# Elliptic Gauss Sums and Hecke L-values at s=1 - Mathematics > Number Theory

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Abstract: The rationality of the elliptic Gauss sum coefficient is shown. The followingis a specific case of our argument.Let fu=sl1-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 thatfu is an elliptic function relative to the period lattice Zi.Let \pi be a primary prime of Zi 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 { u\in S} f u-\pi and\tilde{\pi}:=\prod { u\in S} f u-\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 haveTheorem: 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.

Autor: Tetsuya Asai

Fuente: https://arxiv.org/