arXiv 1603.045282016-03-15math.NT

θ常数值II的代数独立性结果

Algebraic independence results for values of Theta-constants, II

Carsten Elsner, Yohei Tachiya

备注：: 15 pages

摘要：

令$θ_3(τ)=1+2\sum_{ν=1}^{\infty} q^{ν^2}$和$q=e^{iπτ}$表示Jacobi θ函数\[θ(z|τ) \,=\,\sum_{ν=-\infty}^{\infty} e^{πiν^2τ+ 2πiνz} \,.\]的θ nullwert。此外，令$θ_2(τ)=2\sum_{ν=0}^{\infty} q^{{(ν+1/2)}^2}$和$θ_4(τ)=1+2\sum_{ν=1}^{\infty} {(-1)}^νq^{ν^2}$。对代数数$q$和$0<|q|<1$以及对任意$j\in \{ 2,3,4\}$，我们证明了数$θ_j(nτ)$和$θ_j(τ)$在$\mathbb{Q}$上的代数独立性，其中$n\geq 3$是所有奇整数.假设$q$和$τ$上的条件与上述相同，我们利用一个包含结式的准则得到了$θ_j(2mτ)$和$θ_j(τ)$ $(j=2,3,4)$以及$θ_3(4mτ)$和$θ_3(τ)$与奇数$m$在$\mathbb{Q}$上代数独立的充分条件.特别地，我们证明了$θ_3(nτ)$和$θ_3(τ)$对偶数$n$和$2\leq n\leq 22$的代数独立性。本文继续工作的第一个提到的作者，谁已经证明了代数独立的$θ_3(2^mτ)$和$θ_3(τ)$为$m=1,2,\dots$。

英文摘要：

Let $θ_3(τ)=1+2\sum_{ν=1}^{\infty} q^{ν^2}$ with $q=e^{iπτ}$ denote the Thetanullwert of the Jacobi theta function \[θ(z|τ) \,=\,\sum_{ν=-\infty}^{\infty} e^{πiν^2τ+ 2πiνz} \,.\] Moreover, let $θ_2(τ)=2\sum_{ν=0}^{\infty} q^{{(ν+1/2)}^2}$ and $θ_4(τ)=1+2\sum_{ν=1}^{\infty} {(-1)}^νq^{ν^2}$. For algebraic numbers $q$ with $0<|q|<1$ and for any $j\in \{ 2,3,4\}$ we prove the algebraic independence over $\mathbb{Q}$ of the numbers $θ_j(nτ)$ and $θ_j(τ)$ for all odd integers $n\geq 3$. Assuming the same conditions on $q$ and $τ$ as above, we obtain sufficient conditions by use of a criterion involving resultants in order to decide on the algebraic independence over $\mathbb{Q}$ of $θ_j(2mτ)$ and $θ_j(τ)$ $(j=2,3,4)$ and of $θ_3(4mτ)$ and $θ_3(τ)$ with odd positive integers $m$. In particular, we prove the algebraic independence of $θ_3(nτ)$ and $θ_3(τ)$ for even integers $n$ with $2\leq n\leq 22$. The paper continues the work of the first-mentioned author, who already proved the algebraic independence of $θ_3(2^mτ)$ and $θ_3(τ)$ for $m=1,2,\dots$.

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摘要：

英文摘要：

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