Translating auxiliary symmetries between Schottky uniformization and Jacobi parametrization

The explicit description and computation of functions defined on Riemann surfaces of various genera depends on the choice of language: while the Jacobi parametrization is widely known and used, the Schottky uniformization has been proven to provide an alternative approach, useful in particular for (but not limited to) numerical calculations. Despite capturing the geometry of the Riemann surface completely, the two languages are subject to rather different sets of auxiliary symmetries. In this article we translate and compare the symplectic transformations inherent in the Jacobi parametrization to the freedom in choosing Möbius transformations generating the Schottky group for the Schottky uniformization. Our results are aimed at transferring functional relations expressed in the Schottky language to the Jacobi language and vice versa. An immediate application would be the efficient numerical evaluation of special functions in a physics context by favorably tuning the Schottky cover leading to quicker convergence.

💡 Research Summary

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The paper “Translating auxiliary symmetries between Schottky uniformization and Jacobi parametrization” addresses a fundamental problem in the computational theory of functions defined on compact Riemann surfaces of arbitrary genus: the same geometric data can be encoded in two very different languages, each equipped with its own set of auxiliary symmetries. The Jacobi parametrization (often called the Abel–Jacobi map) embeds the surface into its Jacobian variety and records the complex structure in a period matrix τ∈H_g, the Siegel upper half‑space. This period matrix is not unique; any symplectic transformation M∈Sp(2g,ℤ) acting by τ↦(Aτ+B)(Cτ+D)^{-1} yields a different τ that describes exactly the same surface. The authors review the three standard generators of Sp(2g,ℤ): the exchange matrix J_{2g}, the “D‑type” matrices D_A induced by GL(g,ℤ), and the “T‑type” matrices T_B built from symmetric integer matrices B.

In contrast, the Schottky uniformization describes a genus‑g surface as a quotient Ω/Γ, where Ω⊂ℂ̂ is the complement of a collection of 2g Jordan curves and Γ is a free group generated by g loxodromic Möbius transformations γ_i. The Schottky description carries three distinct sources of freedom: (i) the choice of generators (the “marking” γ), (ii) the choice of fundamental domain (the “decoration” S, a symmetric integer matrix encoding which side of each Jordan curve is taken as interior), and (iii) the possibility of swapping A‑ and B‑cycles, which is not directly realizable as a simple change of generators.

The central achievement of the paper is to map each of these Schottky freedoms onto a concrete symplectic transformation on the Jacobi side. A change of marking α (permutations, inversions, or compositions of the γ_i) corresponds to a matrix Ψ_g(α)∈Sp(2g,ℤ) such that τ(α(γ))=Ψ_g(α)·τ(γ). A change of decoration r (simultaneous change of marking and decoration) is encoded by Φ_g(r)∈Sp(2g,ℤ) with τ(r·(S,γ))≈Φ_g(r)·τ(S,γ). Finally, the exchange of A‑ and B‑cycles, represented by J_{2g}, cannot be implemented directly on the Schottky side; instead the authors introduce a “dual Schottky group” j(S,γ) that produces a new marked‑decorated pair (S′,γ′) satisfying τ(j(S,γ))=J_{2g}·τ(S,γ). This construction closes the triangle of correspondences: every symplectic operation on τ has a precise counterpart in the Schottky language, and vice versa.

To illustrate the theory, the authors work out a detailed genus‑3 example. They start from a concrete set of three Jordan‑curve pairs and associated Möbius generators γ_1,γ_2,γ_3. By applying a specific marking change α (e.g., swapping γ_1 and γ_2, inverting γ_2), they compute the associated Ψ_3(α) matrix and obtain the transformed period matrix τ′. Next, they modify the fundamental domain by moving the interior of one curve, which yields a decoration change r; the corresponding Φ_3(r) matrix is applied to τ′, producing τ″. Finally, they construct the dual Schottky group j(S,γ) to realize the J_6 transformation, arriving at τ‴=J_6·τ″. Throughout, they demonstrate how each entry of the period matrix can be expressed both as a convergent Schottky automorphic series (summing over words in the generators) and as a theta‑function or multi‑logarithm expression familiar from the Jacobi side.

The practical payoff of this translation is highlighted in the context of modern theoretical physics. Many observables—Wilson loops, scattering amplitudes, modular graph functions—are naturally expressed as iterated integrals on Riemann surfaces. Numerical evaluation of these objects often suffers from slow convergence when the underlying period matrix is poorly conditioned. By exploiting the Schottky freedom, one can “tune” the covering map so that the associated automorphic series converges rapidly, while the symplectic correspondence guarantees that the same physical quantity is recovered when re‑expressed in the Jacobi language. Thus the paper provides a systematic recipe for optimizing numerical calculations: select a marking and decoration that minimize the imaginary part of τ, compute the corresponding τ via the explicit formulas, and then evaluate the desired function using the most efficient representation.

The authors conclude with several open questions. They suggest developing algorithmic tools for automated selection of optimal Schottky data at higher genus, extending the correspondence to non‑compact surfaces or surfaces with punctures, and exploring the impact on the theory of modular forms and string perturbation theory. Overall, the work bridges a gap between classical algebraic geometry (period matrices, symplectic groups) and modern computational techniques (Schottky automorphic series), offering both conceptual insight and concrete computational benefits for physicists and mathematicians working with high‑genus Riemann surfaces.