Counting points on curves and Abelian varieties over finite fields

Authors:
Leonard M. Adleman;Ming-Deh Huang
Affiliations:
University of Southern California, Los Angeles;University of Southern California, Los Angeles
Venue:
Journal of Symbolic Computation
Year:
2001

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Abstract

We develop efficient methods for deterministic computations with semi-algebraic sets and apply them to the problem of counting 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((q)) time where is polynomial in g as well as in N. For hyper elliptic 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 (q)O(g2g)time.