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)


14K15 Arithmetic ground fields for abelian varieties
14G15 Finite ground fields in algebraic geometry
14H40 Jacobians, Prym varieties
Full Text: Numdam EuDML


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