# turkmath.org

Türkiye'deki Matematiksel Etkinlikler

08 Nisan 2022, 14:30

### İzmir Yüksek Teknoloji Enstitüsü Matematik Bölümü Seminerleri

K. İlhan İkeda
Boğaziçi Üniversitesi - Feza Gürsey Fizik ve Matematik Uygulama Araştırma Merkezi, Türkiye

I shall briefly describe my reflections on the Langlands reciprocity and functoriality principles.

Let $K$ be a number field. The local Langlands group $L_{K_\nu}$ of $K_\nu$ is defined by $L_{K_\nu}=WA_{K_\nu}=W_{K_\nu}\times\mathsf{SU}(2)$ if $\nu\in\mathbb h_K$, and by $L_{K_\nu}=W_{K_\nu}$ if $\nu\in\mathbb a_K$, where $W_{K_\nu}$ denotes the local Weil group of $K_\nu$. For each $\nu\in\mathbb h_K$, fix a Lubin-Tate splitting $\varphi_{K_\nu}$. The local non-abelian norm residue symbol
\begin{equation*}
\{\bullet,K_\nu\}_{\varphi_\nu}^{\mathrm{Langlands}}:{}_\mathbb Z\nabla_{K_\nu}^{(\varphi_{K_\nu})}\times
\mathsf{SU}(2)\xrightarrow{\sim}L_{K_\nu}
\end{equation*}
of $K_\nu$ "in the sense of Langlands'' has been defined and studied in the papers of the speaker, where
${}_\mathbb Z\nabla_{K_\nu}^{(\varphi_{K_\nu})}$ is a certain non-commutative topological group depending only on the ground field $K_\nu$ and constructed, using Fontaine-Wintenberger theory of fields of norms.

Fix $\underline{\varphi}=\{\varphi_{K_\nu}\}_{\nu\in\mathbb h_K}$ and define a non-commutative topological group $\mathscr {WA}_K^{\underline{\varphi}}$ depending only on the ground field $K$ by the "restricted free topological product''
$\mathscr {WA}_K^{\underline{\varphi}}:= {\ast_{\nu\in\mathbb h_K}}' \left({}_\mathbb Z\nabla_{K_\nu}^{(\varphi_{K_\nu})}\times\mathsf{SU}(2): {}_1{\nabla_{K_\nu}^{(\varphi_{K_\nu})}}^{\underline 0}\times\mathsf{SU}(2) \right)\ast W_\mathbb R^{\ast r_1}\ast W_\mathbb C^{\ast r_2}.$
Here, $r_1$ and $r_2$ denote the numbers of real and pairs of complex conjugate embeddings of the number field $K$ in $\mathbb C$. Note that, ${\mathscr {WA}_K^{\underline{\varphi}}}^{ab}=\mathbb J_K$, the idèle group of $K$. So, $\mathscr {WA}_K^{\underline{\varphi}}$ can be viewed as a non-abelian generalisation of $\mathbb J_K$.

Let $L_K$ denote the hypothetical Langlands group $L_K$ of $K$. The existence problem of $L_K$ is one of the major conjectures in the Langlands Program, and according to Arthur, this conjecture is the most fundamental and mysterious one.

For $\nu\in\mathbb h_K\cup\mathbb a_K$, an embedding  $e_\nu:K^{sep}\hookrightarrow K_\nu^{sep}$ determines a continuous homomorphism $e_\nu^{\mathrm{Langlands}}:L_{K_\nu}\rightarrow L_K$ unique up to conjugacy, which in return defines a continuous homomorphism
$\mathsf{NR}_{K_\nu}^{(\varphi_{K_\nu})^{\mathrm{Langlands}}}: {}_\mathbb Z\nabla_{K_\nu}^{\varphi_{K_\nu}}\times\mathsf{SU}(2) \xrightarrow[\sim]{\{\bullet,K_\nu\}_{\varphi_{K_\nu}}\times \mathrm{id}_{\mathsf{SU}(2)}} L_{K_\nu}\xrightarrow{e_\nu^{\mathrm{Langlands}}} L_K$
unique up to conjugacy, for each $\nu\in\mathbb h_K$. Fixing one such morphism for each $\nu\in\mathbb h_K$, the collection $\left\{\mathsf{NR}_{K_\nu}^{(\varphi_{K_\nu})^{\mathrm{Langlands}}}\right\}_{\nu\in\mathbb h_K}$ defines a continuous homomorphism
$\mathsf{NR}_K^{\underline\varphi^{\mathrm{Langlands}}}:\mathscr {WA}_K^{\underline{\varphi}}\rightarrow L_K,$
unique up to "local conjugation'', called the global non-abelian norm residue symbol of $K$ "in the sense of Langlands'', which is also compatible with Arthur's proposed construction of $L_K$. The key philosophical observation is that the source $\mathscr {WA}_K^{\underline{\varphi}}$ is an unconditional object while the target $L_K$ is conjectural!

