Values of a function of Picard at algebraic points
The algebraicity of certain values of the Gauss and Appell hypergeometric functions has been linked by the works of Cohen, Shiga, Wolfart and Wüstholz to arithmetic properties of certain abelian varieties. The geometric tools introduced in these works allow the author to prove the following result: Theorem 1. For every \(a,b,c\in \mathbb{Q}\), \(0<c<1\), \(0<a<c\), \(0<b<c\), and every \(x\in\overline{\mathbb{Q}}\setminus\{0,1\}\), if the value \(F(a,b,c;x)\) of the Gauss hypergeometric function is algebraic and non-zero, then the value of \(F(b+1-c,a+c-1,2-c;x)\) is transcendent or zero. He provides certain nice applications of this results; for instance, he shows that \(F\bigl(5\frac5{12},\frac5{12},\frac43; j(\tau)\bigr)\) is transcendent for every \(\tau\in\mathbb{Q}(\sqrt{-3})\) such that \(| j(\tau)| <1 \), where \(j(\tau)\) is the Dedekind \(j\)-function. It is also shown that \(2^{3\alpha}\cos(\pi \alpha)\) and \(\pi^{-1/2}\frac{\Gamma(\alpha)^2}{\Gamma(2\alpha+\frac12)}\) are transcendent for every \(\alpha\in\mathbb{Q}\,\cap\,]0;\frac14[\). The comparison of the results for the Gauss and Appell hypergeometric functions leads the author to consider the \textit{Picard function}: \[ \Phi_{\lambda}(a,b,b',c;y)= \frac{F_1(a,b,b',c;\lambda,y)}{F(a,b,c;\lambda)}, \] for which he provides the description of an exceptional set in terms of the abelian varieties \(T(\lambda,y)\) associated by Cohen and Wolfart to the Appell function \(F_1(a,b,b',c;\lambda,y)\). In the last part of the paper, the author links, by means of the weak André-Oort conjecture for curves (proven by \textit{B. Edixhoven} and \textit{A. Yafaev} [Ann. Math. (2) 157, No. 2, 621--645 (2003; Zbl 1053.14023)]), the finiteness of the exceptional set of the Gauss hypergeometric function \(F(a,b,c;\lambda)\) with the property of being of Hodge type of the curve \(C_\lambda\) described by the isomorphism classes of the the abelian varieties \(T(\lambda,y)\) as points on a certain Shimura variety.
scientific article; zbMATH DE number 5161937
scientific article