Algebraic results for certain values of the Jacobi theta-constant θ3(τ)

Authors

  • Carsten Elsner
  • Yohei Tachiya

DOI:

https://doi.org/10.7146/math.scand.a-105465

Abstract

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+2ν=1eπiν2τ 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 n2 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 m1. As an application of our main theorem, the algebraic dependence over Q of the three values θ3(τ), θ3(mτ), and θ3(nτ) for integers ,m,n1 is also presented.

References

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Published

2018-09-05

How to Cite

Elsner, C., & Tachiya, Y. (2018). Algebraic results for certain values of the Jacobi theta-constant θ3(τ). MATHEMATICA SCANDINAVICA, 123(2), 249–272. https://doi.org/10.7146/math.scand.a-105465

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