A hyperelliptic curve is a particular kind of algebraic curve. There exist hyperelliptic curves of every genus ${\displaystyle g\geq 1}$. If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves. There is a well-known group structure on the set of points lying on an elliptic curve over some field ${\displaystyle K}$, which we can describe geometrically with chords and tangents. Generalizing this group structure to the hyperelliptic case is not straightforward. We cannot define the same group law on the set of points lying on a hyperelliptic curve, instead a group structure can be defined on the so-called Jacobian of a hyperelliptic curve. The computations differ depending on the number of points at infinity. Imaginary hyperelliptic curves are hyperelliptic curves with exactly 1 point at infinity: real hyperelliptic curves have two points at infinity.

Formal definition

Hyperelliptic curves can be defined over fields of any characteristic. Hence we consider an arbitrary field ${\displaystyle K}$ and its algebraic closure ${\displaystyle {\overline {K}}}$. An (imaginary) hyperelliptic curve of genus ${\displaystyle g}$ over ${\displaystyle K}$ is given by an equation of the form

$${\displaystyle C:y^{2}+h(x)y=f(x)\in K[x,y]}$$

where ${\displaystyle h(x)\in K[x]}$ is a polynomial of degree not larger than ${\displaystyle g}$ and ${\displaystyle f(x)\in K[x]}$ is a monic polynomial of degree ${\displaystyle 2g+1}$. Furthermore, we require the curve to have no singular points. In our setting, this entails that no point ${\displaystyle (x,y)\in {\overline {K}}\times {\overline {K}}}$ satisfies both ${\displaystyle y^{2}+h(x)y=f(x)}$ and the equations ${\displaystyle 2y+h(x)=0}$ and ${\displaystyle h'(x)y=f'(x)}$. This definition differs from the definition of a general hyperelliptic curve in the fact that ${\displaystyle f}$ can also have degree ${\displaystyle 2g+2}$ in the general case. From now on we drop the adjective imaginary and simply talk about hyperelliptic curves, as is often done in literature. Note that the case ${\displaystyle g=1}$ corresponds to ${\displaystyle f}$ being a cubic polynomial, agreeing with the definition of an elliptic curve. If we view the curve as lying in the projective plane ${\displaystyle \mathbb {P} ^{2}(K)}$ with coordinates ${\displaystyle (X:Y:Z)}$, we see that there is a particular point lying on the curve, namely the point at infinity ${\displaystyle (0:1:0)}$ denoted by ${\displaystyle O}$. So we could write ${\displaystyle C=\{(x,y)\in K^{2}|y^{2}+h(x)y=f(x)\}\cup \{O\}}$.

Suppose the point ${\displaystyle P=(a,b)}$ not equal to ${\displaystyle O}$ lies on the curve and consider ${\displaystyle {\overline {P}}=(a,-b-h(a))}$. As ${\displaystyle (-b-h(a))^{2}+h(a)(-b-h(a))}$ can be simplified to ${\displaystyle b^{2}+h(a)b}$, we see that ${\displaystyle {\overline {P}}}$ is also a point on the curve. ${\displaystyle {\overline {P}}}$ is called the opposite of ${\displaystyle P}$ and ${\displaystyle P}$ is called a Weierstrass point if ${\displaystyle P={\overline {P}}}$, i.e. ${\displaystyle h(a)=-2b}$. Furthermore, the opposite of ${\displaystyle O}$ is simply defined as ${\displaystyle {\overline {O}}=O}$.

Example

Figure 1: Example of a hyperelliptic curve

As an example consider ${\displaystyle C:y^{2}=f(x)}$ where ${\displaystyle f(x)=x^{5}-2x^{4}-7x^{3}+8x^{2}+12x=x(x+1)(x-3)(x+2)(x-2)}$ over ${\displaystyle \mathbb {R} }$. As ${\displaystyle f}$ has degree 5 and the roots are all distinct, ${\displaystyle C}$ is a curve of genus ${\displaystyle g=2}$. Its graph is depicted in Figure 1.

