THE ELLIPTIC PI

THE ELLIPTIC PI


 I hereby report a certain number of significant interest that I have recently found in the course of my investigations in the theory of elliptic functions and theta functions.

  It is well-known that 

sin(u+2π) = sin u ,  cos(u+2π) = cos u ;

sn(u+4K, τ) = sn(u, τ) ,  cn(u+4K, τ) = cn(u, τ) ; 

and  lim  sn(u, τ) = sin u ,  lim  cn(u, τ) = cos u  as q approaches 0, where q:=e(iπτ), Im τ > 0.

 It can be shown that lim 2K = π as q approaches 0.

Therefore 2K can be regarded as the 'elliptic pi'. It is known that K= (1/2).π.θ(3)^2, where θ(3) is the special value of the third Jacobi theta function θ(3)(z,τ) at z=0. Hence we have the elliptic pi

2K(τ) = π.θ(3)^2, 

where we have written K(τ) for K. It is well-known that 

π = 3.14159265359 (approximately).

 I have recently calculated the following object:

The Gaussian elliptic pi  2K(i) = 3.70814935458 (approximately).    

Question: Is this transcendental ?

 As τ ranges over the upper half plane, 2K(τ) takes various values. For τ=i, the lattice is that of the Gaussian integers. This explains the name 'Gaussian elliptic pi'.
   
  There is a class of elliptic pi's, each of them corresponding to a specific value of  the modular variable τ in the upper half plane.

Question: Are they all transcendental ?

Somjit Datta, Ph.D

July 8, 2018.
Calcutta, India