q --- date: 2022-04-25 17:40 modification date: Monday 25th April 2022 17:40:34 title: "Untitled" aliases: [Untitled] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #todo/untagged - Refs: - #todo/add-references - Links: - #todo/create-links --- # Motvations ## Motivic Galois groups ![](attachments/Pasted%20image%2020220426011502.png) ![](attachments/Pasted%20image%2020220426005209.png) ![](attachments/Pasted%20image%2020220426005224.png) ![](attachments/Pasted%20image%2020220426005253.png) Interesting open question: ![](attachments/Pasted%20image%2020220426005326.png) ## Inverse Galois Problem: ![](attachments/Pasted%20image%2020220425203818.png) ![](attachments/Pasted%20image%2020220425203831.png) ![](attachments/Pasted%20image%2020220425203631.png) - Known completely if $K = k(t)$ where $k=\kbar$ is any field of characteristic zero. - Cyclic groups: cyclotomic fields ![](attachments/Pasted%20image%2020220425202826.png) - Finite abelian groups: ![](attachments/Pasted%20image%2020220425202958.png) ![](attachments/Pasted%20image%2020220425203854.png) - $S_n$ and $A_n$: ![](attachments/Pasted%20image%2020220425203932.png) - Solvable groups: ![](attachments/Pasted%20image%2020220425204002.png) - Some sporadic groups: ![](attachments/Pasted%20image%2020220425204041.png) - $\PGL$ in some cases: ![](attachments/Pasted%20image%2020220425203232.png) - The monster group: ![](attachments/Pasted%20image%2020220425204130.png) - Open for some finite simple groups of Lie type. Phrased homotopically: ![](attachments/Pasted%20image%2020220425203514.png) Methods: - Hilbert irreducibility ![](attachments/Pasted%20image%2020220425204243.png) ![](attachments/Pasted%20image%2020220425204226.png) - The rigidity method: ![](attachments/Pasted%20image%2020220425204324.png) ![](attachments/Pasted%20image%2020220425204354.png) ![](attachments/Pasted%20image%2020220425204411.png) ![](attachments/Pasted%20image%2020220425204432.png) ![](attachments/Pasted%20image%2020220425204451.png) - Langlands: ![](attachments/Pasted%20image%2020220425204648.png) ![](attachments/Pasted%20image%2020220425204717.png) ![](attachments/Pasted%20image%2020220426000725.png) ![](attachments/Pasted%20image%2020220426005648.png) ## Rigidity ![](attachments/Pasted%20image%2020220426113945.png) ![](attachments/Pasted%20image%2020220426114020.png) ![](attachments/Pasted%20image%2020220425202549.png) ![](attachments/Pasted%20image%2020220425203135.png) ![](attachments/Pasted%20image%2020220426005715.png) ![](attachments/Pasted%20image%2020220426005732.png) ![](attachments/Pasted%20image%2020220426010548.png) ![](attachments/Pasted%20image%2020220426010714.png) ![](attachments/Pasted%20image%2020220426011119.png) ![](attachments/Pasted%20image%2020220426011142.png) ![](attachments/Pasted%20image%2020220426011248.png) ![](attachments/Pasted%20image%2020220426011305.png) ![](attachments/Pasted%20image%2020220426011733.png) ![](attachments/Pasted%20image%2020220426094340.png) ![](attachments/Pasted%20image%2020220426102715.png) ![](attachments/Pasted%20image%2020220426113838.png) # Classical Modular forms ![](attachments/Pasted%20image%2020220425183028.png) ![](attachments/Pasted%20image%2020220425183055.png) ![](attachments/Pasted%20image%2020220425183631.png) ![](attachments/Pasted%20image%2020220425183655.png) ![](attachments/Pasted%20image%2020220425184007.png) ![](attachments/Pasted%20image%2020220425184013.png) ![](attachments/Pasted%20image%2020220425184033.png) ![](attachments/Pasted%20image%2020220425184044.png) ## Adeles Motivation: how do we "do analysis" on $\QQ$ without passing to $\RR$ (loses arithmetic information). In particular: harmonic/Fourier analysis, need local compactness. ![](attachments/Pasted%20image%2020220425185238.png) ![](attachments/Pasted%20image%2020220129175344.png) ![](attachments/Pasted%20image%2020220425185709.png) Motivation from ANT: Let $C_K$ be the idele class group $\AA_k\units/ k\units$. ![](attachments/Pasted%20image%2020220425185825.png) ![](attachments/Pasted%20image%2020220425185926.png) ![](attachments/Pasted%20image%2020220425190444.png) ![](attachments/Pasted%20image%2020220425190503.png) ![](attachments/Pasted%20image%2020220425190607.png) ![](attachments/Pasted%20image%2020220425190741.png) \ ![](attachments/Pasted%20image%2020220425190906.png) # Setup ![](attachments/Pasted%20image%2020220425194602.png) ![](attachments/Pasted%20image%2020220425191933.png) ![](attachments/Pasted%20image%2020220425192029.png) ![](attachments/Pasted%20image%2020220425192206.png) ![](attachments/Pasted%20image%2020220425192237.png) ![](attachments/Pasted%20image%2020220425192307.png) ![](attachments/Pasted%20image%2020220425192318.png) ![](attachments/Pasted%20image%2020220425192549.png) ![](attachments/Pasted%20image%2020220425192652.png) ![](attachments/Pasted%20image%2020220425192704.png) ![](attachments/Pasted%20image%2020220425192857.png) ![](attachments/Pasted%20image%2020220425193135.png) # Hecke algebra Idea: rep theory for finite groups generalized to reductive groups. ![](attachments/Pasted%20image%2020220425194001.png) ![](attachments/Pasted%20image%2020220425193837.png) ![](attachments/Pasted%20image%2020220425193913.png) ![](attachments/Pasted%20image%2020220425193935.png) Character sheaves are a type of perverse sheaf. ![](attachments/Pasted%20image%2020220425194045.png) ![](attachments/Pasted%20image%2020220425194116.png) Character sheaves: ![](attachments/Pasted%20image%2020220425194637.png) ![](attachments/Pasted%20image%2020220425194703.png) ![](attachments/Pasted%20image%2020220425194817.png) ![](attachments/Pasted%20image%2020220425200939.png) ![](attachments/Pasted%20image%2020220425201001.png) ![](attachments/Pasted%20image%2020220425201021.png) ![](attachments/Pasted%20image%2020220425201057.png) ![](attachments/Pasted%20image%2020220425201112.png) ![](attachments/Pasted%20image%2020220425201132.png) # Stacks Why stacks? ![](attachments/Pasted%20image%2020220426012946.png) ![](attachments/Pasted%20image%2020220426013503.png) ![](attachments/Pasted%20image%2020220426014052.png) ![](attachments/Pasted%20image%2020220425230133.png) ![](attachments/Pasted%20image%2020220425234046.png) ![](attachments/Pasted%20image%2020220425234055.png) ![](attachments/Pasted%20image%2020220425233813.png) ![](attachments/Pasted%20image%2020220425230151.png) ![](attachments/Pasted%20image%2020220425230215.png) Moduli of curves: $\Mg = [\Hilb^{(6n-1)(g-1)}(\PP^{5g-5-1})/\PGL_{5g-6}]$. ![](attachments/Pasted%20image%2020220425230449.png) ![](attachments/Pasted%20image%2020220425230722.png) ![](attachments/Pasted%20image%2020220426013106.png) ![](attachments/Pasted%20image%2020220426013211.png) ![](attachments/Pasted%20image%2020220426013255.png) # Bun_g ![](attachments/Pasted%20image%2020220426012436.png) ![](attachments/Pasted%20image%2020220426013659.png) # Level structure ![](attachments/Pasted%20image%2020220425202450.png) ![](attachments/Pasted%20image%2020220425225403.png) ![](attachments/Pasted%20image%2020220425201853.png) ![](attachments/Pasted%20image%2020220425215050.png) ![](attachments/Pasted%20image%2020220425220819.png) Ideas: - $\Gamma(N) = \ker (\SL_2(\ZZ) \mapsvia{\pi_N} \SL_2(\ZZ/N))$ - $\Gamma_0(N) = \pi_N\inv(B_{\SL_2(\ZZ/N)})$ - $\Gamma_1(N) = \pi_N\inv(R_u B_{\SL_2(\ZZ/N)})$ where $U$ is the unipotent radical of a parabolic. ![](attachments/Pasted%20image%2020220425215423.png) ![](attachments/Pasted%20image%2020220425215436.png) ![](attachments/Pasted%20image%2020220425215449.png) ![](attachments/Pasted%20image%2020220425215537.png) ![](attachments/Pasted%20image%2020220425215621.png) # BunG ![](attachments/Pasted%20image%2020220425225940.png) ![](attachments/Pasted%20image%2020220425201157.png) ![](attachments/Pasted%20image%2020220425201210.png)