Point Counting in Families of Hyperelliptic Curves

Point Counting in Families of Hyperelliptic Curves

Let EΓ be a family of hyperelliptic curves defined by Y 2 = Q(X,Γ), where Q is defined over a small finite field of odd characteristic. Then with γ in an extension degree n field over this small field, we present a deterministic algorithm for computing the zeta function of the curve Eγ by using Dwork deformation in rigid cohomology. The complexity of the algorithm is O(n) and it needs O(n) bits...

Point Counting in Families of Hyperelliptic Curves in Characteristic 2

Let ĒΓ be a family of hyperelliptic curves over F cl 2 with general Weierstrass equation given over a very small field F. We describe in this paper an algorithm for computing the zeta function of Ēγ̄, with γ̄ in a degree n extension field of F, which has as time complexity Õ(n3) bit operations and memory requirements O(n2) bits. With a slightly different algorithm we can get time O(n2.667) and me...

Point Counting on Elliptic and Hyperelliptic Curves

Counting hyperelliptic curves

We find a closed formula for the number hyp(g) of hyperelliptic curves of genus g over a finite field k = Fq of odd characteristic. These numbers hyp(g) are expressed as a polynomial in q with integer coefficients that depend on g and the set of divisors of q − 1 and q + 1. As a by-product we obtain a closed formula for the number of self-dual curves of genus g. A hyperelliptic curve is self-du...

Families of Hyperelliptic Curves

Throughout this work we deal with a natural number g ≥ 2 and with an algebraically closed field k whose characteristic differs from 2. A hyperelliptic curve of genus g over k is a smooth curve of genus g, that is a double cover of the projective line P. The Riemann-Hurwitz formula implies that this covering should be ramified at 2g + 2 points. Because of this explicit description, hyperelliptic...