Deformed Schwarz Equation

• The deformed Schwarz equation is a nonlinear third-order differential equation on the Poincaré upper half-plane linking the Schwarzian derivative to the holomorphic Eisenstein series E4.
• It differentiates between modular solutions, arising under specific parameter conditions, and non-modular, equivariant solutions through a detailed representation-theoretic framework.
• The equation establishes a bridge between differential equations, modular forms, and projective structures, utilizing explicit constructions via integrals of weight-2 meromorphic modular forms.

The deformed Schwarz equation refers to the nonlinear third-order differential equation on the Poincaré upper half-plane,

$\{f,\tau\} = s\,E_4(\tau),$

where $\{f,\tau\}$ is the Schwarzian derivative of a locally univalent function $f$ with respect to the variable $\tau\in\mathbb{H}$, $s\in\mathbb{C}$ is a complex parameter, and $E_4(\tau)$ is the holomorphic Eisenstein series of weight $4$ for $\mathrm{SL}_2(\mathbb{Z})$. This framework generalizes classical connections between projective structures, modular forms, and the monodromy of second-order linear differential equations and provides explicit classes of non-modular, automorphically behaving solutions when $s$ falls outside the modular regime. The modern development leverages equivariant function theory and the explicit integration of meromorphic modular forms of weight $2$ to produce non-modular solutions, as well as clarifying the rich representation-theoretic content of the equation and its associated Fuchsian ODEs (Sebbar et al., 2020, Sebbar et al., 2020).

1. Schwarzian Derivative, Eisenstein Series, and the Deformed Equation

The Schwarzian derivative for a locally univalent function $f$ is defined as

$$\{f,\tau\} = \frac{f'''(\tau)}{f'(\tau)} - \frac{1}{2} \left(\frac{f''(\tau)}{f'(\tau)}\right)^2,$$

and satisfies the cocycle rule

$$\{f\circ g,\tau\} = \{f,g(\tau)\}(g'(\tau))^2 + \{g,\tau\}.$$

It vanishes precisely for Möbius transformations, providing a finer invariant than ordinary derivatives under the Möbius group.

The Eisenstein series $E_4(\tau)$ is given by

$$E_4(\tau) = 1 + 240 \sum_{n=1}^\infty \sigma_3(n) q^n, \quad q = e^{2\pi i\tau},$$

with $\sigma_3(n) = \sum_{d\mid n} d^3$. It is a holomorphic weight-4 modular form on $\mathrm{SL}_2(\mathbb{Z})$.

The deformed Schwarz equation, $\{f,\tau\}=s\,E_4(\tau)$, thus prescribes a projective structure whose curvature is governed by a modular form, parametrized by the deformation parameter $s$.

2. Modular and Non-Modular Solutions: Representation-Theoretic Classification

Solutions to the deformed Schwarz equation bifurcate into modular and strictly non-modular (equivariant) classes depending on the value of $s$.

• Modular solutions: If the solution $f$ is itself a modular function for a finite index subgroup of $\mathrm{SL}_2(\mathbb{Z})$, it is necessary and sufficient that

$$s = 272\left(\frac{m}{n}\right)^2,\qquad 2 \leq n \leq 5,\, \gcd(m,n) = 1.$$

Here, $f$ becomes a Hauptmodul for the principal congruence subgroup $\Gamma(n)$, and the attached monodromy representation of $\mathrm{SL}_2(\mathbb{Z})$ is irreducible of finite image in $\mathrm{PGL}_2(\mathbb{C})$ (Sebbar et al., 2020).

• Non-modular, equivariant solutions: For parameters $s$ outside this set, solutions exist that are locally meromorphic on $\mathbb{H}$, but they are not invariant under any finite-index subgroup. Instead, these solutions are equivariant under reducible, typically upper-triangular, representations of the modular group.

The table below summarizes the dichotomy:

Nature of Solution	Parameter $s$	Modularity Property
Modular	$272(m/n)^2$, $2\leq n\leq 5$	Modular function
Non-modular (Equivariant)	$272(n/6)^2$, $n\equiv1\bmod12$	$\rho$-equivariant, non-modular

This structure was rigorously established by Sebbar and Saber using representation theory, equivariant function theory, and Fourier analysis (Sebbar et al., 2020, Sebbar et al., 2020).

3. Explicit Construction of Non-Modular Solutions

For the special family $s_n = \frac{272}{36} n^2$, where $n\equiv1\pmod{12}$, explicit, non-modular but automorphically equivariant solutions are constructed via integrals of weight-2 meromorphic modular forms. The procedure is as follows:

1. Weight-2 Meromorphic Forms: For each $n$, select points $w_1, ..., w_n \in \mathbb{H}$ such that

$$f_n(\tau) = \eta(\tau)^4 \prod_{i=1}^n (J(\tau) - J(w_i))^{-2}$$

is a weight-2 meromorphic modular form with double poles at the $w_i$ and zero residues.

