The p-adic representation of the Weil-Deligne group associated to an abelian variety

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• 作者：Rutger Noot ^{noot@math.unistra.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：14K15 ; 14F30 ; 11G10
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：March 2017
• 年：2017
• 卷：172
• 期：Complete
• 页码：301-320
• 全文大小：501 K

文摘

Let A   be an abelian variety defined over a number field e8772613164b03fcc1db9deb" title="Click to view the MathML source">F⊂C$F\subset \mathbf{C}$ and let bf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">G_A$G_A$ be the Mumford–Tate group of 9ba30d744cf3bed2b85d27af" title="Click to view the MathML source">A_/C$A_{/\mathbf{C}}$. After replacing F by a finite extension, we can assume that, for every prime number ℓ  , the action of ${\mathrm{\Gamma }}_F=\mathrm{Gal}(\overline{F}/F)$ on ${\mathrm{H}}_{\text{ét}}^1(A_{/\overline{F}},{\mathbf{Q}}_{\ell })$ factors through a map e951be975e2b2" title="Click to view the MathML source">ρ_ℓ:Γ_F→G_A(Q_ℓ)${\rho }_{\ell }:{\mathrm{\Gamma }}_F\to G_A({\mathbf{Q}}_{\ell })$.

Fix a valuation v of F and let p be the residue characteristic at v  . For any prime number e9d75c30832f6a1218fe3d6cd2" title="Click to view the MathML source">ℓ≠p$\ell \ne p$, the representation ρ_ℓ${\rho }_{\ell }$ gives rise to a representation e9fa17ff61">${}^{\prime }W_{F_v}\to G_{A/{\mathbf{Q}}_{\ell }}$ of the Weil–Deligne group. In the case where A has semistable reduction at v it was shown in a previous paper that, with some restrictions, these representations form a compatible system of Q-rational representations with values in bf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">G_A$G_A$.

The p  -adic representation e8287935340162f6c2b792b100e57" title="Click to view the MathML source">ρ_p${\rho }_p$ defines a representation of the Weil–Deligne group ${}^{\prime }W_{F_v}\to G_{A/F_{v,0}}^{\iota }$, where 8d98ff358b9963e75e34d83096f0fd1" title="Click to view the MathML source">F_v,0$F_{v,0}$ is the maximal unramified extension of e96cd054afc23d98f3c38" title="Click to view the MathML source">Q_p${\mathbf{Q}}_p$ contained in F_v$F_v$ and $G_A^{\iota }$ is an inner form of bf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">G_A$G_A$ over 8d98ff358b9963e75e34d83096f0fd1" title="Click to view the MathML source">F_v,0$F_{v,0}$. It is proved, under the same conditions as in the previous theorem, that, as a representation with values in bf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">G_A$G_A$, this representation is Q-rational and that it is compatible with the above system of representations e9fa17ff61">${}^{\prime }W_{F_v}\to G_{A/{\mathbf{Q}}_{\ell }}$.