One of the most celebrated properties of elliptic curves is that the set of rational points is a group, with a highly geometric explanation of the group law: the ‘chord and tangent’ process. Two points and their sum are linked by consideration of the intersection of straight lines with the curve: as the curve is a cubic, there are three intersections (subject to some technical book-keeping with repeated points and the point ‘at infinity’). Such an approach clearly won’t transfer immediately to curves given by higher degree polynomials, as there will be more intersections, but as there are only finitely many, one would still hope to be able to define relationships between points. For instance, on an elliptic curve, if A, B and ∞ are colinear then B is -A; thus if on a more complicated curve we had A,B,C,∞ colinear it might make sense to think of C as -(A+B), and then A+C as -B and so on. That is to say, there may be relationships between groups of points rather than individual points.
In algebraic geometry, this (and much more) is captured by the notion of a divisor; rather than present them here in full generality, I will consider the specific case of divisors on (hyperelliptic) curves. These will then serve as the building blocks for a group structure connected to the curve which reduces in the special case of an elliptic curve to the familiar group of rational points.
To fix ideas, let K be a field of characteristic other than 2 with algebraic closure A. A curve C is described as a hyperelliptic curve of genus g if there is some degree 2g+1 polynomial f with distinct roots such that v2=f(u) is a model of C: so the familiar elliptic curves are the special cases with genus 1.
A point P on C is a pair (x,y) of elements of A (not K) satisfying y=f(x); or the point at infinity ∞. Then a divisor D of C is a finite formal sum ΣimiPi for integers mi and points Pi on C. D is described as having degree Σimi; if all the mi≥0 then we write D≥0. Formal (that is, pointwise) addition of divisors gives the additive group D of divisors; its identity is the divisor consisting of summing no points and it has a subgroup D0 consisting of divisors with degree zero.
Any polynomial p(u,v) can be considered as a function on C of the form p=a(u)+b(u)v, since v2=f(u). If p vanishes at (x,y) then the order of the zero (x,y) of p is the exponent of the highest power of (u-x) which divides a2-b2f.
Thus we can define functions on C as h=p/q for p,q polynomials from K[u,v] such that v2-f(u) does not divide q(u,v): that is, q is not everywhere zero on C. Then h will have a finite set of zeros (those of p) and of poles (those of q); we associate to h a divisor, (h) = ΣimiPi where the Pi are those zeros and poles and mi their multiplicities:
=\displaystyle\sum_{\mbox{zeros of } p} ord_{P_i}(p)P_i - \displaystyle\sum_{\mbox{zeros of } q} ord_{P_i}(q)P_i)
If there is a nonzero function h on C such that D a divisor is (h), then D is described as principal. the principal divisors form a subgroup P of D0 and hence D: the jacobian J of C is then the quotient D0/P. That is, two divisors correspond to the same element of the jacobian if they differ by a principal divisor. This gives some idea as to how to simplify arbitrary divisors- we work in the jacobian and seek a simplest representative; that is, one comprised of the minimal number of points.
Consider that if (x,y) is a point P of C, then so is P’=(x,-y). The function u-x has zeros P and P’ with a double pole at ∞ so P+P’+∞ = (u-x) is principal and hence equivalent to zero mod P. Hence -P’ is equivalent to P-2∞ so we can rewrite divisors to only feature positive multiples of points other than ∞ Thus in J, where the degree is necessarily 0, any element has a representation
 - r\infty)
such that if Pi appears in D, then no Pj=P’ for any j different to i. Hence, any point of the form (x,0) will appear at most once. Such a representation is called semi-reduced; if r≤g then it is called reduced.
Remarkably, (by the Riemann-Roch theorem) any divisor in the Jacobian will have a unique, reduced representative (in other words, any divisor is the sum of a reduced divisor and a principal divisor). Now we can see what’s really going on with the elliptic curve group law: as a reduced divisor will have r≤1, it takes the form P-&infty; so there is an obvious isomorphism between the set of rational points and the Jacobian. Hence adding two points A,B on the curve gives rise to another point of the curve, by reducing the divisor A+B-2∞ to some representative C-∞ and setting A+B=C.
But with hyperelliptic curves, this needn’t be the case: the sum of two points is a perfectly good reduced divisor in the next simplest case of genus 2, for instance, so we can’t add two points and expect the answer to be a point. Hence we need to consider the divisors corresponding to rational points in the broader setting of the jacobian; to extract useful information about those points, we’ll need to consider the rational divisors. This motivates an alternative notation for divisors, more suitable to computation: I leave all these issues to the next post.