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

Abstract

For any elliptic curve $E$ over $k\subset \Bbb R$ with $E({\Bbb C})={\Bbb C}^\times/q^{\Bbb Z}$, $q=e^{2\pi iz}, \Im(z)>0$, we study the $q$-average $D_{0,q}$, defined on $E({\Bbb C})$, 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 $R \in {\Bbb Q}(X_1(N))$ such that for any non-cuspidal real point $s\in X_1(N)$ (which defines an elliptic curve $E(s)$ over $\Bbb R$ 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 $\Bbb Q$-rational point of $X_1(N)$, a rare occurrence according to Mazur, $R(s)$ is a rational number.