Special functions Hypergeometric functions Theorem

Schwarz's list

Builds on Hypergeometric differential equation Monodromy of the hypergeometric equation Schwarz triangle function Platonic solid

Statement

Let $a, b, c$ be real, and assume that the monodromy of the hypergeometric equation with these parameters is irreducible. Let $\lambda = \abs{1 - c}$ , $\mu = \abs{c - a - b}$ and $\nu = \abs{a - b}$ be the exponent differences at $0$ , $1$ and $\infty$ . Then all solutions of the hypergeometric differential equation are algebraic functions of $z$ if and only if the triple $(\lambda, \mu, \nu)$ belongs to Schwarz's list. Here the triple is taken up to permutation, after a standard normalization by sign changes and integer shifts. This list consists of one infinite family, $(\frac12, \frac12, \nu)$ with $\nu$ rational, and $14$ further triples. Of these, $2$ have tetrahedral, $2$ octahedral and $10$ icosahedral monodromy. Among them are $(\frac12, \frac13, \frac13)$ , $(\frac12, \frac13, \frac14)$ and $(\frac12, \frac13, \frac15)$ .

Remarks

All solutions are algebraic exactly when the monodromy group is finite. The ratio of two solutions, the Schwarz triangle function, maps the upper half-plane onto a circular-arc triangle with angles $\pi\lambda$ , $\pi\mu$ , $\pi\nu$ . The monodromy group modulo scalars is generated by products of reflections in its sides. Finite monodromy forces the triangle to be spherical and the projective monodromy group to be a finite subgroup of $\mathrm{PSL}_2(\C)$ : cyclic, dihedral, or the rotation group of a Platonic solid. The cyclic case is excluded by irreducibility, the dihedral case gives the infinite family, and the tetrahedral, octahedral and icosahedral groups give the remaining entries.

Example: $(\frac12, \frac13, \frac14)$ gives a spherical triangle with angles $\frac{\pi}{2}, \frac{\pi}{3}, \frac{\pi}{4}$ , whose reflections tile the sphere in $48$ triangles, matching the symmetries of the octahedron.

A necessary condition, for the normalized triple, is $\lambda + \mu + \nu > 1$ , which says that the triangle is spherical. Irreducibility fails exactly when one of $a$ , $b$ , $c - a$ , $c - b$ is an integer. That case is treated separately, since then one solution is elementary and the other is an integral of it.

Going deeper

For a student

The proof combines the monodromy of the hypergeometric equation with the geometry of reflection groups on the sphere. It is the first instance of a general question: when does a linear differential equation have a full set of algebraic solutions? Beukers and Heckman answered it for the higher generalized hypergeometric equations. The same triangles, when $\lambda + \mu + \nu < 1$ , lead instead to hyperbolic triangle groups and automorphic functions.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Hypergeometric differential equation concept
Monodromy of the hypergeometric equation concept
Platonic solid concept
Schwarz triangle function concept
Schwarz's list statement

Tag	`W0MJ51242N`
URI	`m2://W0MJ51242N`
Kind	statement, theorem
Writing	W0 stub
MSC	33C05, 34M35, 30F35
Status	This copy of the site has no build data. Run `m2 status W0MJ51242N`.
About	Hypergeometric differential equation, Monodromy of the hypergeometric equation
Uses	Hypergeometric differential equation, Monodromy of the hypergeometric equation, Schwarz triangle function, Platonic solid
Same as	Wikidata Q7433183 Wikipedia Schwarz's list