Elliptic Gauss Sums and Hecke L-values at s=1
| dc.creator | Asai, Tetsuya | |
| dc.date | 2007-07-25 | |
| dc.date.accessioned | 2026-07-07T08:20:11Z | |
| dc.date.available | 2026-07-07T08:20:11Z | |
| dc.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. | |
| dc.description | 39 pages | |
| dc.identifier | https://arxiv.org/abs/0707.3711 | |
| dc.identifier | http://arxiv.org/abs/0707.3711 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/135003 | |
| dc.subject | Number Theory | |
| dc.subject | 11L05, 11R42, 11G15 | |
| dc.title | Elliptic Gauss Sums and Hecke L-values at s=1 | |
| dc.type | text |