On the automorphic side of the K-theoretic Artin symbol

Abstract

Clausen has constructed a homotopical enrichment of the Artin reciprocity symbol in class field theory. On the Galois side, Selmer K-homology replaces the abelianized Galois group, while on the automorphic side the K-theory of locally compact vector spaces replaces classical idelic objects. We supply proofs for some predictions of Clausen regarding the automorphic side.

Appendix A. Auxiliary computations

1.1 A.1. Locally compact modules over finite fields

Proposition A.1

Let ${\mathsf {A}}$ be any stable $\infty$-category and $K:{\mathrm{Cat}}_{\infty }^{{\mathrm{ex}}} \rightarrow {\mathsf {A}}$ be a localizing invariant (in the sense of [3]). Then there is an equivalence $K({\mathsf {LCA}}_{{\mathbb {F}} _{q}})\overset{\sim }{\longrightarrow }\Sigma K({\mathbb {F}}_{q})$.

Proof

(Step 1) Let G be in ${\mathsf {LCA}}_{{\mathbb {F}}_{q}}$. Pick a number field F, denote its ring of integers by ${\mathcal {O}}$ and choose a prime P such that ${\mathcal {O}}/P\cong {\mathbb {F}}_{q}$. Then via ${\mathcal {O}} \rightarrow {\mathcal {O}}/P\cong {\mathbb {F}}_{q}$ we may also regard G as an object in ${\mathsf {LCA}}_{{\mathcal {O}}}$ and Levin’s structure theory applies, [14, Theorem 2 and Theorem 3]. As G is p-torsion (genuinely, not just topologically), there cannot be a vector ${\mathcal {O}}$-module contribution, i.e. there exists a compact clopen ${\mathcal {O}}$-submodule C such that

$$\begin{aligned} C\hookrightarrow G\twoheadrightarrow D \end{aligned}$$

(A.1)

in exact in ${\mathsf {LCA}}_{{\mathcal {O}}}$ with D discrete. As G is an ${\mathcal {O}}$-module annihilated by P, so are C and D. Thus, D is a discrete ${\mathbb {F}}_{q}$-vector space, and C the Pontryagin dual of a discrete ${\mathbb {F}}_{q}$-vector space. (Step 2) There is an exact functor

$$\begin{aligned} \Gamma :\mathsf {Tate}({\mathsf {Vect}}_{fd}({\mathbb {F}}_{q}))\longrightarrow {\mathsf {LCA}}_{{\mathbb {F}}_{q}}\text {,} \end{aligned}$$

sending a formal ind-pro limit of finite-dimensional ${\mathbb {F}}_{q}$-vector spaces to its evaluation in ${\mathsf {LCA}}$. The construction is a mild variation of the analogous functor $\gamma :\mathsf {Tate}({\mathsf {Mod}} _{fin}({\mathcal {O}}))\rightarrow {\mathsf {LCA}}_{{\mathcal {O}}}$ which was set up in [6, Sect. 6], so we shall not repeat it here. It suffices to observe that ${\mathsf {Vect}}_{fd}({\mathbb {F}}_{q})$ can be regarded as a fully exact subcategory of ${\mathsf {Mod}}_{fin}({\mathcal {O}})$ and that the topological ${\mathcal {O}}$-module structure on $\gamma (G)$ is still annihilated by P, so we may regard it as an ${\mathcal {O}}/P\cong {\mathbb {F}}_{q}$-vector space structure. The functor $\Gamma$ is fully faithful. It is also essentially surjective since by Step 1 every G has a presentation as in Eq. A.1, but this corresponds precisely to the property to have a lattice in the sense of [4], see Definition 5.5 and Theorem 5.6 loc. cit. Thus, the computation of the invariant K reduces to the corresponding computation for the Tate category. By Saito’s delooping theorem [18], we obtain

$$\begin{aligned} K({\mathsf {LCA}}_{{\mathbb {F}}_{q}})\underset{\Gamma }{\overset{\sim }{\longrightarrow }}K(\mathsf {Tate}({\mathsf {Vect}}_{fd}({\mathbb {F}}_{q} )))\overset{\sim }{\longrightarrow }\Sigma K({\mathsf {Vect}}_{fd}({\mathbb {F}} _{q}))=\Sigma K({\mathbb {F}}_{q})\text {,} \end{aligned}$$

giving the claim. Saito’s result is only stated for non-connective K-theory, but the proof generalizes.$\square$

1.2 A.2. Locally compact modules over local fields

In this section, exceptionally, let F be a finite extension of the p-adics ${\mathbb {Q}}_{p}$ for some prime number p.

