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

TL;DR

This paper explores arithmetic over the Gaussian number field related to elliptic curves with complex multiplication, extending previous congruence relations to include cases where elliptic Gauss sums vanish, involving power series coefficients of special functions.

Contribution

It extends prior work by establishing new congruence relations for power series coefficients linked to elliptic Gauss sums and their vanishing conditions.

Findings

New congruence relations for lemniscate cosine coefficients

Conditions linking elliptic Gauss sum vanishing to power series coefficients

Extension of previous results to broader cases involving elliptic Gauss sums

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.