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Counting Points on Curves and Abelian Varieties Over Finite Fields

✍ Scribed by Leonard M. Adleman; Ming-Deh Huang

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 343 KB
Volume: 32
Category: Article
ISSN: 0747-7171
DOI: 10.1006/jsco.2001.0470

No coin nor oath required. For personal study only.

✦ Synopsis

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((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 (log q) O(g 2 log g) time.

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