Let $\mathrm{G}$ be a connected, quasisplit reductive group over $K$. There is a bijection between the set of "$WA$-parameters''
$\phi:\mathscr {WA}_K^{\underline{\varphi}}\rightarrow {}^L\mathrm{G}(\mathbb C)= \widehat{\mathrm G}(\mathbb C)\rtimes L_K$
of $\mathrm{G}$ over $K$ and the set $\mathscr P_{\mathrm G}$ whose elements are the collections
$\{\phi_\nu:L_{K_\nu}\rightarrow{}^L\mathrm{G}_\nu(\mathbb C)\}_{\nu\in\mathbb h_K\cup\mathbb a_K}$
consisting of local $L$-parameters of $\mathrm{G}_\nu$ over $K_\nu$ for each $\nu$. Note that, assuming the local reciprocity principle for $\mathrm{G}_\nu$ over $K_\nu$ for all $\nu\in\mathbb h_K\cup\mathbb a_K$, the set  $\mathscr P_{\mathrm G}$ is in bijection with the set whose elements are the collections
$\{\Pi_{\phi_\nu}\}_{\nu\in\mathbb h_K\cup\mathbb a_K}$  of local $L$-packets of $\mathrm{G}_\nu$ over $K_\nu$ for each $\nu$. As global admissible $L$-packets of $\mathrm G$ over $K$ are the restricted tensor products of local $L$-packets of $\mathrm G_\nu$ over $K_\nu$, by Flath's decomposition theorem, we end up having the following theorems
Theorem.
Let $\mathrm{G}$ be a connected quasisplit reductive group over the number field $K$. Assume that the local Langlands reciprocity principle for $\mathrm{G}_\nu$ over $K_\nu$ for all $\nu\in\mathbb h_K\cup\mathbb a_K$ holds. Then, there exists a bijection
$\{\text{WA-parameters of \mathrm G over K}\}\leftrightarrow\{\text{global admissible L-packets of \mathrm G over K}\}$
satisfying the "naturality'' properties.

and

Theorem.
Let $\mathrm{G}$ and $H$ be connected quasisplit reductive groups over the number field $K$. Let
$\rho:{}^LG\rightarrow {}^LH$
be an $L$-homomorphism.Assume that the local Langlands reciprocity principle for $\mathrm G$ over $K$ holds. Then, the $L$-homomorphism $\rho:{}^LG\rightarrow {}^LH$ induces a map (lifting) from the global admissible $L$-packets of $G$ over $K$ to the global admissible $L$-packets of $H$ over $K$  satisfying the "naturality'' properties.

NOT: Get in touch with Dr. Haydar Göral regarding Microsoft Teams connection details

Genel Matematik Araştırmaları İngilizce
Microsoft Teams

fgc 10.04.2022_16:30'de değişiklik yapıldı!

## İLETİŞİM

Akademik biriminizin ya da çalışma grubunuzun ülkemizde gerçekleşen etkinliklerini, ilan etmek istediğiniz burs, ödül, akademik iş imkanlarını veya konuk ettiğiniz matematikçileri basit bir veri girişi ile kolayca turkmath.org sitesinde ücretsiz duyurabilirsiniz. Sisteme giriş yapmak için gerekli bilgileri almak ya da görüş ve önerilerinizi bildirmek için iletişime geçmekten çekinmeyiniz. Katkı verenler listesi için tıklayınız.

Özkan Değer ozkandeger@gmail.com

## DESTEK VERENLER

31. Journees Arithmetiques Konferansı Organizasyon Komitesi

Web sitesinin masraflarının karşılanması ve hizmetine devam edebilmesi için siz de bağış yapmak, sponsor olmak veya reklam vermek için lütfen iletişime geçiniz.

## ONLİNE ZİYARETÇİLER

©2013-2022 turkmath.org
Tüm hakları saklıdır