Elliptic Gauss Sums and Hecke L-values at s=1
Abstract
Description
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 πbe a primary prime of Z[i] such that norm(π)\equiv 13\mod 16. Let S be the quarter set mod πconsisting of quartic residues.
Let us define G(π):=\sum_{ν\in S} f(ν/π) and \tildeπ:=\prod_{ν\in S} f(ν/π).
The former G(π) is a typical example of elliptic Gauss sum; the latter is regarded as a canonical 4-th root of -π: (\tildeπ)^4=-π. Then we have
Theorem: G(π)/(\tildeπ)^3 is a rational odd integer.
G(π) appears naturally in the central value of Hecke L associated to the quartic residue character mod π, 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.
39 pages
39 pages