## Diophantine approximations for a constant related to elliptic functions.(English)Zbl 1069.11031

Let $$\omega(\lambda)$$ and $$\eta(\lambda)$$ be the real period and the real quasi-period of the elliptic curve $$y^2=x(x-1)(x-\lambda)$$, respectively, and let $$G(\lambda)=\eta(\lambda)/\omega(\lambda)$$. By $$\mathbb{Q}_p$$ denote the $$p$$-adic completion of $$\mathbb{Q}$$, where $$p\in \{\infty$$, primes $$p\}$$, in particular $$\mathbb{Q}_\infty=\mathbb{R}$$. For an irrational number $$\theta\in\mathbb{Q}_p$$ define $$m_p (\theta)=\inf\{\mid$$ For any $$\varepsilon>0$$ there is $$H_0=H_0(\varepsilon)$$ such that $$|\theta-P/Q|_p>H^{-m-\varepsilon}$$ for any rational $$P/Q$$ with $$(|P|, |Q|)\geq H_0\}$$ which is called the irrationality measure of $$\theta$$. Moreover, for a rational number $$\lambda$$ with $$|\lambda|_p=p^{-v}$$ let $$v_p(\lambda)=v$$. For $$\alpha\in\mathbb{Z}$$ put $$u(\alpha)=2^{v_2 (\alpha)}$$ (if $$v_2(\alpha)<4)$$, or $$2^4$$ (if $$v_2(\alpha)\geq 4)$$. The authors prove the following two theorems:
Theorem 1. Let $$\lambda= \alpha/\beta$$, $$\alpha,\beta\in\mathbb{Z}$$, $$\beta>0$$, $$(\alpha,\beta)=1$$ and $$0<|\lambda|<1$$. If $$\beta(e(1-\sqrt{1-\lambda}))^2/u(\alpha) <1$$, then $$G(\lambda)$$ is irrational and $m_\infty\bigl(G (\lambda) \bigr)\leq 1-\left(\log\left(\beta\left( 2e\left(1+|\lambda|/4\right) \right)^2/u(\alpha)\right)\right)/ \left(\log\left(\beta\left(e\left(1-\sqrt{1-\lambda)}\right)^2/u(\alpha) \right)\right)\right).$ Theorem 2. Let $$p$$ be a prime, and $$\lambda=\alpha/\beta$$, $$\alpha,\beta\in\mathbb{Z}$$, $$\beta >0$$, $$(\alpha, \beta)=1$$, $$p\not\mid\beta$$. Denote $$R=|\alpha^2/ (16^2 u(\alpha))|_p$$ and put $$Q=4e^2\beta(1+ |\alpha|/(4\beta))^2/u (\alpha)$$ (if $$|\lambda|<1)$$, or $$16e^2| \alpha|/u(\alpha)$$ (if $$|\lambda|>1)$$. If $$_2F_1(\left.\left(\begin{smallmatrix} 1/2, 1/2\\ 1 \end{smallmatrix} \right|\lambda\right)\neq 0$$ (here $$_2F_1\left.\left( \begin{smallmatrix} a,b\\ c\end{smallmatrix}\right| x\right)$$ is the Gauss hypergeometric function) and $$R<1$$, $$QR<1$$, then $$G(\lambda)$$ is irrational and $m_p\bigl(G(\lambda)\bigr)\leq 1-(\log Q)/(\log QR).$ In particular, $m_p \bigl(G_p(p^h)\bigr)\leq (2h\log p)/(h\log p-2-4\log 2)$ for all $$p\neq 2$$ with $$p^h>16e^2$$.
The proof of theorems depends on a monodromy principle for hypergeometric function and certain results of K. Alladi and M. L. Robinson [J. Reine Angew. Math. 318, 137–155 (1980; Zbl 0425.10039)] and A. Heimonen, T. Matala-Aho and K. Väänänen [Manuscr. Math. 81, 183–202 (1993; Zbl 0801.11032)].

### MSC:

 11J89 Transcendence theory of elliptic and abelian functions 11J82 Measures of irrationality and of transcendence 33C05 Classical hypergeometric functions, $${}_2F_1$$ 33C75 Elliptic integrals as hypergeometric functions 11J91 Transcendence theory of other special functions

### Citations:

Zbl 0425.10039; Zbl 0801.11032
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