Definitions and Notation

accessibility 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 F_q 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 F_q[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/F_q; z): Let C be a projective curve defined over F_q and let N_k count the points on the curve C over the field F_q^k. The zeta function of C is the formal power series
Z(C/F_q; z)= exp(&sum N_kz^k/k), where the sum is over k from 1 to infinity.
L-polynomial: P(z), also denoted L_q(T): By a theorem of Emil Artin, Z(C/F_q; 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 p^1/2 in the complex plane. This implies bounds on the coefficients of P(z), known as the Hasse-Weil bounds.

Jacobian: J_k(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, Pic^₀(C) (we won't attempt a more detailed definition here). This is a finite abelian group of size approximately q^g. The group law can be effectively computed with Cantor's algorithm, representing group elements as a pair of monic polynomials over F_q (a Mumford representation). In genus 2 and 3, specializations of Cantor's algorithm give a very efficient group operation. The notation J_k(C) indicates the Jacobian of the curve C over the extension field F_q^k, with J₁(C) = J(C).

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

Frobenius Endomorphism: &phi
The Frobenius automorphism a&rarr a^q on F_q gives rise to a group endomorphism &phi_q on J_k(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 J_k(C) where 1 denotes the identity map.

Trace Zero Variety: T_k(C)
The kernel of the endomorphism &phi^k-1 + &phi^k-2 + ... + 1 in J_k(C) is called the trace zero variety of J_k(C), denoted T_k(C).

Quotient Groups: J_k/d(C)
When d divides k, the image of the endomorphism &phi^d - 1 is a subgroup of J_k(C) isomorphic to the quotient group J_k(C)/J_d(C). The group J_k/1(C) is a subgroup of the trace zero variety T_k(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 J_k/d(C), so if anyone has a suggestion (or if it already has a name), please let me know - drew@math.mit.edu.

Quadratic Twist: C'

If a genus g hyperelliptic curve C defined over F_q has affine part y² = f(x), the quadratic twist of C over F_q, denoted C', may be defined by y² = a^2g+1f(x/a), where a is any non-residue in F_q. 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 F_q^k (typically k is a power of 2) and then use C_k' to denote the quadratic twist of C over F_qk. The group J_2/1(C) is isomorphic to J(C'), and in general J_2k/k(C) is isomorphic to C_k'.
Jacobian: J_k(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, Pic^₀(C) (we won't attempt a more detailed definition here). This is a finite abelian group of size approximately q^g. The group law can be effectively computed with Cantor's algorithm, representing group elements as a pair of monic polynomials over F_q (a Mumford representation). In genus 2 and 3, specializations of Cantor's algorithm give a very efficient group operation. The notation J_k(C) indicates the Jacobian of the curve C over the extension field F_q^k, with J₁(C) = J(C).
Order of the Jacobian: #J_k(C): The size of the finite group J_k(C) is denoted #J_k(C). The L-polynomial P(z) can be used to compute #J_k(C) by summing P(i>z) over the kth roots of unity.
Frobenius Endomorphism: &phi: The Frobenius automorphism a&rarr a^q on F_q gives rise to a group endomorphism &phi_q on J_k(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 J_k(C) where 1 denotes the identity map.
Trace Zero Variety: T_k(C): The kernel of the endomorphism &phi^k-1 + &phi^k-2 + ... + 1 in J_k(C) is called the trace zero variety of J_k(C), denoted T_k(C).
Quotient Groups: J_k/d(C): When d divides k, the image of the endomorphism &phi^d - 1 is a subgroup of J_k(C) isomorphic to the quotient group J_k(C)/J_d(C). The group J_k/1(C) is a subgroup of the trace zero variety T_k(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 J_k/d(C), so if anyone has a suggestion (or if it already has a name), please let me know - drew@math.mit.edu.
Quadratic Twist: C': If a genus g hyperelliptic curve C defined over F_q has affine part y² = f(x), the quadratic twist of C over F_q, denoted C', may be defined by y² = a^2g+1f(x/a), where a is any non-residue in F_q. 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 F_q^k (typically k is a power of 2) and then use C_k' to denote the quadratic twist of C over F_qk. The group J_2/1(C) is isomorphic to J(C'), and in general J_2k/k(C) is isomorphic to C_k'.