# The Galois action on symplectic K-theory

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*}

• Similar relationship between topological K theory and singular cohomology

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

• Isomorphism to $$H^2_{\mathrm{sing}}(X, {\mathbf{Z}}_p(2i)$$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 $${ \mathsf{Gal}} ( { \mkern 1.5mu\overline{\mkern-1.5mu \mathbf{Q} \mkern-1.5mu}\mkern 1.5mu }/ {\mathbf{Q}})$$.

• Interpretation of local system and etale sheaf

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

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

• 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

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

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

• $${\mathbf{B}}\SP_{2g}({\mathbf{Z}})$$ is the etale homotopy type of $${\mathcal{A}_g}$$

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

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

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

• Free quotient of contractible space: homotopy type of $${\mathbf{B}}\SP_{2g}({\mathbf{Z}})$$.

• 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*}

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

• 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 $${ \mathsf{Gal}} ({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbf{Q} \mkern-1.5mu}\mkern 1.5mu }/{\mathbf{Q}})$$.

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

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

• Constructing K-theory :

• Can also do Quillen’s plus construction : $$K_i({\mathbf{Z}}) = \pi_i( {\mathbf{B}}\operatorname{GL}_{\infty}({\mathbf{Z}})^+)$$, 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 $$\pi_1 {\mathbf{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }$$?

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

• 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:

• Informal statement of main theorem:

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

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

#web/quick-notes #web/blog #homotopy #higher-algebra/K-theory #projects/notes/seminars