Zbl 0175.47901 · doi:10.24033/asens.1175 · numdam:ASENS_1969_4_2_1_63_0 · eudml:81844
[6] Ramanujam, C.P. : The Theorem of Tate, Appendix I in [M] .
[7] Tate, J. : Endomorphisms of abelian varieties over finite fields , Invent. Math., Vol. 2 (1966), 134-144. · Zbl 0147.20303 · doi:10.1007/BF01404549 · eudml:141848
[8] Tate, J. : Classes d’isogénie des variétés abéliennes sur un corps fini (d’après T. Honda) , Sem. Bourbaki 358, 1968-1969. · Zbl 0212.25702 · numdam:SB_1968-1969__11__95_0 · eudml:109770
[9] Waterhouse, W.C. : Abelian varieties over finite fields , Ann. scient. Éc. Norm. Sup., (4), t. 2 (1969), 521-560. · Zbl 0188.53001 · doi:10.24033/asens.1183 · numdam:ASENS_1969_4_2_4_521_0 · eudml:81852
[10] Weil, A. : Zum Beweis des Torellischen Satzes , Nachr. Akad. Wiss Göttingen, Math.-Phys. Kl. (1957), 33-53. · Zbl 0079.37002
[11] Weiss, E. : Algebraic Number Theory , McGraw-Hill, New-York (1963). · Zbl 0115.03601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.