--- date: 2021-05-10 tags: [ web/quick-notes ] --- # 2021-05-10 ## 12:34 We haven't been able to classify the rational points on [modular curves](modular%20curves)! ## Kirsten Wickelgren, Zeta functions and a quadratic enrichment. > Reference: Kirsten Wickelgren, Colloquium Presentation: zeta functions and a quadratic enrichment. [Rational Points and Galois Representations workshop](https://www.pitt.edu/~caw203/DioGal2021.html) Tags: #projects/notes/seminars #homotopy #AG #homotopy/stable-homotopy #motivic Refs: [motivic homotopy](motivic%20homotopy.md) - See [dualizable objects in a category](dualizable.md) \begin{tikzcd} R && {V\tensor V\dual} && {V\tensor V\dual} && R \\ 1 && {\sum e_i \tensor e_i\dual} && {v\tensor \alpha} && {\alpha(v)} \arrow[from=1-1, to=1-3] \arrow["{\phi\tensor 1}", from=1-3, to=1-5] \arrow["\ev", from=1-5, to=1-7] \arrow[from=2-1, to=2-3] \arrow[from=2-3, to=2-5] \arrow[from=2-5, to=2-7] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCJSIl0sWzIsMCwiVlxcdGVuc29yIFZcXGR1YWwiXSxbNCwwLCJWXFx0ZW5zb3IgVlxcZHVhbCJdLFs2LDAsIlIiXSxbMCwxLCIxIl0sWzIsMSwiXFxzdW0gZV9pIFxcdGVuc29yIGVfaVxcZHVhbCJdLFs0LDEsInZcXHRlbnNvciBcXGFscGhhIl0sWzYsMSwiXFxhbHBoYSh2KSJdLFswLDFdLFsxLDIsIlxccGhpXFx0ZW5zb3IgMSJdLFsyLDMsIlxcZXYiXSxbNCw1XSxbNSw2XSxbNiw3XV0=) - Works more generally for a [symmetric monoidal category](symmetric%20monoidal%20category) - Finite dimensionality is replaced by objects being [dualizable](dualizable.md), so for \[ \one & \mapsvia{m} A\tensor B \\ B\tensor A & \mapsvia{\eps} \one ,\] require \begin{tikzcd} A && {A\tensor B \tensor A} && A \\ \\ B && {B\tensor A \tensor B} && B \arrow["{m\tensor 1}", from=1-1, to=1-3] \arrow["{1\tensor \eps}", from=1-3, to=1-5] \arrow["\id"', curve={height=-30pt}, from=1-1, to=1-5] \arrow["{1\tensor m}"', from=3-1, to=3-3] \arrow["{\eps \tensor 1}"', from=3-3, to=3-5] \arrow[curve={height=30pt}, from=3-1, to=3-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwwLCJBIl0sWzIsMCwiQVxcdGVuc29yIEIgXFx0ZW5zb3IgQSJdLFs0LDAsIkEiXSxbMCwyLCJCIl0sWzIsMiwiQlxcdGVuc29yIEEgXFx0ZW5zb3IgQiJdLFs0LDIsIkIiXSxbMCwxLCJtXFx0ZW5zb3IgMSJdLFsxLDIsIjFcXHRlbnNvciBcXGVwcyJdLFswLDIsIlxcaWQiLDIseyJjdXJ2ZSI6LTV9XSxbMyw0LCIxXFx0ZW5zb3IgbSIsMl0sWzQsNSwiXFxlcHMgXFx0ZW5zb3IgMSIsMl0sWzMsNSwiIiwwLHsiY3VydmUiOjV9XV0=) - See [Atiyah duality](Atiyah%20duality) : define the dual of $M$ as $M^{-\T M}$, the [Thom space](Thom%20space.md) of (minus) the tangent bundle. - Define the [trace](trace%20(monoidal%20categories).md) : \begin{tikzcd} 1 && {A\tensor \DD A} && {A\tensor \DD A} && {\DD A \tensor A} && 1 \arrow["\eps", from=1-7, to=1-9] \arrow["{\tau \quad \sim}", from=1-5, to=1-7] \arrow["{\phi \tensor 1}", from=1-3, to=1-5] \arrow["m", from=1-1, to=1-3] \arrow["{\Tr(\phi)}"', curve={height=30pt}, from=1-1, to=1-9] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMCwwLCIxIl0sWzIsMCwiQVxcdGVuc29yIFxcREQgQSJdLFs0LDAsIkFcXHRlbnNvciBcXEREIEEiXSxbNiwwLCJcXEREIEEgXFx0ZW5zb3IgQSJdLFs4LDAsIjEiXSxbMyw0LCJcXGVwcyJdLFsyLDMsIlxcdGF1IFxccXVhZCBcXHNpbSJdLFsxLDIsIlxccGhpIFxcdGVuc29yIDEiXSxbMCwxLCJtIl0sWzAsNCwiXFxUcihcXHBoaSkiLDIseyJjdXJ2ZSI6NX1dXQ==) - Then $\Tr(\phi) \in \Endo_{\cat C}(\one, \one)$ is an endomorphism of the unit. - Example: [Lefschetz fixed point theorem](Lefschetz%20fixed%20point%20theorem.md), \[ \Tr(\phi) = \sum_{x\in M, \phi(x) = x} \ind_x \phi \in \Endo_{\ho\Sp}(\one) \mapsvia{\deg \,\, \sim} \ZZ ,\] where we take the degree of a map between spheres. - $\SS = \one \in \ho\Sp$. - Use that $H^*(\wait, \QQ)$ preserves tensor products, and apply the Kunneth formula: \[ H^*(\Tr(\phi)) &= \Tr(H^*(\phi)) \\ \implies \sum (-1)^i \Tr( H^i(\phi); H^i(M) \selfmap ) &= \sum_{x\in M, \phi(x) = x} \ind_x \varphi .\] - Rationality of $\zeta$: ![](attachments/image_2021-05-10-13-36-13.png) - Use hocolims to glue spaces, but may not work in schemes. - Example: take $X \da \PP^n/\PP^{n-1}$, then we'd want $X(\CC) \cong S^{2n}$ and $X(\RR) \cong S^n$ - Problem: this quotient isn't a [scheme](scheme.md). Can freely add these limits. - We want $\PP^i / \PP^{i-1}$ to be the building blocks or cells - Morel and Voevodsky, $\AA^1$ [stable homotopy category](stable%20homotopy%20category) over $k$, denoted $\SH(k)$. - Take an analog of degree, the Morel degree: \[ \deg: [\PP^n/\PP^{n-1}, \PP^n/\PP^{n-1} ] \mapsvia{} \GW(k) .\] - Recovers degree on $X(\CC)$. - [Grothendieck-Witt](Grothendieck-Witt) group: formal differences of isomorphism classes of nondegenerate symmetric [bilinear forms](bilinear%20forms). - Allow orthogonal direct sum and orthogonal direct *difference*. - Special form: the hyperbolic form \[ \gens{1} + \gens{-1} = \gens{a} + \gens{-a} = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} .\] - See [rank](rank), [signature](signature.md), [discriminants](Unsorted/discriminants.md) of [quadratic forms](quadratic%20form.md). - Trace here will take values in $\GW(k)$. - [Lefschetz fixed point theorem](Lefschetz%20fixed%20point%20theorem.md) due to Hoyois: ![attachments/image_2021-05-10-13-49-00.png](attachments/image_2021-05-10-13-49-00.png) - Notation: $dZ^{\AA^1}(t) = \dd{}{t} \log \zeta^{\AA^1}(t) = \sum_{m\geq 1} \Tr(\phi^m)t^{m-1}$. - Prop: \[ \rank dZ^{\AA^1}(t) = \dd{}{t} \log .\] - See Kapranov [motivic zeta function](motivic%20zeta%20function.md) : - Define $\K_0(\Var_k)$ to be the group completion of varieties under cut-and-paste - Define \[ Z_X^m(t) \da \sum_{m\geq 0} [\Sym^m X] t^m \in \K_0 (\Var_k) \fps{t} .\] - Define an [Euler characteristic](Euler%20characteristic) \[ \chi_C^{\AA^1}: K_0(\Var_k) \to \GW(k) .