Modulo Errors

Last time I introduced the idea of divisors on a curve; but an observant reader may have noticed that along the way the idea of rational points seemed to be lost. Further, whilst the Riemann-Roch theorem guarantees that a divisor from the jacobian will have a reduced representative, no indication was given as to how that representative is to be found. In this post I’ll try to clear up both of these issues.

Recall that a semi-reduced divisor D from the jacobian takes the form (Σ_i^rP_i) - r∞ where the P_i are points (x_i,y_i) of C. We will represent this by a pair of polynomials D=div(a(u),b(u)) with the following properties:

such that b has degree less than that of a, and the appropriate multiplicity for repeated points- i.e., if P_i occurs k times in the semi-reduced representation of D, then (u-x_i)^k divides b-y_i. This ensures uniqueness.

The empty divisor (zero element of the jacobian) is denoted by div(1,0); if a is linear (and hence b constant) then the divisor corresponds to a single point of C; for the point (x,y) the divisor is div(u-x,y). The degree of a is described as the weight; ‘most’ reduced divisors will be of weight g. Recall that the co-ordinates of a point were only required to be in A rather than K; we describe a divisor as rational over K if the coefficients of a and b are from K. Beware that a rational divisor may therefore be the sum of points which are not K-rational points of the curve; however, a weight 1 rational divisor obviously corresponds to a K-rational point.

The K-rational principal divisors of C are a subgroup of P₀ and their image in J, J_K, the subgroup of rational divisors of the Jacobian, is the object of computational interest. The K-rational points of C are then identified with a subset of J_K; namely the divisors of weight at most 1; except in genus 1 (where they are J_K), this isn’t a subgroup due to the lack of closure.

So we wish to work with rational divisors from J_K.Given such divisors of the form div(a,b), it is undesirable to construct their sum by ‘unpacking’ the points P_i and forming new polynomials as the co-ordinates (roots of a, evaluations of b) might not be from K but A. Fortunately, it is also unnecessary: the only complications are connected to repeated points or the combination of a point and its negative; careful manipulation of gcds allows for direct computation of the semi-reduced form. See ¹ for details, which also describes moving from semi-reduced to reduced form. To do this, note that for a divisor D=div(a,b), the divisor E=-((b-v)-D)=div(a’,b’) is equivalent to D with deg(a’)=max(2g+1,2)-deg(a); thus by repeated iteration we can move to a reduced representative (the explicit formulae are a’=(f-b²)/a, b’=-b mod a’).

Maple procedures to do all this will be provided in the next post.

Reference
¹Computing in the Jacobian of a Hyperelliptic Curve D.Gantor Mathematics of Comptuation Vol.48.No.177 (Jan., 1987).

This entry was posted on Wednesday, April 18th, 2007 at 9:45 am and is filed under Algebraic Geometry, Group Theory, Number Theory, PhD.