Schwarz's list
Encyclopedia
In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by 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 function
s. 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 surface
s 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.
The numbers λ, μ, ν are half the differences 1 − c, c − a − b, a − b of the exponents of the hypergeometric differential equation
at the three singular points 0, 1, ∞. They are rational numbers if and only if a, b and c are, a point that matters in arithmetic rather than geometric approaches to the theory.
of the differential Galois group of the hypergeometric equation is a solvable group
. 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 quadrature
s.
Another relevant list is that of K. Takeuchi, who classified the (hyperbolic) triangle group
s that are arithmetic group
s (85 examples).
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. This work recovers in the classical case the finiteness of Takeuchi's list, and by means of a characterisation of the lattices they construct that are arithmetic groups, provided new examples of non-arithmetic lattices in PU(1, n).
Baldassari applied the Klein universality, to discuss algebraic solutions of the Lamé equation by means of the Schwarz list.
Algebraic function
In mathematics, an algebraic function is informally a function that satisfies a polynomial equation whose coefficients are themselves polynomials with rational coefficients. For example, an algebraic function in one variable x is a solution y for an equationwhere the coefficients ai are polynomial...
s. It lists 15 cases, divided up by the isomorphism class of the monodromy group (excluding the case of a cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
), 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
Felix Klein
Christian Felix Klein was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory...
, 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
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
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 surface
Riemann surface
In mathematics, particularly in complex analysis, a Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought of as "deformed versions" of the complex plane: locally near every point they look like patches of the...
s 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.
Schwarz's list
| Number | λ | μ | ν | area/π | 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 λ, μ, ν are half the differences 1 − c, c − a − b, a − b of the exponents of the hypergeometric differential equation
Hypergeometric differential equation
In mathematics, the Gaussian or ordinary hypergeometric function 2F1 is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation...
at the three singular points 0, 1, ∞. They are rational numbers if and only if a, b and c are, a point that matters in arithmetic rather than geometric approaches to the theory.
Further work
An extension of Schwarz's results was given by T. Kimura, who dealt with cases where the identity componentIdentity component
In mathematics, the identity component of a topological group G is the connected component G0 of G that contains the identity element of the group...
of the differential Galois group of the hypergeometric equation is a solvable group
Solvable group
In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
. 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
Michio Kuga
Michio Kuga is a mathematician who received his Ph.D. from University of Tokyo in 1960. His work is linked to the Ramanujan conjecture....
. By general differential Galois theory, the resulting Kimura-Schwarz table classifies cases of integrability of the equation by algebraic functions and quadrature
Quadrature
Quadrature may refer to:In signal processing:*Quadrature amplitude modulation , a modulation method of using both an carrier wave and a 'quadrature' carrier wave that is 90° out of phase with the main, or in-phase, carrier...
s.
Another relevant list is that of K. Takeuchi, who classified the (hyperbolic) triangle group
Triangle group
In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangle...
s that are arithmetic group
Arithmetic group
In mathematics, an arithmetic group in a linear algebraic group G defined over a number field K is a subgroup Γ of G that is commensurable with G, where O is the ring of integers of K. Here two subgroups A and B of a group are commensurable when their intersection has finite index in each of them...
s (85 examples).
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
Discrete group
In mathematics, a discrete group is a group G equipped with the discrete topology. With this topology G becomes a topological group. A discrete subgroup of a topological group G is a subgroup H whose relative topology is the discrete one...
in the projective unitary group
Projective unitary group
In mathematics, the projective unitary group PU is the quotient of the unitary group U by the right multiplication of its center, U, embedded as scalars....
PU(1, n). Pierre Deligne
Pierre Deligne
- See also :* Deligne conjecture* Deligne–Mumford moduli space of curves* Deligne–Mumford stacks* Deligne cohomology* Fourier–Deligne transform* Langlands–Deligne local constant- External links :...
and George Mostow
George Mostow
George Mostow is an American mathematician, a member of the National Academy of Sciences, Henry Ford II Professor of Mathematics at Yale University, the 49th President of the American Mathematical Society ,...
used his ideas to construct lattices in the projective unitary group. This work recovers in the classical case the finiteness of Takeuchi's list, and by means of a characterisation of the lattices they construct that are arithmetic groups, provided new examples of non-arithmetic lattices in PU(1, n).
Baldassari applied the Klein universality, to discuss algebraic solutions of the Lamé equation by means of the Schwarz list.