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The rationality of the elliptic Gauss sum coefficient is shown. The followingis a specific case of our argument.  Let f(u)=sl((1-i)\varpi u), where sl() is the Gauss' lemniscatic sine and\varpi=2.62205... is the real period of the elliptic curve y^2=x^3-x, so thatf(u) is an elliptic function relative to the period lattice Z[i].  Let \pi be a primary prime of Z[i] such that norm(\pi)\equiv 13\mod 16. Let Sbe the quarter set mod \pi consisting of quartic residues.  Let us define G(\pi):=\sum_{\nu\in S} f(\nu/\pi) and\tilde{\pi}:=\prod_{\nu\in S} f(\nu/\pi).  The former G(\pi) is a typical example of elliptic Gauss sum; the latter isregarded as a canonical 4-th root of -\pi: (\tilde{\pi})^4=-\pi. Then we have  Theorem: G(\pi)/(\tilde{\pi})^3 is a rational odd integer.  G(\pi) appears naturally in the central value of Hecke L associated to thequartic residue character mod \pi, and our proof is based on the functionalequation of L and an explicit formula of the root number. In fact, the latteris nothing but the Cassels-Matthews formula on the quartic Gauss sum.

Elliptic Gauss Sums and Hecke L-values at s=1