From this picture it is immediately clear that we cannot use the chords and tangents method to define a group law on the set of points of a hyperelliptic curve. The group law on elliptic curves is based on the fact that a straight line through two points lying on an elliptic curve has a unique third intersection point with the curve. Note that this is always true since ${\displaystyle O}$ lies on the curve. From the graph of ${\displaystyle C}$ it is clear that this does not need to hold for an arbitrary hyperelliptic curve. Actually, Bézout's theorem states that a straight line and a hyperelliptic curve of genus 2 intersect in 5 points. So, a straight line through two point lying on ${\displaystyle C}$ does not have a unique third intersection point, it has three other intersection points.

Coordinate ring

The coordinate ring of C over K is defined as

${\displaystyle K[C]=K[x,y]/(y^{2}+h(x)y-f(x)).}$

The polynomial ${\displaystyle r(x,y)=y^{2}+h(x)y-f(x)}$ is irreducible over ${\displaystyle {\overline {K}}}$, so

${\displaystyle {\overline {K}}[C]={\overline {K}}[x,y]/(y^{2}+h(x)y-f(x))}$

is an integral domain.

Note that any polynomial function ${\displaystyle G(x,y)\in {\overline {K}}[C]}$ can be written uniquely as

${\displaystyle G(x,y)=u(x)-v(x)y}$   with ${\displaystyle u(x),v(x)\in {\overline {K}}[x]}$

Norm and degree

The conjugate of a polynomial function G(x, y) = u(x) − v(x)y in ${\displaystyle {\overline {K}}[C]}$ is defined to be

${\displaystyle {\overline {G}}(x,y)=u(x)+v(x)(h(x)+y).}$

The norm of G is the polynomial function ${\displaystyle N(G)=G{\overline {G}}}$. Note that N(G) = u(x)² + u(x)v(x)h(x) − v(x)²f(x), so N(G) is a polynomial in only one variable.

If G(x, y) = u(x) − v(x) ⋅ y, then the degree of G is defined as

${\displaystyle \deg(G)=\max[2\deg(u),2g+1+2\deg(v)].}$

Properties:

${\displaystyle \deg(G)=\deg _{x}(N(G))}$

${\displaystyle \deg(GH)=\deg(G)+\deg(H)}$

${\displaystyle \deg(G)=\deg({\overline {G}})}$

Function field

The function field K(C) of C over K is the field of fractions of K[C], and the function field ${\displaystyle {\overline {K}}(C)}$ of C over ${\displaystyle {\overline {K}}}$ is the field of fractions of ${\displaystyle {\overline {K}}[C]}$. The elements of ${\displaystyle {\overline {K}}(C)}$ are called rational functions on C. For R such a rational function, and P a finite point on C, R is said to be defined at P if there exist polynomial functions G, H such that R = G/H and H(P) ≠ 0, and then the value of R at P is

${\displaystyle R(P)=G(P)/H(P).}$

For P a point on C that is not finite, i.e. P = ${\displaystyle O}$, we define R(P) as:

If ${\displaystyle \deg(G)<\deg(H)}$  then ${\displaystyle R(O)=0}$, i.e. R has a zero at O.

If ${\displaystyle \deg(G)>\deg(H)}$  then ${\displaystyle R(O)}$  is not defined, i.e. R has a pole at O.

If ${\displaystyle \deg(G)=\deg(H)}$  then ${\displaystyle R(O)}$  is the ratio of the leading coefficients of G and H.

For ${\displaystyle R\in {\overline {K}}(C)^{*}}$ and ${\displaystyle P\in C}$,

If ${\displaystyle R(P)=0}$ then R is said to have a zero at P,

If R is not defined at P then R is said to have a pole at P, and we write ${\displaystyle R(P)=\infty }$.

Order of a polynomial function at a point

For ${\displaystyle G=u(x)-v(x)\cdot y\in {\overline {K}}[C]^{2}}$ and ${\displaystyle P\in C}$, the order of G at P is defined as:

${\displaystyle \mathrm {ord} _{P}(G)=r+s}$ if P = (a,b) is a finite point which is not Weierstrass. Here r is the highest power of (x − a) which divides both u(x) and v(x). Write G(x,y) = (x − a)^r(u₀(x) − v₀(x)y) and if u₀(a) − v₀(a)b = 0, then s is the highest power of (x − a) which divides N(u₀(x) − v₀(x)y) = u₀² + u₀v₀h − v₀²f, otherwise, s = 0.

${\displaystyle \mathrm {ord} _{P}(G)=2r+s}$ if P = (a,b) is a finite Weierstrass point, with r and s as above.

