Schwarz's list

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.

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.

[3][4] A general result connecting the differential Galois group G and the monodromy group Γ states that G is the Zariski closure of Γ — this theorem is attributed in the book of Matsuda to Michio Kuga.

By general differential Galois theory, the resulting Kimura-Schwarz table classifies cases of integrability of the equation by algebraic functions and quadratures.

[5] Émile Picard sought to extend the work of Schwarz in complex geometry, by means of a generalized hypergeometric function, to construct cases of equations where the monodromy was a discrete group in the projective unitary group PU(1, n).

Pierre Deligne and George Mostow used his ideas to construct lattices in the projective unitary group.

[6] Baldassari applied the Klein universality, to discuss algebraic solutions of the Lamé equation by means of the Schwarz list.

[7] Other hypergeometric functions which can be expressed algebraically, like those on Schwarz's list, arise in theoretical physics in the context of