Period Matrix Functions

Since the period matrices that are computed for Riemann surfaces describe the analytic Jacobian of the associated algebraic curve, several functions from Section Analytic Jacobians of Hyperelliptic Curves "Analytic Jacobians of Hyperelliptic Curves" can be applied as well. In particular, if τ1, τ2 denote small and Ω1, Ω2 denote big period matrices corresponding to genus g Riemann surfaces, the following intrinsics apply:

(i)
AnalyticHomomorphism (τ1, τ2)
(ii)
IsIsomorphicSmallPeriodMatrices (τ1, τ2)
(iii)
IsIsomorphicBigPeriodMatrices (Ω1, Ω2)
(iv)
IsIsogenousPeriodMatrices (Ω1, Ω2)

Moreover, the endomorphism ring of a big period matrix Ω can be computed by using the intrinsic

(v)
EndomorphismRing(Ω)

Finally, the evaluation of the multidimensional theta function associated to a small period matrix τ with characteristic p (a (2g x 1)-matrix) and at a z (a (g x 1)-matrix) is performed by the intrinsic:

(vi)
Theta(p, z, τ)

Example RieSrf_iso-small-pm-1 (H124E16)

For a hyperelliptic curve we may obtain period matrices in three different ways now: 
> C<I> := ComplexField(100);
> Cx<x> := PolynomialRing(C);
> f := BernoulliPolynomial(10);
> A := AnalyticJacobian(Evaluate(f, x));
> X := RiemannSurface(f, 2 : Precision := 100);
> Qxy<x,y> := PolynomialRing(Rationals(), 2);
> Y := RiemannSurface(y^2 - Evaluate(f, x) : Precision := 100);
> IsIsomorphicSmallPeriodMatrices(SmallPeriodMatrix(A), SmallPeriodMatrix(Y));
true
[ 0  0  0  0  0  0  1 -1]
[ 0  1 -1 -1  0  1 -1  0]
[ 0 -1 -1 -1  0  0  0  0]
[-1 -1 -1 -1 -1  0  0  0]
[ 0  0  0  1  0  0  0  0]
[ 0  0  1  1  0  0  0  0]
[ 0  0  1  1  1 -1  0  0]
[ 1  1  1  1  0  0  0  0]
> IsIsomorphicSmallPeriodMatrices(SmallPeriodMatrix(X), SmallPeriodMatrix(Y));
true
[ 1 -1  0  0  0 -1  0  0]
[ 1  0  1  1  0  0  1  1]
[ 1  0  0  0  0  0  1 -1]
[ 0  0  1  0  1  1  0  0]
[-1  1  0  0  0  0  0  0]
[-1  0 -1 -1  0  0 -1  0]
[-1  0 -1  0  0  0 -1  0]
[-1  0  0  0  0  0  0  0]
V2.28, 13 July 2023