Congruences modulo $\ell$ between $ϵ$ factors for cuspidal representations of $GL\left(2\right)$

Vignéras, Marie-France. "Congruences modulo $\ell $ between $\epsilon $ factors for cuspidal representations of $GL(2)$." Journal de théorie des nombres de Bordeaux 12.2 (2000): 571-580. <http://eudml.org/doc/248493>.

@article{Vignéras2000,
abstract = {Let $\ell \ne p$ be two different prime numbers, let $F$ be a local non archimedean field of residual characteristic $p$, and let $\overline\{\mathbf \{Q\}\}_\ell , \overline\{\mathbf \{Z\}\}_\ell , \overline\{\mathbf \{F\}\}_\ell $ be an algebraic closure of the field of $\ell $-adic numbers $\mathbf \{Q\}_\ell $, the ring of integers of $\overline\{\mathbf \{Q\}\}_\ell $, the residual field of $\overline\{\mathbf \{Z\}\}_\ell $. We proved the existence and the unicity of a Langlands local correspondence over $\overline\{\mathbf \{F\}\}_\ell $ for all $n \ge 2$, compatible with the reduction modulo $\ell $ in [V5], without using $L$ and $\epsilon $factors of pairs. We conjecture that the Langlands local correspondence over $\overline\{\mathbf \{Q\}\}_\ell $ respects congruences modulo $\ell $ between $L$ and $\epsilon $ factors of pairs, and that the Langlands local correspondence over $\overline\{\mathbf \{F\}\}_\ell $ is characterized by identities between new $L$ and $\epsilon $ factors. The aim of this short paper is prove this when $n = 2$.},
author = {Vignéras, Marie-France},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {local Langlands correspondence},
language = {eng},
number = {2},
pages = {571-580},
publisher = {Université Bordeaux I},
title = {Congruences modulo $\ell $ between $\epsilon $ factors for cuspidal representations of $GL(2)$},
url = {http://eudml.org/doc/248493},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Vignéras, Marie-France
TI - Congruences modulo $\ell $ between $\epsilon $ factors for cuspidal representations of $GL(2)$
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 2
SP - 571
EP - 580
AB - Let $\ell \ne p$ be two different prime numbers, let $F$ be a local non archimedean field of residual characteristic $p$, and let $\overline{\mathbf {Q}}_\ell , \overline{\mathbf {Z}}_\ell , \overline{\mathbf {F}}_\ell $ be an algebraic closure of the field of $\ell $-adic numbers $\mathbf {Q}_\ell $, the ring of integers of $\overline{\mathbf {Q}}_\ell $, the residual field of $\overline{\mathbf {Z}}_\ell $. We proved the existence and the unicity of a Langlands local correspondence over $\overline{\mathbf {F}}_\ell $ for all $n \ge 2$, compatible with the reduction modulo $\ell $ in [V5], without using $L$ and $\epsilon $factors of pairs. We conjecture that the Langlands local correspondence over $\overline{\mathbf {Q}}_\ell $ respects congruences modulo $\ell $ between $L$ and $\epsilon $ factors of pairs, and that the Langlands local correspondence over $\overline{\mathbf {F}}_\ell $ is characterized by identities between new $L$ and $\epsilon $ factors. The aim of this short paper is prove this when $n = 2$.
LA - eng
KW - local Langlands correspondence
UR - http://eudml.org/doc/248493
ER -