Arithmetic over the Gaussian Number Field on a Certain Family of Elliptic Curves with Complex Multiplication

Abstract: This work is a sequel of a previous work of one of the authors (Y.^O), which treated certain congruence relation between an elliptic Gauss sum and a coefficient of power series expansion at the origin of the lemniscate sine function. We extend the previous result (in \cite{O}) which concerned only for non-vanishing elliptic Gauss sums. We give new congruence relations between power series coefficients of the lemniscate cosine function, which hold if and only if the corresponding elliptic Gauss sum vanishes.