Definitions and Notation

For simplicity we restrict our attention to hyperelliptic curves with a distinguished Weierstrass point at infinity over finite fields of odd characteristic. The definitions below can all be generalized.

Hyperelliptic Curve: y2 = f(x)
For a finite field Fq of odd characteristic, a hyperelliptic curve C of genus g with exactly one Weierstrass point at infinity may be specified by an affine equation y2 = f(x), where f(x) is a monic degree 2g+1 polynomial in Fq[x] with non-zero discriminant. Strictly speaking, we are interested in the projective curve given by the homogenization of this equation (which will necessarily be non-singular and irreducible), but it suffices to specify the affine part of the curve. Alternatively, one can view the "point at infinity" as an additional solution of the affine equation.

Zeta Function: Z(C/Fq; z)
Let C be a projective curve defined over Fq and let Nk count the points on the curve C over the field Fqk. The zeta function of C is the formal power series
Z(C/Fq; z)= exp(&sum Nkzk/k),
where the sum is over k from 1 to infinity.

L-polynomial: P(z), also denoted Lq(T)
By a theorem of Emil Artin, Z(C/Fq; z)= P(z)/[(1-z)(1-qz)] is a rational function. The polynomial P(z) has integer coefficients and degree 2g, where g is the genus of the curve. We call P(z) the L-polynomial of the curve C. When C is a genus 2 curve, P(z) has the form
P(z) = p2z4 + apz3+ bz2+ az + 1,

and for genus 3 curves, we have
P(z) = p3z6 + ap2z5 + bpz4+ cz3+ bz2 + az + 1. To specify the zeta function of a genus 2 or 3 curve, it suffices to specify the coefficients a, b, and (in genus 3) c of the L-polynomial P(z). From the theorem of Weil (the so-called Riemann hypothesis for curves) the roots of P(z) occur in conjugate pairs on a circle of radius p1/2 in the complex plane. This implies bounds on the coefficients of P(z), known as the Hasse-Weil bounds.

Jacobian: Jk(C)
The notation J(C) denotes the group of Fq-rational points on the Jacobian variety of C, equivalently, the divisor class group of degree 0, Pic0(C) (we won't attempt a more detailed definition here). This is a finite abelian group of size approximately qg. The group law can be effectively computed with Cantor's algorithm, representing group elements as a pair of monic polynomials over Fq (a Mumford representation). In genus 2 and 3, specializations of Cantor's algorithm give a very efficient group operation. The notation Jk(C) indicates the Jacobian of the curve C over the extension field Fqk, with J1(C) = J(C).

Order of the Jacobian: #Jk(C)
The size of the finite group Jk(C) is denoted #Jk(C). The L-polynomial P(z) can be used to compute #Jk(C) by summing P(i>z) over the kth roots of unity.

Frobenius Endomorphism: &phi
The Frobenius automorphism a&rarr aq on Fq gives rise to a group endomorphism &phiq on Jk(C), written simply as &phi. The elements fixed by &phi are precisely the subgroup J(C). Equivalently, J(C) is the kernel of the endomporhism &phi - 1 on Jk(C) where 1 denotes the identity map.

Trace Zero Variety: Tk(C)
The kernel of the endomorphism &phik-1 + &phik-2 + ... + 1 in Jk(C) is called the trace zero variety of Jk(C), denoted Tk(C).

Quotient Groups: Jk/d(C)
When d divides k, the image of the endomorphism &phid - 1 is a subgroup of Jk(C) isomorphic to the quotient group Jk(C)/Jd(C). The group Jk/1(C) is a subgroup of the trace zero variety Tk(C), and the two are equal precisely when J(C) is k-torsion free (equivalently, #J(C) is not divisible by k).

Note: I don't have a good name for Jk/d(C), so if anyone has a suggestion (or if it already has a name), please let me know -

Quadratic Twist: C'
If a genus g hyperelliptic curve C defined over Fq has affine part y2 = f(x), the quadratic twist of C over Fq, denoted C', may be defined by y2 = a2g+1f(x/a), where a is any non-residue in Fq. This definition does not depend on the choice of a, all non-residues give ismorphic curves. We may also take a as a non-residue in an extension field Fqk (typically k is a power of 2) and then use Ck' to denote the quadratic twist of C over Fqk. The group J2/1(C) is isomorphic to J(C'), and in general J2k/k(C) is isomorphic to Ck'.