Notes from Summer School of Langlands Program

Remark: These notes only contain material I could comprehend during the summer school conducted from 11th July,2022-30th July 2022. Also, for my understanding, I have added extra definitions and examples.

Day 1: Shimura varieties
Original notes shared during lecture can be found here.

Definition (Algebraic Matrix Group): Let $k$ be an algebraicallly closed field. We denote $M(n,k)$ the group of $n$ x $n$ matrices with entries in k and $GL(n,k)$ group of all $n$ x $n$ invertible matices. $GL(n,K)$ forms a affine subvariety in $k^{n^2+1}$ given by
$(g_{11},g_{12},\cdots,g_{nn}) \mapsto (g_{11},g_{12},\cdots,g_{nn}, (det$ g $)^{-1})$
The set $M(n, k)$ carries a topology-the Zariski topology-the closed sets being the algebraic subsets of $M(n, k) = k^{n^2}$ .The ring $GL(n, k)$ is an open subset of $M(n, k)$ and carries the induced topology.
Subgroup $G$ of $GL(n,k)$ is called algebraic matrix group if $G$ is closed subset of $GL(n,k)$ .
Let $I_k$ denote the ideal of all polynomials with coefficient in $k$ and vanishing on $G$ .

Let $G$ be a $k$ -group in the sense of above definition. Then for any $k-algebra$ $A$ , we may consider the set $G_A$ of elements of $GL(n, A)$ whose coefficients annihilate the polynomials in $I_k.$

Definition( Semi Simple Group): A connected linear Algebraic Group $G$ over an algebraically closed field is called semi simple if every smooth connected solvable normal subgroup of $G$ is trivial.

Definition (Reductive group): A connected linear algebraic group $G$ over an algebraically closed field is reductive if every smooth connected unipotent normal subgroup of $G$ is trivial.

Definition (Maximal compact subgroup) A maximal compact subgroup $K$ of topological group $G$ is a subgroup $K$ that is compact space in subspace topology and maximal amongst such subgroups.
If $G$ is a connected LIe group, then any compact subgroup of $G$ is contained is contained in some maximal compact subgroup (in particular maximal compact subgroup must exists) and all maximal compact subgroups of $G$ are connected and conjugate to each other.
Example: $GL(n)$ is algebraic group as well as topological group.

What does geometric Langlands mean to a number theorist?

This talk was given by Sam Raskin.

The goal of this is to give arithmetic version of Geometric Langlands conjecture.

Geometric setting:

working over algebraically closed field.

$X/k$ a smooth projective connected curve

$G/k$ split reductive group

$Bun_G$ moduli stack of $G-bundle$ on $X$

Auxiliary notation:

field $e=\overline{\mathbb{Q}}_l$ and dual group $\check{G}/e$

Let $S/k$ be a variety.
$Lisse(S)\subseteq sheaves~ on~ (S)$ is subcategory where all (perverse) cohomologies of $\mathcal{F}\in Shv (S)$ can be written as a colimit of (finite rank) (usual) Lisse sheaves.

Goal: To define moduli space of local systems on X

Representability theorem:

$LS_{\check{G}}^{restr}$ (moduli of restricted local system) is a formal algebraic stack with infinitely many connected components.

$LS_{\check{G}}^{arthm}$ is a quasi-compact algebraic stack.

Goal 1: Construct a certain quasi-coherent sheaf $Drinf\in Qcoh(LS_{\check{G}}^{arthm})$ with global section $\Gamma (Drinf)=Functions(Bun_G(\mathbb{F_q})).$

The thesis is that $Drinf$ is a better object to study than the RHS.
Goal 2: $Drinf$ can be computed using Geomteric Langlands Conjecture.