Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 058, 12 pages      arXiv:1801.08529

Fuchsian Equations with Three Non-Apparent Singularities

Alexandre Eremenko a and Vitaly Tarasov bc
a) Purdue University, West Lafayette, IN 47907, USA
b) Indiana University - Purdue University Indianapolis, Indianapolis, IN 46202, USA
c) St. Petersburg Branch of Steklov Mathematical Institute, St. Petersburg, 191023, Russia

Received February 02, 2018, in final form June 10, 2018; Published online June 15, 2018

We show that for every second order Fuchsian linear differential equation $E$ with $n$ singularities of which $n-3$ are apparent there exists a hypergeometric equation $H$ and a linear differential operator with polynomial coefficients which maps the space of solutions of $H$ into the space of solutions of $E$. This map is surjective for generic parameters. This justifies one statement of Klein (1905). We also count the number of such equations $E$ with prescribed singularities and exponents. We apply these results to the description of conformal metrics of curvature $1$ on the punctured sphere with conic singularities, all but three of them having integer angles.

Key words: Fuchsian equations; hypergeometric equation; difference equations; apparent singularities; bispectral duality; positive curvature; conic singularities.

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