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Motvations

Motivic Galois groups

Interesting open question:

Inverse Galois

Problem: - Known completely if $K = k(t)$ where $k= { \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu }$ is any field of characteristic zero.

• Cyclic groups: cyclotomic fields

• Finite abelian groups:

• $S_n$ and $A_n$:

  • Solvable groups:

• Some sporadic groups:

• $\operatorname{PGL}$ in some cases:

• The monster group:

• Open for some finite simple groups of Lie type.

Phrased homotopically:

Methods:

• Hilbert irreducibility
• The rigidity method:
• Langlands:

Rigidity

Classical Modular forms

Adeles

Motivation: how do we “do analysis” on ${\mathbf{Q}}$ without passing to ${\mathbf{R}}$ (loses arithmetic information). In particular: harmonic/Fourier analysis, need local compactness.

Motivation from ANT: Let $C_K$ be the idele class group ${\mathbf{A}}_k^{\times}/ k^{\times}$.

Setup

Hecke algebra

Idea: rep theory for finite groups generalized to reductive groups.

Character sheaves are a type of perverse sheaf.

Character sheaves:

Stacks

Why stacks?

Moduli of curves: ${\mathcal{M}_g}= [\operatorname{Hilb}^{(6n-1)(g-1)}({\mathbf{P}}^{5g-5-1})/\operatorname{PGL}_{5g-6}]$.

Bun_g

Level structure

Ideas:

• $\Gamma(N) = \ker ({\operatorname{SL}}_2({\mathbf{Z}}) \xrightarrow{\pi_N} {\operatorname{SL}}_2({\mathbf{Z}}/N))$
• $\Gamma_0(N) = \pi_N^{-1}(B_{{\operatorname{SL}}_2({\mathbf{Z}}/N)})$
• $\Gamma_1(N) = \pi_N^{-1}(R_u B_{{\operatorname{SL}}_2({\mathbf{Z}}/N)})$ where $U$ is the unipotent radical of a parabolic.

BunG