Series and Differential Identities from $\Gamma_0(5)$ and $\Gamma_1(5)$ `

Rob Maier (Arizona)
Abstract: We show how covering relations between genus-zero modular curves, such as $X(1)$, $X_0(4)$, $X_0(5)$, and $X_1(5)$, lead naturally to algebraic identities that are satisfied by certain power series. These identities can be appreciated in their own right, but are most efficiently derived as relationships between series expansions of certain modular functions and forms: either q-series or power series in Hauptmoduls (function field generators). There is another class of modular form identities that involve classical special functions, such as the Gauss hypergeometric and Heun functions. This is because the Picard-Fuchs equations satisfied by periods of the families of elliptic curves parametrized by $X(1)$, $X_0(4)$, $X_0(5)$, and $X_1(5)$ have such special functions as solutions.