Eisenstein series of weight one, $q$-averages of the $0$-logarithm and periods of elliptic curves

Daniel R. Grayson, Dinakar Ramakrishnan

For any elliptic curve $E$ over $k\subset \mathbb{R}$ with $E(\mathbb{C})={\mathbb{C}}^{\times }/q^{\mathbb{Z}}$, $q=e^{2\pi iz},\Im (z)>0$, we study the $q$-average $D_{0,q}$, defined on $E({\Bbb C})…