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

Abstract

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.