Modular correspondences and replicable functions (unabridged version)

TL;DR

This paper explores the nature of solutions to Schwarzian equations related to modular correspondences, revealing their complex structure beyond elliptic curves and hypergeometric functions, with implications for physics and combinatorics.

Contribution

It demonstrates that solutions to Schwarzian equations can reduce to modular correspondence series at roots of unity and introduces examples beyond the elliptic framework.

Findings

One-parameter solutions reduce to modular series at roots of unity.

Two-parameter solutions with compositional properties are constructed.

Examples show series can be non-globally bounded, beyond elliptic curves.

Abstract

Landen transformation, and more generally modular correspondences, can be seen to be exact symmetries of some integrable lattice models, like the square Ising model, or the Baxter model. They are solutions of remarkable Schwarzian equations and have some compositional properties. Most of the known examples correspond, in an elliptic curves framework, to an automorphy property of pullbacked $_2F_1$ hypergeometric functions, associated with modular forms. It is, however, important to underline that these Schwarzian equations go beyond an elliptic curves, and hypergeometric functions framework. The question of a modular correspondence interpretation of the solutions of these ``Schwarzian'' equations was clearly an open question. This paper tries to shed some light on this open question. We first shed some light on the very nature of a one-parameter series solution of the Schwarzian equation. This one-parameter series is not generically a modular correspondence series, but it actually reduces to an infinite set of modular correspondence series for an infinite set of (N-th root of unity) values of the parameter. We also provide an example of two-parameter series, with a compositional property, solution of a Schwarzian equation. We finally provide simple pedagogical examples that are very similar to modular correspondence series, but are far beyond the elliptic curves framework. These last examples show that the modular correspondence-like series, or the nome-like series, are not necessarily globally bounded. The results of that paper can be seen as an incentive to study differentially algebraic series with integer coefficients, in physics or enumeratice combinatorics.