Eisenstein series of weight one, \(q\)-averages of the \(0\)-logarithm and periods of elliptic curves. (English) Zbl 1475.11063

Akbary, Amir (ed.) et al., Geometry, algebra, number theory, and their information technology applications. Toronto, Canada, June, 2016, and Kozhikode, India, August, 2016. Cham: Springer. Springer Proc. Math. Stat. 251, 245-266 (2018).
Summary: 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 sis \(\mathbb{Q}\)-rational point of \(X_1(N)\), a rare occurrence according to Mazur, \(R(s)\) is a rational number.
For the entire collection see [Zbl 1403.11002].


11F03 Modular and automorphic functions
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11G05 Elliptic curves over global fields
11G55 Polylogarithms and relations with \(K\)-theory
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