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Tony Feng, “The Galois action on symplectic K-theory”, EAKTS. https://www.youtube.com/watch?v=Ulm8bCcuW2Q See https://www.mit.edu/~fengt/Galois_action_on_KSp.pdf#page=1
Notes
- Significance of higher \(K\) groups:
- Theorem and consequence of Quillen-Lichtenbaum:
\begin{align*} K_{4i-2}({\mathbf{Z}}) \otimes{\mathbf{Z}}_p \cong H^2_{\text{ét}}( {\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }, {\mathbf{Z}}_p(2i)) \end{align*}
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Similar relationship between topological K theory and singular cohomology
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Related to \(\zeta(1-2i)\) by the Iwasawa main conjecture, see Mazur-Wiles.
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Isomorphism to \(H^2_{\mathrm{sing}}(X, {\mathbf{Z}}_p(2i)\)where \(X\) is an etale \(K(\pi, 1)\) for \(\pi \coloneqq\pi_1({\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] })\) which is a quotient of \({ \mathsf{Gal}} ( { \mkern 1.5mu\overline{\mkern-1.5mu \mathbf{Q} \mkern-1.5mu}\mkern 1.5mu }/ {\mathbf{Q}})\).
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Interpretation of local system and etale sheaf
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This sheaf: \({\mathbf{Z}}_p(1) \coloneqq\underset{n}{\directlim}\, \mu_{p^n}\) and \({\mathbf{Z}}_p(i) \coloneqq{\mathbf{Z}}_p(1)^{\otimes i}\).
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etale homotopy of \({\mathcal{O}}_K\) for \(K\) a global field : a punctured 3-manifold
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Interpret \(H^1({\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }) = H^1_{{\mathsf{Grp}}}(\pi_1 G)\) for \(G \coloneqq{\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] })\), also isomorphic to \(\operatorname{Ext} ^1_G\): space of 2-dimensional Galois representations
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For congruence subgroup \(\Gamma\leq \operatorname{GL}_2({\mathbf{Z}})\), there is a natural Galois action on \(H^*( \Gamma, {\mathbf{Z}}_p)\).
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The Langlands program philosophy: all Galois representations are accounted for by the cohomology of arithmetic groups
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\({\mathbf{B}}\SP_{2g}({\mathbf{Z}})\) is the etale homotopy type of \({\mathcal{A}_g}\)
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PPAV : a complex torus \({\mathbf{C}}^g/ \Lambda\) with a polarization : a symplectic form on the lattice with a positivity condition. Principal: perfect pairing.
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Can recover \(\Lambda = H_1(A; {\mathbf{Z}})\)
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Quotient Siegel half-space (contractible) by \({\mathsf{Sp}}_{2g}({\mathbf{Z}})\) to forget choice of basis for \(\Lambda\). Take stack quotient.
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Free quotient of contractible space: homotopy type of \({\mathbf{B}}\SP_{2g}({\mathbf{Z}})\).
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Some coincidences: \begin{align*} H_{{\mathsf{Grp}}}({\mathsf{Sp}}_{2g}({\mathbf{Z}})) { {}^{ \wedge }_{p} } \cong H_{\operatorname{Sing}}({\mathbb{H}}_g / {\mathsf{Sp}}_{2g}({\mathbf{Z}})){ {}^{ \wedge }_{p} } \cong H_{\operatorname{Sing}}({\mathcal{A}_g}({\mathbf{C}}); {\mathbf{Z}}_p) \cong H_\text{ét}({\mathcal{A}_g}_{g, {\mathbf{C}}}; {\mathbf{Z}}_p) .\end{align*}
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Can define an algebraic stack \(A_g\) over any \(S\in {\mathsf{Sch}}\), classifying flat families of PPAVs.
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Can write \(\mathcal{A}_{g, {\mathbf{C}}} = \mathcal{A}_{g, {\mathbf{Q}}} \underset{\scriptscriptstyle {\operatorname{Spec}{\mathbf{Q}}} }{\times} \operatorname{Spec}{\mathbf{C}}\) which has a natural action of \(\mathop{\mathrm{Aut}}({\mathbf{C}}/{\mathbf{Q}})\) on the 2nd factor, which factors to \({ \mathsf{Gal}} ({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbf{Q} \mkern-1.5mu}\mkern 1.5mu }/{\mathbf{Q}})\).
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Produces an action on \(H_\text{ét}( \mathcal{A}_{g, {\mathbf{C}}}; {\mathbf{Z}}_p)\).
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classifying space of a category: geometric realization of the nerve.
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Constructing K-theory :
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Can also do Quillen’s plus construction : \(K_i({\mathbf{Z}}) = \pi_i( {\mathbf{B}}\operatorname{GL}_{\infty}({\mathbf{Z}})^+)\), which is stable homology?
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Original paper title: Galois action on stable cohomology of \({\mathcal{A}_g}\). Need \(p\gg i\), otherwise proof had many issues. Passing to stable homotopy theory made things easier!
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Question: where are \({\mathbf{B}}\Gamma\) Shimura variety?
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Lie groups are homotopy equivalent to their maximal compact subgroups
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Hodge map: came from taking the Hodge bundle and its Chern class, where the fiber over every \(A\in {\mathcal{A}_g}\) is \(H^0(A, \Omega_A^1)\) the Hodge cohomology
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Galois action unramified except at \(p\) implies it factors through \(\pi_1 {\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\)?
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Guessing the Galois action: trivial on the first factor, the cyclotomic character on the second.
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Room for extensions: the Galois action looks like the following, with a quotient in the bottom-right, a sub in the top-left, and a possible extension in the top-right:
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Informal statement of main theorem:
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Consider the category of extensions and find a universal object.
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\(\operatorname{Spec}{\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\): integers punctured at \(p\), see spec Z as a curve