Divisors on Riemann Surfaces

A rather simple concept of a divisor for a Riemann surface is supported. Here the divisors are realised as finite sums of points with integer coefficients. The type name for these Riemann surface divisors is DivRieSrfElt. The main application of these divisors is to the computation of the Abel--Jacobi map.

Divisor(S,V) : SeqEnum[RieSrfPt], SeqEnum[RngIntElt] -> DivRieSrfElt

Given a sequence S of points P_i belonging to Riemann surface X and a sequence V of integers n_i, construct the formal divisor ∑_i n_iP_i.

ZeroDivisor(X) : RieSrfElt -> DivRieSrfElt

Construct the zero divisor for the Riemann surface X.

RiemannSurface(D) : DivRieSrfElt -> RieSrf

The Riemann surface associated with the divisor D is returned.

Support(D) : DivRieSrfElt -> SeqEnum[RieSrfPt], SeqEnum[RngIntElt]

Given a divisor D= ∑_i n_iP_i with all n_i nonzero, the sequence of points P_i and the sequence of their multiplicities n_i are returned.

Degree(D) : DivRieSrfElt -> RngIntElt

The sum of the multiplicities n_i of the points P_i supporting the divisor D = ∑_I N_Ip_I is returned.

RandomDivisor(X,d) : RieSrf, RngIntElt -> RieSrfDivElt

    Ht: RngIntElt                       Default: 10^5

    Zero: BoolElt                       Default: true

Given a Riemann surface X and a positive integer d return a random divisor for X of degree d. The maximum size of the coefficients can be bounded by assigning a positive integer to the parameter Ht. If parameter Zero is set to true, the degree of the returned divisor will be zero.