--- date: 2021-11-05 23:35 modification date: Friday 5th November 2021 23:35:18 title: Langlands aliases: [Langlands, "the Langlands program", "The Langlands Program", "Langlands program", "Langlands conjectures", "global Langlands", "local Langlands"] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags - #arithmetic-geometry/Langlands #open/problems - Refs: - #resources/videos - Course notes: rep theory of p-adic reductive groups #resources/course-notes - Trace formula course notes: #resources/course-notes - Notes: #resources/notes - Notes for function fields: - Links: - [Geometric Langlands correspondence](Geometric%20Langlands%20correspondence) - [Local Langlands correspondence](Local%20Langlands%20correspondence) - [Global Langlands correspondence](Global%20Langlands%20correspondence) - [arithmetic Langlands program](arithmetic%20Langlands%20program) - [cyclotomic character](cyclotomic%20character.md) - [Tate twist](Tate%20twist) - [Congruence subgroup](Congruence%20subgroup.md) - [algebraic group](algebraic%20group.md) - [locally symmetric space](locally%20symmetric%20space) - [Shimura variety](Shimura%20variety.md) - [perverse sheaf](Unsorted/perverse%20sheaf.md) - [Unsorted/rigidity](Unsorted/rigidity.md) - [Unsorted/tamely ramified](Unsorted/tamely%20ramified.md) - [modular form](Unsorted/modular%20form.md) --- # Langlands ![](attachments/2023-03-05-details.png) Motto: Automorphic reps $\mapstofrom$ Galois reps. Over a curve: Hecke eigensheaves on $\Bun_{\GL_n}(C) \mapstofrom$ $\ell\dash$adic Galois reps $\simeq \ell\dash$adic local systems [Kronecker-Weber](Unsorted/Kronecker-Weber%20theorem.md) yields $\Gal(\QQ(\zeta_n)/\QQ) = C_n\units$ so $\Gal(\QQ^\ab/\QQ) = \varprojlim C_n\units = \prod_p \ZZpadic\units$. For other global fields $K$, one has $\Gal(K^\ab/\QQ)\approx \dcosetl{\GL_1 K}{\GL_1 \AA_K}$ -- for $K=\QQ$ one has $\GL_1\AA_\QQ = \AA_\QQ\units = \RR_{\gt 0}\times \prod_p \ZZpadic\units$. Class field theory says one-dimensional reps of $\Gal(K^\ab/K)$ biject with one-dimensional reps of $\GL_1 \AA_K$. Langlands replaces one with $n$ and *automorphic* reps on $C^0(\dcosetl{\GL_n K}{\GL_n \AA_K}\to ?)$. If Galois reps come from $X$, should relate point counts $X(\FF_p)$ to Fourier coefficients of automorphic forms. Matching $L\dash$functions: for a Galois rep $\rho$, take $\Frob_p \in G_K$ and their images $\rho(\Frob_p) \subset \GL_n$ which are conjugacy classes. Frobenius eigenvalues $(e_1(\rho), \cdots, e_n(\rho)$ are tracked by $L_\rho(s)$. For an automorphic rep $\pi$, take Hecke eigenvalues $(e_1(\pi), \cdots, e_n(\pi))$ and track by $L_\pi(s)$. So Frobenius eigenvalues (Galois) $\mapstofrom$ Hecke eigenvalues (automorphic). $\Rep(\pi_1^\et K) = \Rep(G_K) \mapstofrom \Rep(\GL_n \AA_K)$. Applications: - The Langlands conjectures were used to prove Fermat's Last Theorem. (Wiles) - The Langlands conjectures are used to study the Sato-Tate conjecture for elliptic curves. (Clozel, Harris, Shepherd-Baron, Taylor) - The Langlands conjectures were used to prove Ramanujan's $\tau$-conjecture. (Deligne) - The Langlands conjectures imply Artin's L-function conjecture. (Langlands) - The Langlands conjectures were used to show that the continuous homomorphism $\operatorname{Gal}\left(\overline{\mathbb{Q}_p} / \mathbb{Q}_p\right) \rightarrow \operatorname{Gal}\left(\mathbb{Q}_{\{p, \ell\}} / \mathbb{Q}\right)$ is injective, where $\ell$ and $p$ are distrinct primes and $\mathbb{Q}_{\{p, \ell\}}$ is the maximal extension of $\mathbb{Q}$ unramified outside of $p$ and $\ell$. (ChenevierClozel) - The Langlands conjectures (for function fields) were used to prove finiteness of geometric representations. (Litt) Known cases: - Function fields: known (replace automorphic reps with Hecke eigensheaves) - Number fields: mostly unknown, outside of Shimura-Taniyama. - Local: - $\Res_{F/\QQpadic}\GL_{n, F}$ - $\Res_{F/\QQpadic}\Sp_{2n}$ - $\Res_{F/\QQpadic}\SO_{2n}$ - $\Res_{F/\QQpadic}\U_{E/F}(N)^*$ quasi-split unitary groups (and their inner forms) - $\Res_{F/\QQpadic}\GSp_4$ - Global: - Cases of $\Res_{F/\QQ}\GL_{n, F}$ ![](attachments/Pasted%20image%2020220730125326.png) ![](attachments/Pasted%20image%2020220427102849.png) ![](attachments/Pasted%20image%2020220427102858.png) ![](attachments/Pasted%20image%2020220129172042.png) ![](attachments/Pasted%20image%2020220428002840.png) ![](attachments/Pasted%20image%2020220504003005.png) ![](attachments/Pasted%20image%2020220511235642.png) ## L Functions ![](attachments/Pasted%20image%2020220210175725.png) ![](attachments/Pasted%20image%2020220210175813.png) ![](attachments/Pasted%20image%2020230109142627.png) ![](attachments/Pasted%20image%2020230109142637.png) # Topics ![](attachments/Pasted%20image%2020220210230517.png) ![Pasted image 20211105233522.png](Pasted%20image%2020211105233522.png) ![Pasted image 20211106014758.png](Pasted%20image%2020211106014758.png) - For [deformation spaces](Unsorted/Deformation%20space.md) of abelian varieties. - ## Height pairing ![Pasted image 20211106015333.png](Pasted%20image%2020211106015333.png) ![Pasted image 20211106015305.png](Pasted%20image%2020211106015305.png) # Examples ![Pasted image 20211106015100.png](Pasted%20image%2020211106015100.png) # Global Langlands ![](attachments/Pasted%20image%2020220515001417.png)