The Galois action on symplectic K-theory
Tags: #blog #number_theory #homotopy_theory #k_theory #talk_notes
Tony Feng, “The Galois action on symplectic K-theory”, EAKTS. https://www.youtube.com/watch?v=Ulm8bCcuW2Q
Notes
- Significance of higher \(K\) groups:
- 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*}
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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 \(H^2_{\mathrm{sing}}(X, {\mathbb{Z}}_p(2i)\)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 \({ \mathsf{Gal}} ( { \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }/ {\mathbb{Q}})\).
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Interpretation of local system and etale sheaf
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This sheaf: \({\mathbb{Z}}_p(1) \coloneqq\underset{n}{\directlim}\, \mu_{p^n}\) and \({\mathbb{Z}}_p(i) \coloneqq{\mathbb{Z}}_p(1)^{\otimes i}\).
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etale homotopy of \({\mathcal{O}}_K\) for \(K\) a global field : a punctured [[Three-manifold|3-manifold]]
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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
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For congruence subgroup \(\Gamma\leq \operatorname{GL}_2({\mathbb{Z}})\), there is a natural Galois action on \(H^*( \Gamma, {\mathbb{Z}}_p)\).
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The Langlands Program philosophy: all Galois representations are accounted for by the cohomology of arithmetic groups
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\({\mathbf{B}}{\operatorname{SP}}_{2g}({\mathbb{Z}})\) is the etale homotopy type of \({\mathcal{A}_g}\)
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PPAV : a complex torus \({\mathbb{C}}^g/ \Lambda\) with a polarization : a symplectic form on the lattice with a positivity condition. Principal: perfect pairing.
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Can recover \(\Lambda = H_1(A; {\mathbb{Z}})\)
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Quotient Siegel half-space (contractible) by \({\mathsf{Sp}}_{2g}({\mathbb{Z}})\) to forget choice of basis for \(\Lambda\). Take stack quotient.
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Free quotient of contractible space: homotopy type of \({\mathbf{B}}{\operatorname{SP}}_{2g}({\mathbb{Z}})\).
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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*}
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Can define an algebraic stack \(A_g\) over any \(S\in {\mathsf{Sch}}\), classifying [[Flat Family|flat families]] of PPAVs.
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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 \({ \mathsf{Gal}} ({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }/{\mathbb{Q}})\).
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Produces an action on \(H_\text{ét}( \mathcal{A}_{g, {\mathbb{C}}}; {\mathbb{Z}}_p)\).
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attachments/image_2021-04-29-13-39-01.png❗ -
classifying space of a category: geometric realization of the nerve.
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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?
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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 \(\pi_1 {\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\)?
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Guessing the Galois action: trivial on the first factor, the cyclotomic character on the second.
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.
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\(\operatorname{Spec}{\mathbb{Z}}{ \left[ { \scriptstyle \frac{1}{p} } \right] }\): integers punctured at \(p\)
Yves André (CNRS), “On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)”
Tags: #seminar_notes #blog
Reference: Yves André (CNRS), “On the canonical, fpqc and finite topologies: classical questions, new answers (and conversely)”. Princeton/IAS NT seminar.
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What does it mean for an algebra to be faithfully flat over another algebra?
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p-adic Hilbert functor : see Bhatt-Lurie.
- Use this to get “almost” results, then use prismatic cohomology]] techniques (where one has Frobenius) to remove the “almost”.
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Extracting \(p\)th roots? Passing from \(k[g_1, \cdots, g_n]\) to \(k[g_1^{1/p}, \cdots, g_n^{1/p}]\), I think…
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F-pure and strongly F-regular singularities are characteristic \(p\) analogs of log-canonical and log-terminal singularities in the minimal model program.
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Tilting]] : pass from mixed characteristic to characteristic \(p\). Try to use simpler proofs/theorems from characteristic \(p\) situation.
- Going forward: some limiting process after inverting \(p\)..? Going backward: take Witt Vectors]].
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What properties of schemes descend along faithfully flat morphism? See EGA. However, what properties descend for the fpqc topology?
- What is a faithfully flat morphism?
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See Faltings almost purity theorem.
