The uniformization of certain algebraic hypergeometric functions

Abstract

The hypergeometric functions n F n−1 are higher transcendental functions, but for certain parameter values they become algebraic, because the monodromy of the defining hypergeometric differential equation becomes finite. It is shown that many algebraic n F n−1 's, for which the finite monodromy is irreducible but imprimitive, can be represented as combinations of certain explicitly algebraic functions of a single variable; namely, the roots of trinomials. This generalizes a result of Birkeland, and is derived as a corollary of a family of binomial coefficient identities that is of independent interest. Any tuple of roots of a trinomial traces out a projective algebraic curve, and it is also determined when this so-called Schwarz curve is of genus zero and can be rationally parametrized. Any such parametrization yields a hypergeometric identity that explicitly uniformizes a family of algebraic n F n−1 's. Many examples of such uniformizations are worked out explicitly. Even when the governing Schwarz curve is of positive genus, it is shown how it is sometimes possible to construct explicit single-valued or multivalued parametrizations of individual algebraic n F n−1 's, by parametrizing a quotiented Schwarz curve. The parametrization requires computations in rings of symmetric polynomials.

Modular forms, Schwarzian conditions, and symmetries of differential equations in physics

“…This representation of modular forms in terms of hypergeometric functions with many pullbacks, is well described in Maier's papers [57,58]. It is different from the "mainstream" mathematical definition of modular forms as (complex) analytic functions on the upper half-plane satisfying functional equations with respect to the group action of the modular group.…”

“…Recalling (57), (58), (59), and specifying the Heun identity (48), or (52), for M = 2, M = −1, and M = 1/2 respectively, one gets three identities on the hypergeometric function 2 F 1 ([1/4, 3/4], [5/4], x). These three identities are in fact consequences of the simple identity:…”

“…Remark: For these selected values of M , one could be surprised that the function Q(x) in the case of the Heun function is such that Q(x) = Y (x) 2 , when the Q(x) for the hypergeometric function (61) closely related to this Heun function (see identities (57), (58), (59)) is such that Q(x) = Y (x) 4 . This difference comes from the pullbacks (65): the pullbacked hypergeometric functions (57), (58), (59) also correspond to…”

Schwarzian conditions for linear differential operators with selected differential Galois groups

“…Condition (94) is a result whose scope transcends the hypergeometric functions framework. In order to show this, let us apply this result on the open problem of finding Heun functions 30 that could be modular forms [38], or pullbacked 2 F 1 functions [16,50]. The Heun function HeunG (a, q, α, β, γ, δ, x) is solution of a linear differential operator of order two…”

“…is a much more general problem corresponding to massive calculations even if one restricts to operators that are homomorphic to their adjoint (thus corresponding to selected, orthogonal or symplectic, differential Galois groups) 50 . Performing such calculations will require new tools and ideas.…”