Real hyperelliptic curve

Definition

A real hyperelliptic curve of genus g over K is defined by an equation of the form C : y 2 + h ( x ) y = f ( x ) where h ( x ) ∈ K has degree not larger than g+1 while f ( x ) ∈ K must have degree 2g+1 or 2g+2. This curve is a non singular curve where no point ( x , y ) in the algebraic closure of K satisfies the curve equation y 2 + h ( x ) y = f ( x ) and both partial derivative equations: 2 y + h ( x ) = 0 and h ′ ( x ) y = f ′ ( x ) . The set of (finite) K –rational points on C is given by

C ( K ) = { ( a , b ) ∈ K 2 | b 2 + h ( a ) b = f ( a ) } ∪ S

Where S is the set of points at infinity. For real hyperelliptic curves, there are two points at infinity, ∞ 1 and ∞ 2 . For any point P ( a , b ) ∈ C ( K ) , the opposite point of P is given by P ¯ = ( a , − b − h ) ; it is the other point with x-coordinate a that also lies on the curve.

Example

Let C : y 2 = f ( x ) where

f ( x ) = x 6 + 3 x 5 − 5 x 4 − 15 x 3 + 4 x 2 + 12 x = x ( x − 1 ) ( x − 2 ) ( x + 1 ) ( x + 2 ) ( x + 3 )

over R . Since deg ⁡ f ( x ) = 2 g + 2 and f ( x ) has degree 6, thus C is a curve of genus g = 2.

The homogeneous version of the curve equation is given by

Y 2 Z 4 = X 6 + 3 X 5 Z − 5 X 4 Z 2 − 15 X 3 Z 3 + 4 X 2 Z 4 + 12 X Z 5 .

It has a single point at infinity given by (0:1:0) but this point is singular. The blowup of C has 2 different points at infinity, which we denote ∞ 1 and ∞ 2 . Hence this curve is an example of a real hyperelliptic curve.

In general, every curve given by an equation where f has even degee has two points at infinity and is a real hyperelliptic curve while those where f has odd degree have only a single point in the blowup over (0:1:0) and are thus imaginary hyperelliptic curves. In both cases this assumes that the affine part of the curve is nonsingular (see the conditions on the derivatives above)

Arithmetic in a real hyperelliptic curve

In real hyperelliptic curve, addition is no longer defined on points as in elliptic curves but on divisors and the Jacobian. Let C be a hyperelliptic curve of genus g over a finite field K. A divisor D on C is a formal finite sum of points P on C . We write

D = ∑ P ∈ C n P P where n P ∈ Z and n p = 0 for almost all P .

The degree of D = ∑ P ∈ C n P P is defined by

deg ⁡ ( D ) = ∑ P ∈ C n P .

D is said to be defined over K if D σ = ∑ P ∈ C n P P σ = D for all automorphisms σ of K ¯ over K . The set D i v ( K ) of divisors of C defined over K forms an additive abelian group under the addition rule

∑ a P P + ∑ b P P = ∑ ( a P + b P ) P .

The set D i v 0 ( K ) of all degree zero divisors of C defined over K is a subgroup of D i v ( K ) .

We take an example:

Let D 1 = 6 P 1 + 4 P 2 and D 2 = 1 P 1 + 5 P 2 . If we add them then D 1 + D 2 = 7 P 1 + 9 P 2 . The degree of D 1 is deg ⁡ ( D 1 ) = 6 + 4 = 10 and the degree of D 2 is deg ⁡ ( D 2 ) = 1 + 5 = 6 . Then, deg ⁡ ( D 1 + D 2 ) = d e g ( D 1 ) + d e g ( D 2 ) = 16.

For polynomials G ∈ K [ C ] , the divisor of G is defined by

d i v ( G ) = ∑ P ∈ C o r d P ( G ) P . If the function

G has a pole at a point P then − o r d P ( G ) is the order of vanishing of G at P . Assume G , H are polynomials in K [ C ] ; the divisor of the rational function F = G / H is called a principal divisor and is defined by d i v ( F ) = d i v ( G ) − d i v ( H ) . We denote the group of principal divisors by P ( K ) , i.e. P ( K ) = d i v ( F ) | F ∈ K ( C ) . The Jacobian of C over K is defined by J = D i v 0 / P . The factor group J is also called the divisor class group of C . The elements which are defined over K form the group J ( K ) . We denote by D ¯ ∈ J ( K ) the class of D in D i v 0 ( K ) / P ( K ) .

