Hyperelliptic Curves - Elliptic Curves: Number Theory and Cryptography - page 410

Then

div( y + V )=
j

d j ([( a j , −b j )] − [ ∞ ]) =
j

d j ([ w ( a j ,b j )] − [ ∞ ]) .

Since
div( f −V 2 )=div( y + V )+div( y−V )=
j

d j ([( a j ,b j )]+[( a j , −b j )] − 2[ ∞ ]) ,

V 2 = j ( x a j ) d j , by part (a). Therefore, every root of we must have f − −

V 2 yields a term in div( y f − − V ). This completes the proof.

Part (a) of the proposition has a converse. If D = c j [ P j ] is a divisor, let

w ( D )= c j [ w ( P j )].

PROPOSITION 13.3
Let D be a divisor of degree 0. T hen D + w ( D ) isaprincipaldivisor; infact,
itisthe divisor of a rationalfunction in x .

Write D = j c j [ P j ], where possibly some P j is ∞ .Sincedeg D = PROOF

0, we have j c j =0,so D = j c j ([ P j ] − [ ∞ ]). If some P j = ∞ ,that
term can now be omitted, so we may assume P j = ∞ for all j . Therefore,

D + w ( D )= j c j ([ P j ]+[ w ( P j )] − 2[ ∞ ]), which is the divisor of a polynomial

in x , by Proposition 13.2.

A divisor of the form D = j c j ([ P j ] − [ ∞ ]), with P j =( a j ,b j ), is called
semi-reduced if the following hold:

1. c j ≥ 0 for all j

2. if b j =0,then c j =0or1

3. if [ P j ]with b j = 0 occurs in the sum (that is, c j > 0), then [ w ( P j )] does

not occur.
If, in addition, j c j ≤ g ,then D is called reduced .
Proposition 13.2 implies that div ( y − V ( x )) is semi-reduced for every poly-
nomial V ( x ).

Let D 1 = j c j ([ P j ] − [ ∞ ]) and D 2 = j d j ([ P j ] − [ ∞ ]) be two divisors
with c j ≥ 0and d j ≥ 0. Define
gcd( D 1 ,D 2 )=
j

min { c j ,d j } ([ P j ] − [ ∞ ]) .