Hyperelliptic Curves - Elliptic Curves: Number Theory and Cryptography - page 407

generalized Weierstrass form. It can be shown that the genus of C (in the
sense of Theorem 11.15) is g .
If the characteristic of K is not 2, we can complete the square on the left
side and put the curve into the form

C : y 2 = f ( x ) ,

(13.2)

with f monic of degree 2 g + 1. The nonsingularity condition states that no
point on the curve satisfies 2 y =0= f ( x ). Since y =0meansthat f ( x )=0,
this is equivalent to saying that no x satisfies f ( x )=0= f ( x ). In other
words, f ( x ) has no multiple roots, just as for the Weierstrass form of an
elliptic curve. For simplicity, we work with the form (13.2) throughout this
chapter.
There is one point at infinity, given by (0 : 1 : 0) and denoted ∞ .If g ≥ 2,
this point is singular, but this has no effect on what we do in this chapter.

Technical point: Variousresultsthatweneedtoapplyto C require that
the curve be nonsingular. Therefore, it is necessary to remove the singularity
at infinity. This is done by taking what is called the normalization of C .
Fortunately, the resulting nonsingular curve agrees with C at the a ne (that
is, non-infinite) points and has a unique point at infinity (see, for example,
[106]), which we again denote ∞ . In the following, we will be working with
functions and divisors. It will be easy to see what happens at the ane points.
The behavior at ∞ , which might seem harder to understand, is forced. For
exam ple, the function x − a ,where a is a constant, has two zeros, namely

( a, f ( a )) and ( a, − f ( a )). Since the function has no other zeros or poles in
the a ne plane, and the degree of its divisor is 0, it must have a double pole at
∞

. Similarly, any polynomial in x, y gives a function that has no poles in the
a ne plane, so the poles are at . In fact, it can be shown that the rational

functions on C with no poles except possibly at

∞

∞ are the polynomials in x, y .

In summary, in most situations we can work with without worry. However,

the point (more accurately, a neighborhood of the point) is more complicated
than it might appear.

∞

REMARK 13.1 There are also hyperelliptic curves given by equations of
the forms (13.1) and (13.2) with deg f =2 g + 2. However, these will not be
used in this chapter. Therefore, throughout this chapter, hyperellipticcurve
will mean a curve with deg f =2 g +1.

Let P =( x, y )beapointon C . Define

w ( P )=( x, − y ) ,

which is also a point on C .Themap w : P → w ( P ) is called the hyperelliptic
involution .Itsatisfies w ( w ( P )) = P for all points P on C . On elliptic curves,
w is multiplication by − 1.