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Commutative algebra: see excellent regular domains, integral vs algebraic closures. - Cohen-Macaulay rings and modules
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Can have \(\mathrm{\operatorname{fpqc}}\) coverings that are not fppf coverings.
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What is a regular scheme?
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Theorem: any finite covering of a regular scheme is an \(\mathrm{\operatorname{fpqc}}\) covering.
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Very nontrivial in characteristic zero.
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Noether normalization can show some finite coverings of \({\mathbb{A}}^3_{/k}\) are not \(\mathrm{\operatorname{fppf}}\) coverings.
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Sometimes local or coherent cohomology classes
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Grothendieck’s descent?
faithfully flat implies something is an equivalence.
Ribet, “Class groups and Galois representations”
Tags: #seminar_notes
Reference: Ribet, “Class Groups and Galois Representations”. https://math.berkeley.edu/~ribet/herbrand.pdf
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Alternate definition of ideal class group : the group of fractional ideals.
- Defined as \({\mathbb{Z}}[\operatorname{mSpec}{\mathcal{O}}_K]\) (the free \({\mathbb{Z}}{\hbox{-}}\)module on maximal ideals) modulo principal fractional ideal
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What is the maximal unramified extension, i.e. the Hilbert class field?
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The Artin map from class field theory : \({ \operatorname{Cl}} (K) \xrightarrow{\sim} { \mathsf{Gal}} (H/K)\) where \({\mathfrak{p}}\mapsto \mathop{\mathrm{Frob}}_{{\mathfrak{p}}}\).
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Set \(G_k \coloneqq{ \mathsf{Gal}} ({ \mkern 1.5mu\overline{\mkern-1.5mu \mathbb{Q} \mkern-1.5mu}\mkern 1.5mu }/K)\), then \({ \operatorname{Cl}} (K)\) is a quotient of \(G_k^{{\mathsf{Ab}}}\).
- Equivalently, \({ \operatorname{Cl}} (K) {}^{ \vee }\leq G_k {}^{ \vee }\) where \(({-}) {}^{ \vee }\coloneqq\mathop{\mathrm{Hom}}_{{\mathsf{Top}}{\mathsf{Grp}}}({-}, {\mathbb{C}}^{\times})\). I.e. take continuous characters.
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Open question: are there infinitely many quadratic fields \(K\) for which \({ \operatorname{Cl}} (K) = 0\)
\begin{align*} \zeta_K \coloneqq\prod_{{\mathfrak{p}}\in \operatorname{mSpec}{\mathcal{O}}_K}(1 - N({\mathfrak{p}})^{-s} )^{-1} .\end{align*}
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Note: guessing about the indexing set here. Original source just indexes over \({\mathfrak{p}}\)…
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\(\mathop{\mathrm{Res}}_{s=1} \zeta_K\) involves \(h_k \coloneqq{\sharp}{ \operatorname{Cl}} (K)\).
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Serre stresses: use functional equation to look at \(s=0\) instead of \(s=1\)! Leads to cleaner/simpler formulas.
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Kummer theory proved FLT for exponent \(p\) for regular primes, i.e. \(\gcd(h_K, p) = 1\).
- Kummer’s criterion: \(p\) is regular iff \(p\) divides none of the numerators of some Bernoulli numbers.
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What is the Teichmuller character?
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See Birch and Swinnerton-Dyer conjecture
- Define the Tate-Shafarevich group as \(\Sha(E/{\mathbb{Q}})\) for an elliptic curve, then
\begin{align*} {\sharp}\Sha(E/{\mathbb{Q}}) \underset{?}{=} \qty{ L(E, 1) \over \Omega} \qty{ {\sharp}(E({\mathbb{Q}}))^2 \over \prod_\ell w_\ell} ,\end{align*}
where \(\Omega\) is a period and \(w_\ell\) are the local Tamagawa numbers.
- Need lower bounds of sizes of [[class group|class groups]]. Might be able to use elliptic curves, congruences between Eisenstein series and cusp forms on \(\operatorname{U}(2, 2)\) – can obtain 4-dimensional Galois representations that lead to nontrivial elements of \(\Sha\).