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)].


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
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