---
tag: W0MJ51242N
kind: statement
role: theorem
name: Schwarz's list
msc: ["33C05", "34M35", "30F35"]
uses: [W8VE25R5A7, T8SBE28DM1, XMD3Q49Z2C, 017E5PRS1Q]
about: [W8VE25R5A7, T8SBE28DM1]
sources:
  - url: https://en.wikipedia.org/wiki/Schwarz%27s_list
same_as:
  wikidata: Q7433183
  wikipedia: Schwarz's list
---

## Statement

Let $a, b, c$ be real, and assume that the [[monodromy of the hypergeometric equation|T8SBE28DM1]] 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|W8VE25R5A7]] 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|XMD3Q49Z2C]], 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|017E5PRS1Q]].
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.

## Explanations

### L1 human

Usually the hypergeometric function is transcendental: it cannot be defined by a polynomial equation with its variable.
Schwarz asked when it is algebraic, like $\sqrt{1 - z}$.
He found that this happens only for a short explicit list of parameter values, and the list matches the symmetry groups of the regular polyhedra.

### L2 human

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.