There are two canonical ways of representing divisor classes for real hyperelliptic curves C which have two points infinity S = { ∞ 1 , ∞ 2 } . The first one is to represent a degree zero divisor by D ¯ such that D = ∑ i = 1 r P i − r ∞ 2 , where P i ∈ C ( F ¯ q ) , P i ≠ ∞ 2 , and P i ≠ P j ¯ if i ≠ j The representative D of D ¯ is then called semi reduced. If D satisfies the additional condition r ≤ g then the representative D is called reduced. Notice that P i = ∞ 1 is allowed for some i. It follows that every degree 0 divisor class contain a unique representative D ¯ with

D = D x − d e g ( D x ) ∞ 2 + v 1 ( D ) ( ∞ 1 − ∞ 2 ) ,

where D x is divisor that is coprime with both

∞ 1 and ∞ 2 , and 0 ≤ d e g ( D x ) + v 1 ( D ) ≤ g .

The other representation is balanced at infinity. Let D ∞ = ∞ 1 + ∞ 2 , note that this divisor is K -rational even if the points ∞ 1 and ∞ 2 are not independently so. Write the representative of the class D ¯ as D = D 1 + D ∞ , where D 1 is called the affine part and does not contain ∞ 1 and ∞ 2 , and let d = deg ⁡ ( D 1 ) . If d is even then

D ∞ = d 2 ( ∞ 1 + ∞ 2 ) .

If d is odd then

D ∞ = d + 1 2 ∞ 1 + d − 1 2 ∞ 2 .

For example, let the affine parts of two divisors be given by

D 1 = 6 P 1 + 4 P 2 and D 2 = 1 P 1 + 5 P 2

then the balanced divisors are

D 1 = 6 P 1 + 4 P 2 − 5 D ∞ 1 − 5 D ∞ 2 and D 2 = 1 P 1 + 5 P 2 − 3 D ∞ 1 − 3 D ∞ 2

Transformation from real hyperelliptic curve to imaginary hyperelliptic curve

Let C be a real quadratic curve over a field K . If there exists a ramified prime divisor of degree 1 in K then we are able to perform a birational transformation to an imaginary quadratic curve. A (finite or infinite) point is said to be ramified if it is equal to its own opposite. It means that P = ( a , b ) = P ¯ = ( a , − b − h ( a ) ) , i.e. that h ( a ) + 2 b = 0 . If P is ramified then D = P − ∞ 1 is a ramified prime divisor.

The real hyperelliptic curve C : y 2 + h ( x ) y = f ( x ) of genus g with a ramified K -rational finite point P = ( a , b ) is birationally equivalent to an imaginary model C ′ : y ′ 2 + h ¯ ( x ′ ) y ′ = f ¯ ( x ′ ) of genus g , i.e. deg ⁡ ( f ¯ ) = 2 g + 1 and the function fields are equal K ( C ) = K ( C ′ ) . Here:

x ′ = 1 x − a and y ′ = y + b ( x − a ) g + 1 … (i)

In our example C : y 2 = f ( x ) where f ( x ) = x 6 + 3 x 5 − 5 x 4 − 15 x 3 + 4 x 2 + 12 x , h(x) is equal to 0. For any point P = ( a , b ) , h ( a ) is equal to 0 and so the requirement for P to be ramified becomes b = 0 . Substituting h ( a ) and b , we obtain f ( a ) = 0 , where f ( a ) = a ( a − 1 ) ( a − 2 ) ( a + 1 ) ( a + 2 ) ( a + 3 ) , i.e. a ∈ { 0 , 1 , 2 , − 1 , − 2 , − 3 } .

From (i), we obtain x = a x ′ + 1 x ′ and y = y ′ x ′ g + 1 . For g=2, we have y = y ′ x ′ 3

For example, let a = 1 then x = x ′ + 1 x ′ and y = y ′ x ′ 3 , we obtain

( y ′ x ′ 3 ) 2 = x ′ + 1 x ′ ( x ′ + 1 x ′ + 1 ) ( x ′ + 1 x ′ + 2 ) ( x ′ + 1 x ′ + 3 ) ( x ′ + 1 x ′ − 1 ) ( x ′ + 1 x ′ − 2 ) .

To remove the denominators this expression is multiplied by x 6 , then:

y ′ 2 = ( x ′ + 1 ) ( 2 x ′ + 1 ) ( 3 x ′ + 1 ) ( 4 x ′ + 1 ) ( 1 ) ( 1 − x ′ )

giving the curve

C ′ : y ′ 2 = f ¯ ( x ′ ) where f ¯ ( x ′ ) = ( x ′ + 1 ) ( 2 x ′ + 1 ) ( 3 x ′ + 1 ) ( 4 x ′ + 1 ) ( 1 ) ( 1 − x ′ ) = − 24 x ′ 5 − 26 x ′ 4 + 15 x ′ 3 + 25 x ′ 2 + 9 x ′ + 1 .

C ′ is an imaginary quadratic curve since f ¯ ( x ′ ) has degree 2 g + 1 .