The Galois action on symplectic K-theory

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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:

attachments/image_2021-04-29-13-18-09.png

  • Theorem and consequence of Quillen-Lichtenbaum:

\begin{align*} K_{4i-2}({\mathbb{Z}}) \otimes{\mathbb{Z}}_p \cong H^2_{\text{ét}}( {\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }, {\mathbb{Z}}_p(2i)) \end{align*}

  • Similar relationship between topological K theory and singular cohomology

  • Related to \(\zeta(1-2i)\) by the Iwasawa main conjecture, see Mazur-Wiles.

  • Isomorphism to \(H^2_{\mathrm{sing}}(X, {\mathbb{Z}}_p(2i)\)where \(X\) is an etale \(K(\pi, 1)\) for \(\pi \coloneqq\pi_1({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] })\) which is a quotient of \({ \mathsf{Gal}} ( { \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }/ {\mathbb{Q}})\).

  • Interpretation of local system and etale sheaf

  • This sheaf: \({\mathbb{Z}}_p(1) \coloneqq\underset{n}{\directlim}\, \mu_{p^n}\) and \({\mathbb{Z}}_p(i) \coloneqq{\mathbb{Z}}_p(1)^{\otimes i}\).

  • etale homotopy of \({\mathcal{O}}_K\) for \(K\) a global field : a punctured 3-manifold

  • Interpret \(H^1({\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }) = H^1_{{\mathsf{Grp}}}(\pi_1 G)\) for \(G \coloneqq{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] })\), also isomorphic to \(\operatorname{Ext} ^1_G\): space of 2-dimensional Galois representations

  • For congruence subgroup \(\Gamma\leq \operatorname{GL}_2({\mathbb{Z}})\), there is a natural Galois action on \(H^*( \Gamma, {\mathbb{Z}}_p)\).

  • The Langlands program philosophy: all Galois representations are accounted for by the cohomology of arithmetic groups

  • \({\mathbf{B}}{\operatorname{SP}}_{2g}({\mathbb{Z}})\) is the etale homotopy type of \({\mathcal{A}_g}\)

  • PPAV : a complex torus \({\mathbb{C}}^g/ \Lambda\) with a polarization : a symplectic form on the lattice with a positivity condition. Principal: perfect pairing.

    • Can recover \(\Lambda = H_1(A; {\mathbb{Z}})\)

    • Quotient Siegel half-space (contractible) by \({\mathsf{Sp}}_{2g}({\mathbb{Z}})\) to forget choice of basis for \(\Lambda\). Take stack quotient.

    • Free quotient of contractible space: homotopy type of \({\mathbf{B}}{\operatorname{SP}}_{2g}({\mathbb{Z}})\).

  • Some coincidences: \begin{align*} H_{{\mathsf{Grp}}}({\mathsf{Sp}}_{2g}({\mathbb{Z}})) { {}^{ \wedge }_{p} } \cong H_{\operatorname{Sing}}({\mathbb{H}}_g / {\mathsf{Sp}}_{2g}({\mathbb{Z}})){ {}^{ \wedge }_{p} } \cong H_{\operatorname{Sing}}({\mathcal{A}_g}({\mathbb{C}}); {\mathbb{Z}}_p) \cong H_\text{ét}({\mathcal{A}_g}_{g, {\mathbb{C}}}; {\mathbb{Z}}_p) .\end{align*}

  • Can define an algebraic stack \(A_g\) over any \(S\in {\mathsf{Sch}}\), classifying flat families of PPAVs.

    • Can write \(\mathcal{A}_{g, {\mathbb{C}}} = \mathcal{A}_{g, {\mathbb{Q}}} { \underset{\scriptscriptstyle {\operatorname{Spec}{\mathbb{Q}}} }{\times} } \operatorname{Spec}{\mathbb{C}}\) which has a natural action of \(\mathop{\mathrm{Aut}}({\mathbb{C}}/{\mathbb{Q}})\) on the 2nd factor, which factors to \({ \mathsf{Gal}} ({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }/{\mathbb{Q}})\).

    • Produces an action on \(H_\text{ét}( \mathcal{A}_{g, {\mathbb{C}}}; {\mathbb{Z}}_p)\).

  • attachments/image_2021-04-29-13-39-01.png

  • classifying space of a category: geometric realization of the nerve.

  • Constructing K-theory :

    attachments/image_2021-04-29-13-40-18.png

  • Can also do Quillen’s plus construction : \(K_i({\mathbb{Z}}) = \pi_i( {\mathbf{B}}\operatorname{GL}_{\infty}({\mathbb{Z}})^+)\), which is stable homology?

  • 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!

  • Question: where are \({\mathbf{B}}\Gamma\) Shimura variety?

  • Lie groups are homotopy equivalent to their maximal compact subgroups

  • 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

  • Galois action unramified except at \(p\) implies it factors through \(\pi_1 {\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\)?

  • Guessing the Galois action: trivial on the first factor, the cyclotomic character on the second.

    attachments/image_2021-04-29-13-53-53.png

  • 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:

    attachments/image_2021-04-29-13-54-39.png

  • Informal statement of main theorem:

    attachments/image_2021-04-29-13-55-03.png

  • Consider the category of extensions and find a universal object.

  • \(\operatorname{Spec}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\): integers punctured at \(p\), see spec Z as a curve

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