Resumen de Algebraic results for certain values of the Jacobi theta-constant θ3(τ)
Carsten Elsner, Yohei Tachiya
In its most elaborate form, the Jacobi theta function is defined for two complex variables z and τ by θ(z|τ)=∑∞ν=−∞eπiν2τ+2πiνz, which converges for all complex number z, and τ in the upper half-plane. The special case θ3(τ):=θ(0|τ)=1+2∑ν=1∞eπiν2τ is called a Jacobi theta-constant or Thetanullwert of the Jacobi theta function θ(z|τ). In this paper, we prove the algebraic independence results for the values of the Jacobi theta-constant θ3(τ). For example, the three values θ3(τ), θ3(nτ), and Dθ3(τ) are algebraically independent over Q for any τ such that q=eπiτ is an algebraic number, where n≥2 is an integer and D:=(πi)−1d/dτ is a differential operator. This generalizes a result of the first author, who proved the algebraic independence of the two values θ3(τ) and θ3(2mτ) for m≥1. As an application of our main theorem, the algebraic dependence over Q of the three values θ3(ℓτ), θ3(mτ), and θ3(nτ) for integers ℓ,m,n≥1 is also presented.
