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.