Elliptic Gauss Sums and Hecke L - values at s = 1 Dedicated to Professor Tomio

where p (> 3) is a rational prime such that p ≡ 3 (mod 4) and p ≡ 1 (mod 4), respectively ; ( p ) is the Legendre symbol and h(−p) is the class number of the quadratic field Q( √−p). The formulas are apparently related to the Dirichlet L-values at s = 1. To get a typical elliptic Gauss sum, we have only to replace the Legendre symbol by the cubic or the quartic residue character, and the trigonometric function by a suitable elliptic function. The notion of elliptic Gauss sum was first introduced by G.Eisenstein for a concern of higher reciprocity laws, but since then it has been regarded seemingly as a minor object of study. (cf. [L], p.311) We shall here try to reconsider it. Especially, we treat the problem of rationality of the coefficient, so we call, of the elliptic Gauss sum, which is an analogy of the coefficient h(−p) in the above classical case. A typical example is as follows. Let sl(u) be the lemniscatic sine of Gauss so that sl((1− i)̟u) is an elliptic function