Hyperelliptic Curves - Elliptic Curves: Number Theory and Cryptography - page 414

for some points P j and some integers c j .Since D 1 and w ( D 2 ) are reduced,
[ P j ] occurs in one of D 1 and w ( D 2 ), and [ w ( P j )] occurs in the other. By
switching the names of P j and w ( P j ), if necessary, we may assume that
D 1 =
j

w ( D 2 )=
j

([ P j ] − [ ∞ ]) and ([ w ( P j )] − [ ∞ ]) .

This implies that D 1 = D 2 , as desired.

We refer to elements of the group of divisors of degree 0 modulo principal
divisors as divisor classes of degree 0 . The set of divisor classes of degree
0 can be given the structure of an algebraic variety, called the Jacobian
variety J of C . Over the complex numbers, the Jacobian has the structure
of a g -dimensional complex torus C g /L ,where L is a lattice in g -dimensional
complex space (the case g = 1 is the case of elliptic curves treated in Chapter
9). The addition of divisor classes corresponds to addition of points in C g /L .
Let P 1 ,P 2 be points on C .Since[ P 1 ] − [ ∞ ]and[ P 2 ] − [ ∞ ] are reduced, the
uniqueness part of Proposition 13.6 implies that these two divisors are not
equivalent modulo principal divisors. Therefore, the map

C −→ J

P −→ [ P ] − [ ∞ ]

gives an injective mapping of C into its Jacobian. In the case of elliptic curves,
this is an isomorphism (Corollary 11.4).

THEOREM 13.7
Thereisaone-to-one correspondence between divisor classes of degree 0 on
C and pairs ( U ( x ) ,V ( x )) of polynom ialssatisfying

1. U ismonic.

2. deg V< deg U

≤ g .

3. V 2

− f ( x ) isamultipleof U .

PROOF By Proposition 13.6, every divisor class of degree 0 is represented
by a unique reduced divisor. By Theorem 13.5, these divisors are in one-to-
one correspondence with the pairs ( U, V ) as in the statement of the present
theorem.

REMARK 13.8 The pair ( U, V ) is called the Mumford representation
of the corresponding divisor class. In many situations, it is easier to work with
the Mumford representations than directly with the divisor classes. In the next