Counting Points on Curves and Abelian Varieties Over Finite Fields

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 overFq, we improve Pila’s result and show that the problem can be solved in O(( q)δ) time where δ is polynomial in g as well as in N. For hyperelliptic curves of genus goverFq 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(g2 g)time.