2016/09/13 by Carsten Elsner, Elsner, Carsten, Yohei Tachiya +1
Mathematics · #11F27 #11J85 #11J91 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1609.03660
openalex publication_date 2016/09/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let θ3(τ)=1+2∑ν=1∞ qν2 with q=eiπτ and \Im (τ)>0 denote the Thetanullwert of the Jacobi theta function θ(z|τ) = ∑ν=-∞∞ eπiν2τ+ 2πiνz . Moreover, let θ2(τ)=2∑ν=0∞ q^(ν+1/2)2 and θ4(τ)=1+2∑ν=1∞ (-1)νqν2. For every even integer n≥ 6, which is not a power of two, we prove constructively the existence of a nontrivial integer polynomial Qn(X,Y) such that Qn( (θ34(nτ))/(θ34(τ)),(θ24(τ))/(θ34(τ)) ) = 0 holds for all complex numbers τ from the upper half plane of ℂ. These polynomials are used to prove the algebraic independence of θ3(nτ) and θ3(τ) for all algebraic numbers q=eiπτ with 0