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.