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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 $\\Omega^+(E(s))R(s)$. 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Grayson, Dinakar Ramakrishnan","submitted_at":"2018-06-13T10:12:18Z","abstract_excerpt":"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 $\\Omega^+(E(s))R(s)$. 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