Algebraic hypergeometric transformations of modular origin

Abstract

Abstract. It is shown that Ramanujan's cubic transformation of the Gauss hypergeometric function 2 F 1 arises from a relation between modular curves, namely the covering of X 0 (3) by X 0 (9). In general, when 2 N 7, the N -fold cover of X 0 (N ) by X 0 (N 2 ) gives rise to an algebraic hypergeometric transformation. The N = 2, 3, 4 transformations are arithmetic-geometric mean iterations, but the N = 5, 6, 7 transformations are new. In the final two cases the change of variables is not parametrized by rational functions, since X 0 (6), X 0 (7) are of genus 1. Since their quotients X + 0 (6), X + 0 (7) under the Fricke involution (an Atkin-Lehner involution) are of genus 0, the parametrization is by two-valued algebraic functions. The resulting hypergeometric transformations are closely related to the two-valued modular equations of Fricke and H. Cohn.

Functional equations for Mahler measures of genus-one curves

“…We point out that Maier obtained several results along these lines in [Maier 2006]. The functional equation for φ(t) (after substituting z = k/(1 − k 2 )) implies a new hypergeometric transformation:…”

On a class of new hypergeometric transformations

“…Algebraic transformations of hypergeometric functions of modular origin, are related to the monodromy of the underlying linear differential equations. This piece of research was started by the seminal contribution of Goursat, see [6] as the starting point, we highlight [7,8,9] for some recent developments for this kind of approach. In [10] the theory of Theta functions is used to derive hypergeometric transformation formulae , while [11] presents an approach referring to functional identities inspired by the work of [12].…”

Hypergeometric transformation formulas of degrees 3, 7, 11 and 23

“…The transformation formulas (1.2), (1.3) and (1.4) have been used to produce iterations that are analogues of the AGM in [7][8][9][10]. Uniform proofs of all of (1.1)-(1.4) have since been given in [15] and [23].…”

“…Hypergeometric transformation formulas of degrees 5, 6 and 7 were given by R. Maier [23]. Other hypergeometric transformation formulas of degree 5 appear in [16,Section 4] and [20,Theorem 6.5], a different formula of degree 7 is given in [20,Theorem 4.5], and a hypergeometric transformation formula of degree 13 is given in [19,Theorem 5.10].…”