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

Abstract

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 $E(\mathbb{C})$, of the function $D_0(z)=\Im (z/(1-z))$. Let ${\Omega }^{+}(E)$ denote the real period of $E$. We show that there is a rational function $R\in \mathbb{Q}(X_1(N))$ such that for any non-cuspidal real point $s\in X_1(N)$ (which defines an elliptic curve $E(s)$ over $\mathbb{R}$ together with a point $P(s)$ of order $N$), $\pi D_{0,q}(P(s))$ equals ${\Omega }^{+}(E(s))R(s)$. In particular, if $s$ is $\mathbb{Q}$-rational point of $X_1(N)$, a rare occurrence according to Mazur, $R(s)$ is a rational number.