  The rationality of the elliptic Gauss sum coefficient is shown. The following
is 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 that
f(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 S
be 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 is
regarded 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 the
quartic residue character mod \pi, and our proof is based on the functional
equation of L and an explicit formula of the root number. In fact, the latter
is nothing but the Cassels-Matthews formula on the quartic Gauss sum.