\] - See [Euler class](Euler%20class.md), [Hopf map](Hopf%20map) - Major point: this is genuinely something new, isn't just recovered by taking the compactly supported euler characteristic: ![](attachments/image_2021-05-10-13-56-27.png) - Defines a [zeta function](zeta%20function) for any endomorphism of any variety. Doesn't need to be over $\FF_p$, and doesn't need to have Frobenius! ## Foling Zou, Nonabelian Poincare duality theorem and equivariant factorization homology of Thom spectra Tags: #higher-algebra/THH #homotopy/factorization-homology #projects/notes/seminars Refs: [nonabelian Poincare duality](nonabelian%20Poincare%20duality.md), [factorization homology](factorization%20homology.md) > Reference: Foling Zou, Nonabelian Poincare duality theorem and equivariant factorization homology of Thom spectra. MIT Topology Seminar. - [factorization homology](factorization%20homology.md) setup: ![attachments/image_2021-05-10-16-35-34.png](attachments/image_2021-05-10-16-35-34.png) - Goal: want to formulate [monads](monads) and [operads](operads) categorically. - See [lambda sequences](lambda%20sequences), something like a functor $\Finset\op \to \cat{C}$? - See [Day convolution](Day%20convolution.md) as an example of a monoidal product. - Another example: the [Kelly product](Kelly%20product) : ![](attachments/image_2021-05-10-16-45-36.png) - Can define operads and reduced [operads](operads) as [monoids](monoids) in certain categories: ![](attachments/image_2021-05-10-16-47-55.png) - See [monadic bar construction](monadic%20bar%20construction) and monoidal [bar construction](bar%20construction.md). - Examples of [factorization homology](factorization%20homology.md) : \[ \int_{S^1}A &&\homotopic \mTHH(A) \\ \int_{T^n}A &&\homotopic \mTHH^n(A) && \text{iterated THH} .\] - For $\sigma$ the [sign representation](sign%20representation), $\int_{S^\sigma} A \homotopic \operatorname{THR}(A)$ for $E_\sigma\dash C_2$ spectra. See Horev, Hessolholt-Madsen. - Axiomatic approach to factorization homology: take a left [Kan extension](Kan%20extension.md) of the following: ![](attachments/image_2021-05-10-17-05-53.png) - Can compute [Kan extension](Kan%20extension.md) via the [bar construction](bar%20construction.md). - Theorem: [equivariant](equivariant.md) [nonabelian Poincare duality](nonabelian%20Poincare%20duality.md) : ![](attachments/image_2021-05-10-17-09-08.png) - What is the [virtual dimension](virtual%20dimension) of a bundle? - $\Pic$: subcategory of invertible objects, [PIcard groupoid](Picard%20group.md) - [Thom spectrum](Thom%20spectrum) functor: ![](attachments/image_2021-05-10-17-13-16.png) where $R\dash$line is the $\infty\dash$category of line bundles up to equivalence? - Preserves $G\dash$colimits, so formally the Thom spectrum functor commutes with factorization homology. - In proof of theorem, use [nonabelian Poincare duality](nonabelian%20Poincare%20duality.md) to reduce a complicated gadget to a mapping space. - Also appears as a step in a later proof identifying $\mTHH_{C_2} (\mH\FF_2) \approx \mH \FF_2 \smashprod (\Loop S^3)_+$. ![](attachments/image_2021-05-10-17-27-00.png) > For $\operatorname{THR}$ on the algebra side, see [Teena Gerhardt's work](https://users.math.msu.edu/users/gerhar18/home.html)? Haynes Miller suggests looking at the [Unsorted/de Rham-Witt](Unsorted/de%20Rham-Witt.md)?