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

Abstract

For any elliptic curve E over k⊂ ℝ with E(ℂ) = ℂ×/qℤ, q= e2πiz, Im (z) > 0, we study the q-average D0,q, defined on E(ℂ), of the function D0(z) = Im (z/ (1 - z)). Let Ω +(E) denote the real period of E. We show that there is a rational function R∈ ℚ(X1(N)) such that for any non-cuspidal real point s∈ X1(N) (which defines an elliptic curve E(s) over ℝ together with a point P(s) of order N), πD0,q(P(s)) equals Ω +(E(s)) R(s). In particular, if s is ℚ-rational point of X1(N), a rare occurrence according to Mazur, R(s) is a rational number.