Fuchsian equations with unitary monodromy

Definitions and mathematical statement

Prescribe $n\geq4$ distinct points on the Riemann sphere and real exponent differences $\alpha_1,\ldots,\alpha_n$. After a change of coordinate put the last point at infinity and write the others as $a_1,\ldots,a_{n-1}\in\mathbb C$. In projective normal form the second-order equation is \[ y''+Q(z)y=0,\qquad Q(z)=\sum_{j=1}^{n-1} \left(\frac{1-\alpha_j^2}{4(z-a_j)^2} +\frac{\beta_j}{z-a_j}\right). \] The accessory parameters $\beta_j$ satisfy \[ \sum_j\beta_j=0,\qquad \sum_j\left(\frac{1-\alpha_j^2}{4}+a_j\beta_j\right) =\frac{1-\alpha_n^2}{4}. \] These conditions impose the prescribed regular-singular behavior at infinity, leaving an affine parameter space of dimension $n-3$. The exponent difference is the difference of the two local indicial roots; its sign does not affect this normal form.

Analytic continuation of the ratio of two independent local solutions around loops avoiding the marked points acts by Möbius transformations. This is the projective monodromy group, defined up to conjugacy. Let $E$ be the accessory parameters for which this group is conjugate into $\operatorname{PSU}(2)=\operatorname{SU}(2)/\{\pm I\}$, the rotation group of the round sphere.

For every prescribed configuration and real exponent data, is $E$ discrete in the accessory parameter space? Is it finite? Discreteness means every point of $E$ is isolated; finiteness also excludes infinitely many isolated parameters escaping to infinity. The source establishes finiteness for $n=4$; the general remaining question concerns $n\geq5$.

Short English statement

For fixed singularities and exponent differences, are there only finitely many Fuchsian equations with spherical unitary monodromy?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 3700001 (AMR-036-0001), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Eremenko - My favorite unsolved problems, Fuchsian equations, two questions. Original source indexed by AMR-036-0001. https://www.math.purdue.edu/~eremenko/uns1.html.
• S3 A. Eremenko, Fuchsian equations with unitary monodromy. Author-maintained primary problem note. https://www.math.purdue.edu/~eremenko/dvi/fuchsian.pdf.