The local Langlands correspondence in families and Ihara's lemma for U(n)

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• 作者：Claus M. Sorensen ^{csorensen@ucsd.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11F33
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：July 2016
• 年：2016
• 卷：164
• 期：Complete
• 页码：127-165
• 全文大小：713 K

文摘

The goal of this paper is to reformulate the conjectural Ihara lemma for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si1.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=392e90231cd5286a84121fffc96b1854" title="Click to view the MathML source">U(n)class="mathContainer hidden">class="mathCode">$U(n)$ in terms of the local Langlands correspondence in families class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si2.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=4db273d3c250d6e134037768dd9f2dfd">class="imgLazyJSB inlineImage" height="16" width="37" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16000433-si2.gif">class="mathContainer hidden">class="mathCode">${\tilde{\pi }}_{\mathrm{\Sigma }}(\cdot )$, as developed by Emerton and Helm. The reformulation roughly takes the following form. Suppose we are given an irreducible mod ℓ   Galois representation class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si3.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=77fb4cdb8c69cf27157a9375eddc1e75">class="imgLazyJSB inlineImage" height="13" width="49" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16000433-si3.gif">class="mathContainer hidden">class="mathCode">$\overline{r}={\overline{r}}_{\mathfrak{m}}$, which is modular of full level (and small weight), and a finite set of places Σ, none of which divides ℓ  . Then class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si4.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=640b9544b714ea12822b7857a0b0c9e9">class="imgLazyJSB inlineImage" height="16" width="51" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16000433-si4.gif">class="mathContainer hidden">class="mathCode">${\tilde{\pi }}_{\mathrm{\Sigma }}(r_{\mathfrak{m}})$ exists, and has a global realization as a localized module of ℓ  -integral algebraic modular forms, where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si5.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=c22b2d4357c3e38fe978e7a5b6b81504" title="Click to view the MathML source">r_mclass="mathContainer hidden">class="mathCode">$r_{\mathfrak{m}}$ is the universal deformation of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si6.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=20de4f55003f3b101e20ffdd961d4557">class="imgLazyJSB inlineImage" height="10" width="10" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16000433-si6.gif">class="mathContainer hidden">class="mathCode">$\overline{r}$ of type Σ. This is unconditional for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si7.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=94606eeb6cb21582f2b69bebb041e14b" title="Click to view the MathML source">n=2class="mathContainer hidden">class="mathCode">$n=2$, where Ihara's lemma is an almost trivial consequence of the strong approximation theorem. We emphasize that throughout we will work with banal   primes class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si8.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=63f405044537b871c554682dd55d2359" title="Click to view the MathML source">ℓ>nclass="mathContainer hidden">class="mathCode">$\ell >n$. That is, those for which class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si9.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=b1a6e0d97f50814c1348a6a2108fa435" title="Click to view the MathML source">#k(v)ⁱ≢1class="mathContainer hidden">class="mathCode">$\mathrm{\#}k{(v)}^i\not\equiv 1$ (mod ℓ  ), where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si10.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=2570ba4573ebde15cd9b464c73999962" title="Click to view the MathML source">1≤i≤nclass="mathContainer hidden">class="mathCode">$1\le i\le n$, for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si11.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=3b25ac4cc3996761bd83282bd4c982e6" title="Click to view the MathML source">v∈Σclass="mathContainer hidden">class="mathCode">$v\in \mathrm{\Sigma }$. A weakening of this assumption (quasi-banal) is ubiquitous in [CHT] (particularly in their Chapter 5 on the non-minimal case and Ihara), where the mod ℓ   representation theory of class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16000433&_mathId=si12.gif&_user=111111111&_pii=S0022314X16000433&_rdoc=1&_issn=0022314X&md5=a09edde11e99d9e7c38df1a7be95e6af">class="imgLazyJSB inlineImage" height="16" width="62" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16000433-si12.gif">class="mathContainer hidden">class="mathCode">${\text{GL}}_n(F_{\tilde{v}})$ plays a pivotal role. We will need actual banality, to ensure the functors we define are exact. Banality also removes many of the subtleties in the Emerton–Helm correspondence. For example, in yet unpublished work [Hel13], Helm shows that local Langlands in families do exist in the banal case, although we do not use his result. We should stress that our main result has been common knowledge among experts for some time, but its proof has never appeared in print.