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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