On Hecke groups, Schwarzian triangle functions and a class of hyper-elliptic functions

Abstract

Let m be a positive integer ≥3 and λ = 2 cos π m . The Hecke group G(λ) is generated by the fractional linear transformations τ + λ and − 1 τ for τ in the upper half plane H of the complex plane C. We consider a set of functions f 0 , f i and f ∞ automorphic with respect to G(λ), constructed from the conformal mapping of the fundamental domain of G(λ) to the upper half plane H, and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of λ = 1 and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of π in terms of the hypergeometric series.

Quantum geometry of resurgent perturbative/nonperturbative relations

“…Another aspect of our approach is that we find a deep connection between the allorders P/NP relations (1.5) and (1.6), and Ramanujan's theory of elliptic functions with respect to alternative bases [93][94][95][96][97][98][99][100][101][102], and extensions to modular functions [103]. These number theoretic functions are also associated with topological c = 3 Landau-Ginzburg models [104,105] and certain superconformal quantum field theories [106].…”

“…This modular structure can be formulated within Ramanujan's theory of elliptic functions in alternative bases [93][94][95][96][97][98][99][100][101][102][103] and it plays an important role in the corresponding quantum theories, as we discuss below in section 4. But here we first review some number theoretic results concerning the classical theories.…”

“…In the number theory literature, these values of M (or r) for which the Hecke group is commensurable with the modular group are referred to as the "arithmetic cases" [96]. For these arithmetic cases, Jacobi's inversion formula also generalizes [94][95][96][97][98][99][100][101][102]. Physically, this means that one can invert and express the energy u, and also the classical period ω 0 (u) and classical action a 0 (u), as explicit number-theoretic functions of the modular parameter τ 0 defined in (3.46)…”

Concepts of polymer statistical topology

“…Second step consists in mapping the auxiliary upper half-plane Im ζ > 0 onto the circular triangle ABC with angles {α, α, 0} -the fundamental domain of the Hecke group [39] in r, where we are intersted in the specific case {α, α, 0} = {0, 0, 0} -see is realized by the function ζ(r), constructed as follows [40]. Let ζ(r) be the inverse function of r(ζ) written as a quotient…”

From geometric optics to plants: the eikonal equation for buckling

“…Second step consists in mapping the auxiliary upper halfplane ℑζ > 0 onto the circular triangle ABC with angles {α, α, 0} -the fundamental domain of the Hecke group 23 in r, where we are intersted in the specific case {α, α, 0} = {0, 0, 0} -see Fig.9b-c. This mapping is realized by the function ζ (r), constructed as follows 26 .…”