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We haven’t been able to classify the rational points on modular curves!
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
Tags: #projects/notes/seminars #homotopy #AG #homotopy/stable-homotopy #motivic Refs: motivic homotopy
- See dualizable objects in a category
- Works more generally for a symmetric monoidal category
- Finite dimensionality is replaced by objects being dualizable, so for \begin{align*} \one & \xrightarrow{m} A\otimes B \\ B\otimes A & \xrightarrow{{\varepsilon}} \one ,\end{align*} require
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See Atiyah duality : define the dual of \(M\) as \(M^{-{\mathbf{T}}M}\), the Thom space of (minus) the tangent bundle.
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Define the trace :
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Then \(\operatorname{Tr}(\phi) \in { \operatorname{End} }_{\mathsf{C}}(\one, \one)\) is an endomorphism of the unit.
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Example: Lefschetz fixed point theorem, \begin{align*} \operatorname{Tr}(\phi) = \sum_{x\in M, \phi(x) = x} \operatorname{Ind}_x \phi \in { \operatorname{End} }_{{\mathsf{ho}}{\mathsf{Sp}}}(\one) \xrightarrow{\deg \,\, \sim} {\mathbf{Z}} ,\end{align*} where we take the degree of a map between spheres.
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\({\mathbb{S}}= \one \in {\mathsf{ho}}{\mathsf{Sp}}\).
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Use that \(H^*({-}, {\mathbf{Q}})\) preserves tensor products, and apply the Kunneth formula: \begin{align*} H^*(\operatorname{Tr}(\phi)) &= \operatorname{Tr}(H^*(\phi)) \\ \implies \sum (-1)^i \operatorname{Tr}( H^i(\phi); H^i(M) {\circlearrowleft}) &= \sum_{x\in M, \phi(x) = x} \operatorname{Ind}_x \varphi .\end{align*}
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Rationality of \(\zeta\):
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Use hocolims to glue spaces, but may not work in schemes.
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Example: take \(X \coloneqq{\mathbf{P}}^n/{\mathbf{P}}^{n-1}\), then we’d want \(X({\mathbf{C}}) \cong S^{2n}\) and \(X({\mathbf{R}}) \cong S^n\)
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Problem: this quotient isn’t a scheme. Can freely add these limits.
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We want \({\mathbf{P}}^i / {\mathbf{P}}^{i-1}\) to be the building blocks or cells
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Morel and Voevodsky, \({\mathbf{A}}^1\) stable homotopy category over \(k\), denoted \({\mathsf{SH}}(k)\).
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Take an analog of degree, the Morel degree: \begin{align*} \deg: [{\mathbf{P}}^n/{\mathbf{P}}^{n-1}, {\mathbf{P}}^n/{\mathbf{P}}^{n-1} ] \xrightarrow{} {\operatorname{GW}}(k) .\end{align*}
- Recovers degree on \(X({\mathbf{C}})\).
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Grothendieck-Witt group: formal differences of isomorphism classes of nondegenerate symmetric bilinear forms.
- Allow orthogonal direct sum and orthogonal direct difference.
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Special form: the hyperbolic form \begin{align*} \left\langle{1}\right\rangle + \left\langle{-1}\right\rangle = \left\langle{a}\right\rangle + \left\langle{-a}\right\rangle = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} .\end{align*}
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See rank, signature, discriminants of quadratic forms.
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Trace here will take values in \({\operatorname{GW}}(k)\).
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Lefschetz fixed point theorem due to Hoyois:
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Notation: \(dZ^{{\mathbf{A}}^1}(t) = {\frac{\partial }{\partial t}\,} \log \zeta^{{\mathbf{A}}^1}(t) = \sum_{m\geq 1} \operatorname{Tr}(\phi^m)t^{m-1}\).
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Prop: \begin{align*} \operatorname{rank}dZ^{{\mathbf{A}}^1}(t) = {\frac{\partial }{\partial t}\,} \log .\end{align*}
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See Kapranov motivic zeta function :
- Define \({\mathsf{K}}_0({\mathsf{Var}}_k)\) to be the group completion of varieties under cut-and-paste
- Define \begin{align*} Z_X^m(t) \coloneqq\sum_{m\geq 0} [\operatorname{Sym}^m X] t^m \in {\mathsf{K}}_0 ({\mathsf{Var}}_k) {\left[\left[ t \right]\right] } .\end{align*}
- Define an Euler characteristic \begin{align*} \chi_C^{{\mathbf{A}}^1}: K_0({\mathsf{Var}}_k) \to {\operatorname{GW}}(k) .\end{align*}
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See Euler class, Hopf map
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Major point: this is genuinely something new, isn’t just recovered by taking the compactly supported euler characteristic:
- Defines a zeta function for any endomorphism of any variety. Doesn’t need to be over \({ \mathbf{F} }_p\), 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, factorization homology
Reference: Foling Zou, Nonabelian Poincare duality theorem and equivariant factorization homology of Thom spectra. MIT Topology Seminar.
- factorization homology setup:
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Goal: want to formulate monads and operads categorically.
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See lambda sequences, something like a functor \({\mathsf{FinSet}}^{\operatorname{op}}\to \mathsf{C}\)?
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See Day convolution as an example of a monoidal product.
- Another example: the Kelly product :
- Can define operads and reduced operads as monoids in certain categories:
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See monadic bar construction and monoidal bar construction.
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Examples of factorization homology : \begin{align*} \int_{S^1}A &&\simeq{\operatorname{THH}}(A) \\ \int_{T^n}A &&\simeq{\operatorname{THH}}^n(A) && \text{iterated THH} .\end{align*}
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For \(\sigma\) the sign representation, \(\int_{S^\sigma} A \simeq\operatorname{THR}(A)\) for \(E_\sigma{\hbox{-}}C_2\) spectra.
See Horev, Hessolholt-Madsen.
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Axiomatic approach to factorization homology: take a left Kan extension of the following:
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Can compute Kan extension via the bar construction.
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Theorem: equivariant nonabelian Poincare duality :
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What is the virtual dimension of a bundle?
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\(\operatorname{Pic}\): subcategory of invertible objects, PIcard groupoid
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Thom spectrum functor:
where \(R{\hbox{-}}\)line is the \(\infty{\hbox{-}}\)category of line bundles up to equivalence?
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Preserves \(G{\hbox{-}}\)colimits, so formally the Thom spectrum functor commutes with factorization homology.
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In proof of theorem, use nonabelian Poincare duality to reduce a complicated gadget to a mapping space.
- Also appears as a step in a later proof identifying \({\operatorname{THH}}_{C_2} ({ \mathsf{H} }{ \mathbf{F} }_2) \approx { \mathsf{H} }{ \mathbf{F} }_2 \wedge({\Omega}S^3)_+\).
For \(\operatorname{THR}\) on the algebra side, see Teena Gerhardt’s work? Haynes Miller suggests looking at the Unsorted/de Rham-Witt?