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Motvations
Motivic Galois groups
Interesting open question:
Inverse Galois
Problem:
- Known completely if K=k(t) where k=¯k is any field of characteristic zero.
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Cyclic groups: cyclotomic fields
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Finite abelian groups:
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Sn and An:
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Solvable groups:
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Solvable groups:
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Some sporadic groups:
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PGL in some cases:
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The monster group:
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Open for some finite simple groups of Lie type.
Phrased homotopically:
Methods:
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Hilbert irreducibility
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The rigidity method:
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Langlands:
Rigidity
Classical Modular forms
Adeles
Motivation: how do we “do analysis” on Q without passing to R (loses arithmetic information). In particular: harmonic/Fourier analysis, need local compactness.
Motivation from ANT: Let CK be the idele class group A×k/k×.
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: Mg=[Hilb(6n−1)(g−1)(P5g−5−1)/PGL5g−6].
Bun_g
Level structure
Ideas:
- Γ(N)=ker(SL2(Z)πN→SL2(Z/N))
- Γ0(N)=π−1N(BSL2(Z/N))
- Γ1(N)=π−1N(RuBSL2(Z/N)) where U is the unipotent radical of a parabolic.
BunG