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 $E({\mathbb {C}})={\mathbb {C}}^\times /q^{{\mathbb {Z}}}$, $q=e^{2\pi iz}, \mathrm{Im}(z)>0$, we study the q-average $D_{0,q}$, defined on $E({\mathbb {C}})$, of the function $D_0(z) = \mathrm{Im}(z/(1-z))$. 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.

To Kumar Murty: Hyapi aravai

D. R. Grayson and D. Ramakrishnan Research supported by the NSF; D. Ramakrishnan supported by a Simons Fellowship