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 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.
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