## 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
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### References:

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