Abstract
We develop efficient methods for deterministic computations with semi-algebraic sets and apply them to the problem of counting rational points on curves and abelian varieties over finite fields. For abelian varieties of dimension g in projective N space over Fq, we improve Pila's result and show that the problem can be solved in O((log q)δ) time where δ is polynomial in g as well as in N. For hyperelliptic curves of genus g over Fq we show that the number of rational points on the curve and the number of rational points on its Jacobian can be computed in time \(O((\log q)^{O(g^6 )} )\)time.
Research supported by NSF through grant CCR-9403662.
Research supported by NSF through grant CCR-9412383.
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© 1996 Springer-Verlag Berlin Heidelberg
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Adleman, L.M., Huang, MD.A. (1996). Counting rational points on curves and abelian varieties over finite fields. In: Cohen, H. (eds) Algorithmic Number Theory. ANTS 1996. Lecture Notes in Computer Science, vol 1122. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61581-4_36
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DOI: https://doi.org/10.1007/3-540-61581-4_36
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