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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si2.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=08e6947ee8772613164b03fcc1db9deb" title="Click to view the MathML source">F⊂Cclass="mathContainer hidden">class="mathCode">$F\subset \mathbf{C}$ and let class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si3.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6edbf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">G_Aclass="mathContainer hidden">class="mathCode">$G_A$ be the Mumford–Tate group of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si4.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=b0274ff49ba30d744cf3bed2b85d27af" title="Click to view the MathML source">A_/Cclass="mathContainer hidden">class="mathCode">$A_{/\mathbf{C}}$. After replacing F by a finite extension, we can assume that, for every prime number ℓ  , the action of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si5.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=dd71083b6b95a0644a397f5fbc0479a4">class="imgLazyJSB inlineImage" height="19" width="114" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1630213X-si5.gif">class="mathContainer hidden">class="mathCode">${\mathrm{\Gamma }}_F=\mathrm{Gal}(\overline{F}/F)$ on class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si6.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=1ca5c09203798f209df11c23a272284c">class="imgLazyJSB inlineImage" height="21" width="94" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1630213X-si6.gif">class="mathContainer hidden">class="mathCode">${\mathrm{H}}_{\text{ét}}^1(A_{/\overline{F}},{\mathbf{Q}}_{\ell })$ factors through a map class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si7.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=51f086fd16b086ed279e951be975e2b2" title="Click to view the MathML source">ρ_ℓ:Γ_F→G_A(Q_ℓ)class="mathContainer hidden">class="mathCode">${\rho }_{\ell }:{\mathrm{\Gamma }}_F\to G_A({\mathbf{Q}}_{\ell })$.

Fix a valuation v of F and let p be the residue characteristic at v  . For any prime number class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si225.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=4eb649e9d75c30832f6a1218fe3d6cd2" title="Click to view the MathML source">ℓ≠pclass="mathContainer hidden">class="mathCode">$\ell \ne p$, the representation class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si9.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=528275032338b4090101d91be212c478" title="Click to view the MathML source">ρ_ℓclass="mathContainer hidden">class="mathCode">${\rho }_{\ell }$ gives rise to a representation class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si10.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6af2bdab1a04d72afc7f43e9fa17ff61">class="imgLazyJSB inlineImage" height="19" width="108" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1630213X-si10.gif">class="mathContainer hidden">class="mathCode">${}^{\prime }W_{F_v}\to G_{A/{\mathbf{Q}}_{\ell }}$ 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 class="boldFont">Q-rational representations with values in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si3.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6edbf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">G_Aclass="mathContainer hidden">class="mathCode">$G_A$.

The p  -adic representation class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si11.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=91de8287935340162f6c2b792b100e57" title="Click to view the MathML source">ρ_pclass="mathContainer hidden">class="mathCode">${\rho }_p$ defines a representation of the Weil–Deligne group class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si12.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=e407aa7cf8b8f8686307036dff8581a0">class="imgLazyJSB inlineImage" height="22" width="115" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1630213X-si12.gif">class="mathContainer hidden">class="mathCode">${}^{\prime }W_{F_v}\to G_{A/F_{v,0}}^{\iota }$, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si13.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=18d98ff358b9963e75e34d83096f0fd1" title="Click to view the MathML source">F_v,0class="mathContainer hidden">class="mathCode">$F_{v,0}$ is the maximal unramified extension of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si123.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=45d4a71d0aee96cd054afc23d98f3c38" title="Click to view the MathML source">Q_pclass="mathContainer hidden">class="mathCode">${\mathbf{Q}}_p$ contained in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si15.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=d0017b6d89f51badcddd5b5566fe2dc5" title="Click to view the MathML source">F_vclass="mathContainer hidden">class="mathCode">$F_v$ and class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si16.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=e377f7f92b5f52dc572d82c0a7162f80">class="imgLazyJSB inlineImage" height="16" width="24" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1630213X-si16.gif">class="mathContainer hidden">class="mathCode">$G_A^{\iota }$ is an inner form of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si3.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6edbf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">G_Aclass="mathContainer hidden">class="mathCode">$G_A$ over class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si13.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=18d98ff358b9963e75e34d83096f0fd1" title="Click to view the MathML source">F_v,0class="mathContainer hidden">class="mathCode">$F_{v,0}$. It is proved, under the same conditions as in the previous theorem, that, as a representation with values in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si3.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6edbf6c9b7689c7c290234c6ccdb5848" title="Click to view the MathML source">G_Aclass="mathContainer hidden">class="mathCode">$G_A$, this representation is class="boldFont">Q-rational and that it is compatible with the above system of representations class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X1630213X&_mathId=si10.gif&_user=111111111&_pii=S0022314X1630213X&_rdoc=1&_issn=0022314X&md5=6af2bdab1a04d72afc7f43e9fa17ff61">class="imgLazyJSB inlineImage" height="19" width="108" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X1630213X-si10.gif">class="mathContainer hidden">class="mathCode">${}^{\prime }W_{F_v}\to G_{A/{\mathbf{Q}}_{\ell }}$.