Computing period matrices and the Abel-Jacobi map of superelliptic curves

Explicit arithmetic intersection theory and computation of Néron-Tate heights

“…This code makes it possible to numerically approximate these objects efficiently to any desired precision. If C is superelliptic, then we instead use Neurohr's implementation of the specialised algorithms of Molin-Neurohr [MN19] (https: //github.com/pascalmolin/hcperiods). The code requires as input a (possibly singular) plane model of C; this is easy to produce in practice, for instance via projection or by computing a primitive element of the function field of C.…”

“…The archimedean local Néron pairings D, E ∞ are computed in essentially the same way as in in [Hol12] and [Mül14], by pulling back a translate of the Riemann theta function to C(C). This requires explicitly computing period matrices and Abel-Jacobi maps on Riemann surfaces; we use the recent algorithms of Neurohr [Neu18, Chapter 4] and Molin-Neurohr [MN19].…”

Numerical computation of endomorphism rings of Jacobians

“…In practice it is limited to about 2000 digits. Recent work by Molin-Neurohr [23] can reach higher accuracy and also applies to superelliptic curves.…”

“…This is no coincidence, since the first author suggested its use to Van Wamelen at the time, while sharing an office in Sydney, and was eager to see its use tested for general curves. Dealing with hyperelliptic and superelliptic curves, [28] and [23] use a shortcut in determining homotopy generators. The explicit use of a graph cycle basis in Step 1 above, while directly suggested by basic topological arguments, is to our knowledge new for an implementation in arbitrary precision.…”

Numerical Evaluation of Elliptic Functions, Elliptic Integrals and Modular Forms

“…We will, in particular, focus on arbitrary-precision arithmetic and omit techniques that only matter in machine precision. A good overview and a comprehensive bibliography can be found in chapters 19,20,22 and 23 of the NIST Handbook of Mathematical Functions [23] or its online counterpart, the Digital Library of Mathematical Functions 3 . Cohen's book on computational number theory [5] is also a useful resource.…”