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Counting rational points on curves and abelian varieties over finite fields. (English) Zbl 0898.11045
Cohen, Henri (ed.), Algorithmic number theory. Second international symposium, ANTS-II, Talence, France, May 18-23, 1996. Proceedings. Berlin: Springer. Lect. Notes Comput. Sci. 1122, 1-16 (1996).
Let $$A$$ be an abelian variety of dimension $$g$$ over a finite field $${\mathbb F}_q$$. Suppose that $$A$$ is given as a closed subvariety of projective $$n$$-space. The authors exhibit a deterministic algorithm that computes the characteristic polynomial of the Frobenius endomorphism of $$A$$ that runs in time $$O((\text{log} q)^c)$$, where $$c$$ is a polynomial expression in $$g$$ as well as $$n$$. This improves upon an earlier result of J. Pila [Math. Comput. 55, 745-763 (1990; Zbl 0724.11070)], who obtained a similar result but with the constant $$c$$ depending exponentially on $$n$$.
By applying this to the Jacobian varieties of curves $$X$$ over $${\mathbb F}_q$$, one obtains a deterministic algorithm to count the number of $${\mathbb F}_q$$-rational points of $$X$$ that runs in time $$O((\text{log} q)^c)$$, where $$c$$ is a polynomial expression in $$n$$, as well as the genus $$g$$ of $$X$$. In the special case of hyperelliptic curves of genus $$g$$, the authors show that the number of $${\mathbb F}_q$$-rational points on $$X$$ can be counted deterministically in time $$(\text{log} q)^{O(g^6)}$$. This case is of interest in view of the primality test decribed by the authors in their monograph [Primality testing and abelian varieties over finite fields, Lect. Notes Math. 1512 (Springer-Verlag, 1992; Zbl 0744.11065)].
For the entire collection see [Zbl 0852.00023].

##### MSC:
 11Y16 Number-theoretic algorithms; complexity 11G25 Varieties over finite and local fields 11G20 Curves over finite and local fields 14K15 Arithmetic ground fields for abelian varieties 14G05 Rational points 14G15 Finite ground fields in algebraic geometry 14Q05 Computational aspects of algebraic curves 14Q15 Computational aspects of higher-dimensional varieties 14H25 Arithmetic ground fields for curves