--- date: 2021-04-29 tags: [ web/quick-notes ] --- # The Galois action on symplectic K-theory Tags: #web/blog #homotopy #higher-algebra/K-theory #projects/notes/seminars > Tony Feng, "The Galois action on symplectic K-theory", EAKTS. > > See ### Notes - Significance of higher $K$ groups: ![attachments/image_2021-04-29-13-18-09.png](attachments/image_2021-04-29-13-18-09.png) - Theorem and consequence of Quillen-Lichtenbaum: \[ K_{4i-2}(\ZZ) \tensor \ZZ_p \cong H^2_{\et}( \ZZ \invert{p}, \ZZ_p(2i)) \] - Similar relationship between [topological K theory](topological%20K%20theory.md) and singular cohomology - Related to $\zeta(1-2i)$ by the [Iwasawa main conjecture](Iwasawa%20main%20conjecture), see [Mazur-Wiles](Mazur-Wiles). - Isomorphism to $H^2_\sing(X, \ZZ_p(2i)$where $X$ is an etale $K(\pi, 1)$ for $\pi \da \pi_1(\ZZ\invert{p})$ which is a quotient of $\Gal( \Qbar / \QQ)$. - Interpretation of [local system](local%20system.md) and [etale sheaf](etale%20sheaf) - This sheaf: $\ZZ_p(1) \da \underset{n}{\directlim}\, \mu_{p^n}$ and $\ZZ_p(i) \da \ZZ_p(1)^{\tensor i}$. - [etale homotopy](etale%20homotopy) of $\OO_K$ for $K$ a [global field](global%20field.md) : a punctured [3-manifold](Unsorted/Three-manifolds%20MOC.md) - Interpret $H^1(\ZZ\invert{p}) = H^1_{\Grp}(\pi_1 G)$ for $G \da \ZZ\invert{p})$, also isomorphic to $\Ext^1_G$: space of 2-dimensional [Galois representations](Galois%20representations.md) - For [congruence subgroup](congruence%20subgroup.md) \( \Gamma\leq \GL_2(\ZZ) \), there is a natural Galois action on $H^*( \Gamma, \ZZ_p)$. - [The Langlands program](Unsorted/Langlands.md) philosophy: all [Galois representations](Galois%20representations.md) are accounted for by the cohomology of [arithmetic groups](arithmetic%20groups) - $\B \SP_{2g}(\ZZ)$ is the [etale homotopy type](etale%20homotopy%20type) of $\Ag$ - [PPAV](PPAV) : a complex torus $\CC^g/ \Lambda$ with a [polarization](polarization.md) : a [symplectic form](symplectic%20form) on the lattice with a positivity condition. Principal: perfect pairing. - Can recover \( \Lambda = H_1(A; \ZZ) \) - Quotient Siegel half-space (contractible) by $\Sp_{2g}(\ZZ)$ to forget choice of basis for \( \Lambda \). Take [stack quotient](stack%20quotient). - Free quotient of contractible space: homotopy type of $\B \SP_{2g}(\ZZ)$. - Some coincidences: \[ H_{\Grp}(\Sp_{2g}(\ZZ)) \pcomplete \cong H_\Sing (\HH_g / \Sp_{2g}(\ZZ))\pcomplete \cong H_\Sing(\Ag(\CC); \ZZ_p) \cong H_\et (\Ag_{g, \CC}; \ZZ_p) .\] - Can define an [algebraic stack](Unsorted/stacks%20MOC.md) $A_g$ over any $S\in \Sch$, classifying [flat families](Flat%20Family.md) of PPAVs. - Can write \( \mathcal{A}_{g, \CC} = \mathcal{A}_{g, \QQ} \fiberprod{\spec \QQ} \spec \CC \) which has a natural action of $\Aut(\CC/\QQ)$ on the 2nd factor, which factors to $\Gal(\Qbar/\QQ)$. - Produces an action on $H_\et( \mathcal{A}_{g, \CC}; \ZZ_p)$. - ![attachments/image_2021-04-29-13-39-01.png](attachments/image_2021-04-29-13-39-01.png) - [classifying space](classifying%20space.md) of a category: [geometric realization](geometric%20realization.md) of the [nerve](nerve.md). - Constructing [K-theory](K-theory.md) : ![attachments/image_2021-04-29-13-40-18.png](attachments/image_2021-04-29-13-40-18.png) - Can also do Quillen's [plus construction](plus%20construction) : $K_i(\ZZ) = \pi_i( \B\GL_{\infty}(\ZZ)^+)$, which is stable homology? - Original paper title: Galois action on stable *cohomology* of $\Ag$. Need $p\gg i$, otherwise proof had many issues. Passing to stable homotopy theory made things easier! - Question: where are $\B \Gamma$ [Shimura variety](Shimura%20variety.md)? - Lie groups are homotopy equivalent to their maximal compact subgroups - Hodge map: came from taking the [Hodge bundle](Hodge%20bundle.md) and its [Chern class](Chern%20class.md), where the fiber over every $A\in \Ag$ is $H^0(A, \Omega_A^1)$ the [Hodge cohomology](Hodge%20cohomology) - Galois action [unramified](unramified.md) except at $p$ implies it factors through $\pi_1 \ZZ\invert{p}$? - Guessing the Galois action: trivial on the first factor, the [cyclotomic character](cyclotomic%20character.md) on the second. ![attachments/image_2021-04-29-13-53-53.png](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](attachments/image_2021-04-29-13-54-39.png) - Informal statement of main theorem: ![attachments/image_2021-04-29-13-55-03.png](attachments/image_2021-04-29-13-55-03.png) - Consider the category of extensions and find a universal object. - $\spec \ZZ\invert{p}$: integers punctured at $p$, see [spec Z as a curve](Unsorted/spec%20Z%20as%20a%20curve.md)