Modular correspondences and replicable functions (unabridged version)

TL;DR

This work introduces Schwarzian differential equations as a unifying framework for modular correspondences and replicable-like structures arising in physics and mathematics. It shows that one-parameter solution series $y(a,x)$ often reproduce modular correspondences at $N$-th roots of unity, while more general instances extend beyond elliptic curves to broader automorphic contexts via mirror and nome maps. The paper constructs explicit modular equations for $q\to q^N$ (e.g., $N=2,3,4,5$), derives associated linear differential operators, and analyzes both algebraic and differential-algebraic series, including epsilon- and alpha-parameter extensions that yield two-parameter families and Heun-type examples. It also provides polynomial and truncation-based examples to illustrate how modular-like behavior can arise in non-classical settings, including Heun functions and beyond, with implications for physics (Ising/Baxter models) and combinatorics. Overall, the results suggest a rich landscape of differential-algebraic series governed by Schwarzian conditions, offering a path to generalizing modular correspondences and exploring replicable-like structures beyond traditional modular forms.

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.