2. Integral Representation: Define the non-modular solution by

$h_n(\tau) = \int_{i}^{\tau} f_n(z)\, dz,$

where the integration path lies in $\mathbb{H}$. The function $h_n$ is meromorphic, with simple poles only at the $w_i$.

3. Schwarzian Evaluation: It follows that

$\{h_n, \tau\} = \frac{272}{36} n^2\, E_4(\tau),$

with the automorphic but non-modular transformation property controlled by a triangular multiplier system for $\mathrm{SL}_2(\mathbb{Z})$. The construction depends on the solvability of an explicit algebraic system governing the location of the $w_i$ (Sebbar et al., 2020).

4. Connection to Fuchsian Differential Equations

If $y_1$ and $y_2$ are two linearly independent solutions to the second-order ODE

$y''(\tau) + F(\tau)\, y(\tau) = 0,$

then $f(\tau) = y_2(\tau)/y_1(\tau)$ satisfies

$\{f,\tau\} = 2 F(\tau).$

For the non-modular case,

$$\{h_n, \tau\} = 2 F(\tau),\qquad F(\tau) = \frac{136}{36}n^2 E_4(\tau),$$

and the corresponding general ODE is

$$y''(\tau) + \frac{136}{36} n^2 E_4(\tau)\, y(\tau) = 0.$$

Upon normalization with $t=2\pi i \tau$, this translates to

$$y''(t) + \frac{\pi^2 n^2}{36} E_4\left(\frac{t}{2 \pi i}\right) y(t) = 0,$$

whose explicit closed-form solutions are expressible via the construction outlined above. The basis functions

$$y_1(\tau) = \frac{1}{\sqrt{h_n'(\tau)}},\qquad y_2(\tau) = \frac{h_n(\tau)}{\sqrt{h_n'(\tau)}}$$

realize the required two-dimensional representation (Sebbar et al., 2020, Sebbar et al., 2020).

5. Equivariant Functions and Automorphic Properties

A meromorphic function $f:\mathbb{H} \to \mathbb{P}^1(\mathbb{C})$ is $\rho$-equivariant with respect to a representation $\rho:\Gamma\to\mathrm{GL}_2(\mathbb{C})$ if

$f(\gamma\cdot \tau) = \rho(\gamma) f(\tau),$

for all $\gamma\in\Gamma$, with $\gamma\cdot\tau$ the usual Möbius action. The cases of interest have $\rho$ upper-triangular but nontrivial, implying $f$ is not modular but still satisfies a finite-order functional equation. These non-modular solutions arise precisely for the reducible representation cases and are characterized by their transformation behavior under the modular group, typically with level $6$ in the multiplier system (i.e., $\Gamma(6) \subset \ker \rho$).

Despite their non-modularity, the Schwarzian derivative $\{h_n, \tau\}$ remains a genuine weight-4 modular form for $\mathrm{SL}_2(\mathbb{Z})$. The interplay between the analytic aspects (singularities, residues, Fourier expansions) of the underlying meromorphic weight-2 integrands and the automorphic characteristics of the equivariant function is central to the construction and automorphic analysis (Sebbar et al., 2020).

6. Broader Context and Deformation of the Classical Schwarzian

The deformed Schwarzian equation is a particular instance of more general algebraic and analytic deformations of the classical Schwarzian structure, as encountered in the study of the covariance of differential operators under pullbacks—central both in the theory of modular forms and various applications in mathematical physics: $v(y(x))\, [y'(x)]^2 - v(x) + \{y(x), x\} = 0$ for a given function $v$. In the modular case, $v(x)$ and $y(x)$ reflect modular equations and relations among modular forms or automorphic functions. The classical equation $\{f,x\}=0$ yields Möbius invariants; the deformed case encodes richer projective and automorphic phenomena, notably in the context of $\,_2F_1$ and Heun functions, isogenies of elliptic curves, and the theory of modular equations (Abdelaziz et al., 2016).

7. Significance and Outlook

The explicit construction and classification of solutions to the deformed Schwarz equation with modular or non-modular data elucidate the deep connections between automorphic forms, differential equations, and representation theory. The methods developed not only generalize classical results by Hurwitz and Klein but also bridge to modern aspects of equivariant function theory and symmetry analysis in physics. The analytic realization via integrals of modular forms of weight $2$ marks a substantial refinement in constructing explicit, non-modular automorphic objects. This synthesis continues to prompt further developments in both the theory of modular forms and differential Galois theory (Sebbar et al., 2020, Sebbar et al., 2020, Abdelaziz et al., 2016).