HEXAGONbeta

A polynomial-time algorithm to compute Theta functions of Schottky groups

Primarily AI-generated textHuman understanding: some partsmath.NA — Numerical Analysismath.AG — Algebraic Geometrymath.CV — Complex Variables

Contributed by Marc Masdeu ↗

Version 1 / Nov 12, 2026 / CC BY 4.0

Abstract

Abelian integrals, holomorphic differentials, period matrices and the Schottky--Klein prime function of a Riemann surface uniformized by a classical Schottky group are given by Poincar\'e theta series, which are sums over the elements of a free group. The algorithms that evaluate these series sum a truncated, possibly adaptively pruned, portion of the Cayley tree of the group, and the number of terms needed grows exponentially with the number of correct digits. We observe that the sum over all group elements starting with a fixed generator is a holomorphic function on the complement of a disc, and that these functions satisfy an exact linear functional equation coming from the self-similarity of the Cayley tree, given by a transfer operator built from the generators. Representing them by truncated Taylor expansions turns the evaluation of the series into a finite-dimensional linear problem, which we solve iteratively. The resulting algorithm computes the integrals Ωn\Omega_n, the period matrix and the prime function to PP bits of precision; for a fixed Schottky datum the number of arithmetic operations is cubic in PP and in the genus. We prove that it converges geometrically under a standard Schottky-type hypothesis on the group, which is closely related to the absolute convergence of the series. Along the way we give a complete proof of the cross-ratio formula for the period matrix, including a term that is missing from one version of it in the literature. We compare with the existing algorithms, and illustrate the method by reproducing a published value of the prime function and computing it to 3030 digits, and by computing period matrices of genus up to six to about 2525 digits.

Provenance statement

The initial idea and first prototype of the code is original from the author, who planned to develop an complex-analytic analogue to the p-adic construction of Masdeu--Xarles. This paper has been produced after some conversations with Claude's Opus 5.5 during the course of two days. All computations have been carried out on local servers.

Tools used

SageMath
SageMathVersion 10.6
Anthropic
Claude OpusVersion 5.5
Anthropic
Claude SonnetVersion 5.5
Anthropic
Claude HaikuVersion 5.5

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Version history

  1. v1 · 24 pagesSubmitted by Marc MasdeuInitial depositCurrentNov 05, 2026