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:***y*^{2}=*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*y*^{2}=*f*(*x*), where*f*(*x*) is a monic degree 2*g*+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_{qk}. The zeta function of C is the formal power series

Z(C/F where the sum is over_{q};*z*)= exp(&sum N_{k}*z*/^{k}*k*),*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 2*g*, 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*) =*p*^{2}*z*^{4}+*apz*^{3}+*bz*^{2}+*az*+ 1,

and for genus 3 curves, we have

P( *z*) =*p*^{3}*z*^{6}+*ap*^{2}*z*^{5}+*bpz*^{4}+*cz*^{3}+*bz*^{2}+*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 F
q-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*. The group law can be effectively computed with Cantor's algorithm, representing group elements as a pair of monic polynomials over F^{g}_{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, with J_{qk}_{1}(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*k*th roots of unity.

**Frobenius Endomorphism: &phi**- The Frobenius automorphism
*a*&rarr*a*on F^{q}_{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*y*^{2}=*f*(*x*), the quadratic twist of C over F_{q}, denoted C', may be defined by*y*^{2}=*a*^{2g+1}*f*(*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_{qk}(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}'.