${\displaystyle \mathrm {ord} _{P}(G)=-\deg(G)}$ if P = O.

The divisor and the Jacobian

In order to define the Jacobian, we first need the notion of a divisor. Consider a hyperelliptic curve ${\displaystyle C}$ over some field ${\displaystyle K}$. Then we define a divisor ${\displaystyle D}$ to be a formal sum of points in ${\displaystyle C}$, i.e. ${\textstyle D=\sum _{P\in C}{c_{P}[P]}}$ where ${\displaystyle c_{P}\in \mathbb {Z} }$ and furthermore ${\displaystyle \{c_{P}\mid c_{P}\neq 0\}}$ is a finite set. This means that a divisor is a finite formal sum of scalar multiples of points. Note that there is no simplification of ${\displaystyle c_{P}[P]}$ given by a single point (as one might expect from the analogy with elliptic curves). Furthermore, we define the degree of ${\displaystyle D}$ as ${\textstyle \deg(D)=\sum _{P\in C}{c_{P}}\in \mathbb {Z} }$. The set of all divisors ${\displaystyle \mathrm {Div} (C)}$ of the curve ${\displaystyle C}$ forms an Abelian group where the addition is defined pointwise as follows ${\textstyle \sum _{P\in C}{c_{P}[P]}+\sum _{P\in C}{d_{P}[P]}=\sum _{P\in C}{(c_{P}+d_{P})[P]}}$. It is easy to see that ${\textstyle 0=\sum _{P\in C}{0[P]}}$ acts as the identity element and that the inverse of ${\textstyle \sum _{P\in C}{c_{P}[P]}}$ equals ${\textstyle \sum _{P\in C}{-c_{P}[P]}}$. The set ${\displaystyle \mathrm {Div} ^{0}(C)=\{D\in \mathrm {Div} (C)\mid \deg(D)=0\}}$ of all divisors of degree 0 can easily be checked to be a subgroup of ${\displaystyle \mathrm {Div} (C)}$.
Proof. Consider the map ${\displaystyle \varphi$ :\mathrm {Div} (C)\rightarrow \mathbb {Z} } defined by ${\displaystyle \varphi (D)=\deg(D)}$, note that ${\displaystyle \mathbb {Z} }$ forms a group under the usual addition. Then ${\textstyle \varphi (\sum _{P\in C}{c_{P}[P]}+\sum _{P\in C}{d_{P}[P]})=\varphi (\sum _{P\in C}{(c_{P}+d_{P})[P]})=\sum _{P\in C}{c_{P}+d_{P}}=\sum _{P\in C}{c_{p}}+\sum _{P\in C}{d_{p}}=\varphi (\sum _{P\in C}{c_{P}[P]})+\varphi (\sum _{P\in C}{d_{P}[P]})}$ and hence ${\displaystyle \varphi }$ is a group homomorphism. Now, ${\displaystyle \mathrm {Div} ^{0}(C)}$ is the kernel of this homomorphism and thus it is a subgroup of ${\displaystyle \mathrm {Div} (C)}$.

