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)
- 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
- 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 - firstname.lastname@example.org.
- 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'.