Schwarz's list

In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by Hermann Schwarz (1873, p. 323) when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the cases in which the hypergeometric equation has a finite monodromy group, or equivalently has two independent solutions that are algebraic functions. It lists 15 cases, divided up by the isomorphism class of the monodromy group (excluding the case of a cyclic group), and was first derived by Schwarz by methods of complex analytic geometry. Correspondingly the statement is not directly in terms of the parameters specifying the hypergeometric equation, but in terms of quantities used to describe certain spherical triangles.

The wider importance of the table, for general second-order differential equations in the complex plane, was shown by Felix Klein, who proved a result to the effect that cases of finite monodromy for such equations and regular singularities could be attributed to changes of variable (complex analytic mappings of the Riemann sphere to itself) that reduce the equation to hypergeometric form. In fact more is true: Schwarz's list underlies all second-order equations with regular singularities on compact Riemann surfaces having finite monodromy, by a pullback from the hypergeometric equation on the Riemann sphere by a complex analytic mapping, of degree computable from the equation's data.[1][2]

Number	${\displaystyle \lambda }$	${\displaystyle \mu }$	${\displaystyle \nu }$	area/${\displaystyle \pi }$	polyhedron
1	1/2	1/2	p/n (≤ 1/2)	p/n	Dihedral
2	1/2	1/3	1/3	1/6	Tetrahedral
3	2/3	1/3	1/3	2/6	Tetrahedral
4	1/2	1/3	1/4	1/12	Cube/octahedron
5	2/3	1/4	1/4	2/12	Cube/octahedron
6	1/2	1/3	1/5	1/30	Icosahedron/Dodecahedron
7	2/5	1/3	1/3	2/30	Icosahedron/Dodecahedron
8	2/3	1/5	1/5	2/30	Icosahedron/Dodecahedron
9	1/2	2/5	1/5	3/30	Icosahedron/Dodecahedron
10	3/5	1/3	1/5	4/30	Icosahedron/Dodecahedron
11	2/5	2/5	2/5	6/30	Icosahedron/Dodecahedron
12	2/3	1/3	1/5	6/30	Icosahedron/Dodecahedron
13	4/5	1/5	1/5	6/30	Icosahedron/Dodecahedron
14	1/2	2/5	1/3	7/30	Icosahedron/Dodecahedron
15	3/5	2/5	1/3	10/30	Icosahedron/Dodecahedron

The numbers ${\displaystyle \lambda ,\mu ,\nu }$ are (up to permutations, sign changes and addition of ${\displaystyle (\ell ,m,n)\in \mathbb {Z} ^{3}}$ with ${\displaystyle \ell +m+n}$ even) the differences ${\displaystyle 1-c,c-a-b,b-a}$ of the exponents of the hypergeometric differential equation at the three singular points ${\displaystyle 0,1,\infty }$. They are rational numbers if and only if ${\displaystyle a,b}$ and ${\displaystyle c}$ are, a point that matters in arithmetic rather than geometric approaches to the theory.