# 2021-05-10

## 12:34

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

• See Atiyah duality : define the dual of $$M$$ as $$M^{-{\mathbf{T}}M}$$, the Thom space of (minus) the tangent bundle.

• Define the trace :

• Then $$\operatorname{Tr}(\phi) \in { \operatorname{End} }_{\mathsf{C}}(\one, \one)$$ is an endomorphism of the unit.

• 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.

• $${\mathbb{S}}= \one \in {\mathsf{ho}}{\mathsf{Sp}}$$.

• 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*}

• Rationality of $$\zeta$$:

• Use hocolims to glue spaces, but may not work in schemes.

• Example: take $$X \coloneqq{\mathbf{P}}^n/{\mathbf{P}}^{n-1}$$, then we’d want $$X({\mathbf{C}}) \cong S^{2n}$$ and $$X({\mathbf{R}}) \cong S^n$$

• Problem: this quotient isn’t a scheme. Can freely add these limits.

• We want $${\mathbf{P}}^i / {\mathbf{P}}^{i-1}$$ to be the building blocks or cells

• Morel and Voevodsky, $${\mathbf{A}}^1$$ stable homotopy category over $$k$$, denoted $${\mathsf{SH}}(k)$$.

• 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 $$X({\mathbf{C}})$$.
• Grothendieck-Witt group: formal differences of isomorphism classes of nondegenerate symmetric bilinear forms.

• Allow orthogonal direct sum and orthogonal direct difference.
• 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*}

• See rank, signature, discriminants of quadratic forms.

• Trace here will take values in $${\operatorname{GW}}(k)$$.

• Lefschetz fixed point theorem due to Hoyois:

• Notation: $$dZ^{{\mathbf{A}}^1}(t) = {\frac{\partial }{\partial t}\,} \log \zeta^{{\mathbf{A}}^1}(t) = \sum_{m\geq 1} \operatorname{Tr}(\phi^m)t^{m-1}$$.

• Prop: \begin{align*} \operatorname{rank}dZ^{{\mathbf{A}}^1}(t) = {\frac{\partial }{\partial t}\,} \log .\end{align*}

• 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*}
• See Euler class, Hopf map

• 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:

• Goal: want to formulate monads and operads categorically.

• See lambda sequences, something like a functor $${\mathsf{FinSet}}^{\operatorname{op}}\to \mathsf{C}$$?

• 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:

• See monadic bar construction and monoidal bar construction.

• 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*}

• For $$\sigma$$ the sign representation, $$\int_{S^\sigma} A \simeq\operatorname{THR}(A)$$ for $$E_\sigma{\hbox{-}}C_2$$ spectra.

• Axiomatic approach to factorization homology: take a left Kan extension of the following:

• Can compute Kan extension via the bar construction.

• Theorem: equivariant nonabelian Poincare duality :

• What is the virtual dimension of a bundle?

• $$\operatorname{Pic}$$: subcategory of invertible objects, PIcard groupoid

• Thom spectrum functor:

where $$R{\hbox{-}}$$line is the $$\infty{\hbox{-}}$$category of line bundles up to equivalence?

• Preserves $$G{\hbox{-}}$$colimits, so formally the Thom spectrum functor commutes with factorization homology.

• 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?

#web/quick-notes #projects/notes/seminars #homotopy #AG #homotopy/stable-homotopy #motivic #higher-algebra/THH #homotopy/factorization-homology