Arithmetic of Ramanujan's continued fractions and Hypergeometric series

Abstract

In this thesis we study three topics. First we treat certain Ramanujan`s continued fractions. One of the famous continued fractions which were studied by Ramanujan is the Rogers-Ramanujan continued fraction $R(\tau)$ . Through the works of Gee and Honesbeek([15]), Duke([13]), Cais and Conrad([2]), we see that there are some interesting facts about modularity of $R(\tau)$ , its modular equations and application to the construction of ray class fields. So we will investigate these topics with other Ramanujan`s continued fractions such as $v(\tau)$ and $C(\tau)$ . Second we will treat the growth of the coefficients of the modular equations for a modular function. P. Cohen first found some growth condition of the coefficients of modular equation for $j(\tau)$ ([7]) and Cais and Conrad showed that the ratio of the logarithmic heights of $j(\tau)$ and $j_5 (\tau)$ , which is the Hauptmodul of $\Gamma(5)$ , goes to the group index $[\overline {\Gamma(1)}: \overline {\Gamma(5)}]$ as n approaches $\infty$ . And we extend it to the case of somewhat general Hauptmoduln. Finally we introduce some identities of basic hypergeometric series and prove them.