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9.3 Divisors

Let Z be a proper closed subscheme of T and W be its image in 1. As before W is defined as the vanishing locus of a homogeneous polynomial F(U, V). Let F = VkF1k1 ... Frnr be a factorisation of F with Fi irreducible and distinct from each other and V. Clearly W is the disjoint union of closed subschemes Wi each defined by the vanishing of Fi(U, V)ki and the scheme W0 defined by Vk = 0. As before, we write fi(x) = Fi(U, V)/Vdeg(Fi), where x = U/V; let Qi denote the closed point in 1 defined by fi and Q0 be the point in 1 defined by V = 0. We can decompose Z into the components Zi that lie over the component Wi of W. We can then classify Zi according to the classification of the polynomials fi as above. In cases (1), (2) and (3) above there is exactly one closed point that lies over Qi, thus the schemes Zi are thickenings'' of the corresponding closed points Pi. In case (4) there are two closed points corresponding to the distinct roots; we denote these by Pi, 1 and Pi, 2. Let Pi, 1 correspond to the solution y = g(x) or y2 + a(x)y + b(x) = 0 in [x]/(fi(x)). By Hensel's lemma we can find gki(x) in [x]/(fi(x)ki) which is a lift'' of the solution g(x). Thus we have the closed subscheme Zi, 1 of Zi defined by the solution y = gki(x). Similarly, we have Zi, 2 and it is clear that Zi is the union of these two schemes. Thus each proper closed subscheme of T is the disjoint union of thickened'' closed points.

For any such closed subscheme Z of T we have a vector space scheme given by (a)Z extended by zero on the rest of T. We denote this vector space scheme by (P) when Z is the subscheme associated to the a closed point P. The vector space scheme associated with the thickened'' closed points is equivalent, in the K-group, to n(P) for some integer n. This can be shown by a composition series argument''. A similar Jordan-Hölder composition series can be used to show that the K-group of T is generated by (a)T and the elements (P). Moreover, if we consider an element D of the form ni(Pi) of the K-group then the number deg(D) = nideg(Pi) can be shown to be well-defined (independent of the representation of D). Thus the important group becomes the group of divisors of degree 0'' of the subgroup of the K-group consisting of elements of the form ni(Pi) where nideg(Pi) = 0. This group is denoted Pic0(T). An important theorem of Weil states that there is a group scheme J (called the Jacobian variety of T) such that Pic0(T) can be naturally identified with J(). There is also a natural analogy of this with the divisor class group for quadratic number fields that we will consider in the next subsection.

To compute the group Pic0(T) of divisors of degree 0, it enough to work modulo () which is of degree 1, since any divisor can be converted to one of degree 0 by subtracting a suitable multiple of (). Thus, we see that this group is generated by the elements [P] = (P) - deg(P)(). For a divisor D of degree d we introduce the notation [D] = D - d () to denote the corresponding element in Pic0(T).

Next: 9.4 Computing with the Up: 9 Hyperelliptic Cryptosystems Previous: 9.2 Closed points
Kapil Hari Paranjape 2002-10-20