Consider a function ${\displaystyle f\in {\overline {K}}(C)^{*}}$, then we can look at the formal sum div${\textstyle (f)=\sum _{P\in C}{\mathrm {ord} _{P}(f)[P]}}$. Here ${\displaystyle \mathrm {ord} _{P}(f)}$ denotes the order of ${\displaystyle f}$ at ${\displaystyle P}$. We have that ord${\displaystyle _{P}(f)<0}$ if ${\displaystyle f}$ has a pole of order ${\displaystyle -\mathrm {ord} _{P}(f)}$ at ${\displaystyle P}$, ${\displaystyle \mathrm {ord} _{P}(f)=0}$ if ${\displaystyle f}$ is defined and non-zero at ${\displaystyle P}$ and ${\displaystyle \mathrm {ord} _{P}(f)>0}$ if ${\displaystyle f}$ has a zero of order ${\displaystyle \mathrm {ord} _{P}(f)}$ at ${\displaystyle P}$.^[1] It can be shown that ${\displaystyle f}$ has only a finite number of zeroes and poles,^[2] and thus only finitely many of the ord${\displaystyle _{P}(f)}$ are non-zero. This implies that div${\displaystyle (f)}$ is a divisor. Moreover, as ${\textstyle \sum _{P\in C}{\mathrm {ord} _{P}(f)}=0}$,^[2] it is the case that ${\displaystyle \operatorname {div} (f)}$ is a divisor of degree 0. Such divisors, i.e. divisors coming from some rational function ${\displaystyle f}$, are called principal divisors and the set of all principal divisors ${\displaystyle \mathrm {Princ} (C)}$ is a subgroup of ${\displaystyle \mathrm {Div} ^{0}(C)}$.
Proof. The identity element ${\textstyle 0=\sum _{P\in C}{0[P]}}$ comes from a constant function which is non-zero. Suppose ${\textstyle D_{1}=\sum _{P\in C}{\mathrm {ord} _{P}(f)[P]},D_{2}=\sum _{P\in C}{\mathrm {ord} _{P}(g)[P]}\in \mathrm {Princ} (C)}$ are two principal divisors coming from ${\displaystyle f}$ and ${\displaystyle g}$ respectively. Then ${\textstyle D_{1}-D_{2}=\sum _{P\in C}{(\mathrm {ord} _{P}(f)-\mathrm {ord} _{P}(g))[P]}}$ comes from the function ${\displaystyle f/g}$, and thus ${\displaystyle D_{1}-D_{2}}$ is a principal divisor, too. We conclude that ${\displaystyle \mathrm {Princ} (C)}$ is closed under addition and inverses, making it into a subgroup.

We can now define the quotient group ${\displaystyle J(C):=\mathrm {Div} ^{0}(C)/\mathrm {Princ} (C)}$ which is called the Jacobian or the Picard group of ${\displaystyle C}$. Two divisors ${\displaystyle D_{1},D_{2}\in \mathrm {Div} ^{0}(C)}$ are called equivalent if they belong to the same element of ${\displaystyle J(C)}$, this is the case if and only if ${\displaystyle D_{1}-D_{2}}$ is a principal divisor. Consider for example a hyperelliptic curve ${\displaystyle C:y^{2}+h(x)y=f(x)}$ over a field ${\displaystyle K}$ and a point ${\displaystyle P=(a,b)}$ on ${\displaystyle C}$. For ${\displaystyle n\in \mathbb {Z} _{>0}}$ the rational function ${\displaystyle f(x)=(x-a)^{n}}$ has a zero of order ${\displaystyle n}$ at both ${\displaystyle P}$ and ${\displaystyle {\overline {P}}}$ and it has a pole of order ${\displaystyle 2n}$ at ${\displaystyle O}$. Therefore, we find ${\displaystyle \operatorname {div} (f)=nP+n{\overline {P}}-2nO}$ and we can simplify this to ${\displaystyle \operatorname {div} (f)=2nP-2nO}$ if ${\displaystyle P}$ is a Weierstrass point.

The Jacobian of a hyperelliptic curve

The general hyperelliptic case is a bit more complicated. Consider a hyperelliptic curve ${\displaystyle C:y^{2}+h(x)y=f(x)}$ of genus ${\displaystyle g}$ over a field ${\displaystyle K}$. A divisor ${\displaystyle D}$ of ${\displaystyle C}$ is called reduced if it has the form ${\textstyle D=\sum _{i=1}^{k}{[P_{i}]}-k[O]}$ where ${\displaystyle k\leq g}$, ${\displaystyle P_{i}\not =O}$ for all ${\displaystyle i=1,\dots ,k}$ and ${\displaystyle P_{i}\not ={\overline {P_{j}}}}$ for ${\displaystyle i\not =j}$. Note that a reduced divisor always has degree 0, also it is possible that ${\displaystyle P_{i}=P_{j}}$ if ${\displaystyle i\not =j}$, but only if ${\displaystyle P_{i}}$ is not a Weierstrass point. It can be proven that for each divisor ${\displaystyle D\in \mathrm {Div} ^{0}(C)}$ there is a unique reduced divisor ${\displaystyle D'}$ such that ${\displaystyle D}$ is equivalent to ${\displaystyle D'}$.^[3] Hence every class of the quotient group ${\displaystyle J(C)}$ has precisely one reduced divisor. Instead of looking at ${\displaystyle J(C)}$ we can thus look at the set of all reduced divisors.