/hcperiods

Period matrice and Abel-Jacobi map of hyperelliptic and superperelliptic curves

Primary LanguageCMIT LicenseMIT

hcperiods

Numerical evaluation of periods matrices of hyperelliptic and superelliptic curves to arbitrary precision by numerical integration.

Structure

  • gp

    Initial projet as gp script, period matrix + Abel-Jacobi map for hyperelliptic curves

  • magma

    Generalization for superelliptic curves with Christian Neurohr, complete Abel-Jacobi map. Written in magma.

  • arb

    Rigorous Arb implementation, currently only period matrices.

  • hcperiods.pdf description of the algorithm and proofs

How to use

See the magma intructions to use the magma package.

The arb version can be used as follows (needs arb>=v2.12.0 installed)

cd arb && make example

then

build/examples/periods -m 2 --pol 4 1 0 -2 3 0

outputs a period matrix for the curve y^2 = x^4 -2x^2 + 3x.

Use options -m 5 to switch to the curve y^5 = x^4 -2x^2 + 3x, or --prec 1024 for 1024 bits precision.

Other options:

  • --gp output fo pari/gp
  • --de force use of double exponential integration (instead of Gauss, if m = 2).
  • --big return big period matrices instead of reduced matrix tau.