# grössencharacter

Let $K$ be a number field^{} and let ${A}_{K}$ be idele group of $K$, i.e.

$${A}_{K}=\prod _{\nu}{K}_{\nu}^{\ast}$$ |

where the product is a restricted direct product^{} running over all places (infinite and finite) of $K$ (see entry on http://planetmath.org/node/Ideleideles). Recall that ${K}^{\ast}$ embeds into ${A}_{K}$ diagonally:

$$x\in {K}^{\ast}\mapsto {({x}_{\nu})}_{\nu}$$ |

where ${x}_{\nu}$ is the image of $x$ under the embedding of $K$ into its completion at the place $\nu $, ${K}_{\nu}$.

###### Definition 1.

A Grössencharacter $\psi $ on $K$ is a continuous homomorphism^{}:

$$\psi :{A}_{K}\u27f6{\u2102}^{\ast}$$ |

which is trivial on ${K}^{\mathrm{\ast}}$, i.e. if $x\mathrm{\in}{K}^{\mathrm{\ast}}$ then $\psi \mathit{}\mathrm{(}{\mathrm{(}{x}_{\nu}\mathrm{)}}_{\nu}\mathrm{)}\mathrm{=}\mathrm{1}$. We say that $\psi $ is unramified at a prime $\mathrm{\wp}$ of $K$ if $\psi \mathit{}\mathrm{(}{\mathrm{O}}_{\mathrm{\wp}}^{\mathrm{\ast}}\mathrm{)}\mathrm{=}\mathrm{1}$, where ${\mathrm{O}}_{\mathrm{\wp}}$ is the ring of integers^{} inside ${K}_{\mathrm{\wp}}$. Otherwise we say that $\psi $ is ramified at $\mathrm{\wp}$.

Let ${\mathcal{O}}_{K}$ be the ring of integers in $K$. We may define a homomorphism on the (multiplicative) group of non-zero fractional ideals^{} of $K$ as follows. Let $\mathrm{\wp}$ be a prime of $K$, let $\pi $ be a uniformizer of ${K}_{\mathrm{\wp}}$ and let ${\alpha}_{\mathrm{\wp}}\in {A}_{K}$ be the element which is $\pi $ at the place $\mathrm{\wp}$ and $1$ at all other places. We define:

$$\psi (\mathrm{\wp})=\{\begin{array}{cc}0,\text{if}\psi \text{is ramified at}\mathrm{\wp};\hfill & \\ \psi ({\alpha}_{\mathrm{\wp}}),\text{otherwise}.\hfill & \end{array}$$ |

###### Definition 2.

The Hecke L-series attached to a Grössencharacter $\psi $ of $K$ is given by the Euler product^{} over all primes of $K$:

$$L(\psi ,s)=\prod _{\mathrm{\wp}}{\left(1-\frac{\psi (\mathrm{\wp})}{{({N}_{\mathbb{Q}}^{K}(\mathrm{\wp}))}^{s}}\right)}^{-1}.$$ |

Hecke L-series of this form have an analytic continuation and satisfy a certain functional equation. This fact was first proved by Hecke himself but later was vastly generalized by Tate using Fourier analysis on the ring ${A}_{K}$ (what is usually called Tate’s thesis).

Title | grössencharacter |
---|---|

Canonical name | Grossencharacter |

Date of creation | 2013-03-22 15:45:19 |

Last modified on | 2013-03-22 15:45:19 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 5 |

Author | alozano (2414) |

Entry type | Definition |

Classification | msc 11R56 |

Related topic | GrossencharacterAssociatedToACMEllipticCurve |

Defines | grossencharacter |