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The main goal was to put together a p-adic heights calculator for hyperelliptic curves. Before we have extensions of p-adics, we could:
- Write a tiny integrals calculator for Weierstrass residue discs.
- Expand a rational function with poles only in Weierstrass residue discs as a power series.
- Implement Besser's techniques for calculating with local indices.
After that, we could:
- Implement full Coleman integration between points defined over extension fields. (Beware: this requires a semilinear version of the fundamental linear system.)
Longer term projects could include:
- Develop a version of Harvey's algorithm for computing Coleman integrals.
- Come up with a good framework for doing iterated Coleman integrals (for use in the nonabelian Chabauty method).