Definition A.2

Let ${\mathsf {LCA}}_{F,{\mathrm{top}}}$ be the category of locally compact topological F-modules. Unlike in the rest of this text, we demand that the scalar multiplicaton

$$\begin{aligned} F\times M\longrightarrow M \end{aligned}$$

gives a topological F-module structure, where F is equipped with its valuation topology (rather than the discrete one!).

Lemma A.3

The category ${\mathsf {LCA}}_{F,{\mathrm{top}}}$ is quasi-abelian, and in particular naturally an exact category. If ${\mathsf {LCA}}_{F}$ denotes (as usual in this paper) the category of locally compact F-modules, but where F is read with the discrete topology, then there is an exact functor

$$\begin{aligned} {\mathsf {LCA}}_{F,{\mathrm{top}}}\longrightarrow {\mathsf {LCA}}_{F}\text {.} \end{aligned}$$

Proof

Being quasi-abelian can be proven as by Hoffmann and Spitzweck [10, Prop. 1.2] for ${\mathsf {LCA}}$ plain, and the exact structure then stems from [7, Prop. 4.4].$\square$

Proposition A.4

Suppose F is a finite extension of the p-adics ${\mathbb {Q}}_{p}$ for some prime number p. Then there is an exact equivalence of exact categories

$$\begin{aligned} {\mathsf {LCA}}_{F,{\mathrm{top}}}\overset{\sim }{\longrightarrow }{\mathsf {Vect}}_{fd}(F)\text {.} \end{aligned}$$

Proof

(Step 1) Although a direct proof is possible, we will prove this by reducing it to our formalism for number fields. Let $F_{0}$ be a number field such that

$$\begin{aligned} F_{0}\otimes _{{\mathbb {Q}}}{\mathbb {Q}}_{p}\cong F\text {.} \end{aligned}$$

(Proof: Such a number field always exists. Let f be a minimal polynomial generating the extension $F/{\mathbb {Q}}_{p}$. Then $f\in {\mathbb {Q}}_{p}[X]$ and we may pick a sequence of polynomials $f_{n} \in {\mathbb {Q}}[X]$ of the same degree with rational coefficients such that $\lim _{n\rightarrow \infty }f_{n}=f$ in the p-adic topology on the coefficients. Such a sequence exists as ${\mathbb {Q}}$ is dense in ${\mathbb {Q}}_{p}$ and the space of coefficients is ${\mathbb {Q}}_{p}^{\deg f}$. Even though $(f_{n})_{n}$ need not become stationary as a sequence of polynomials, nor the sequence of number fields $T_{n}$ obtained by adjoining all roots of $f_{n}$, the sequence $T_{n}\cdot {\mathbb {Q}}_{p}$ must become stationary by Krasner’s Lemma, and by convergence the limit must be F; take $F_{0}$ to be any number field $T_{n}$ such that $T_{n}\cdot {\mathbb {Q}}_{p}=F$). Next, pick an element $\pi \in F_{0}$ which has the property that under $F_{0}\hookrightarrow F$ it becomes a uniformizer with respect to the natural valuation on F. (Step 2) Let $G\in {\mathsf {LCA}}_{F,{\mathrm{top}}}$ be an arbitrary object. Since for ${\mathsf {LCA}}_{F,{\mathrm{top}}}$ we had assumed the F-module structure to be continuous with regards to the natural valuation topology on F (instead of just the discrete topology), we have $\lim _{n\rightarrow \infty } \pi ^{n}=0$ in F, and thus for all $g\in G$ we get

$$\begin{aligned} \lim _{n\rightarrow \infty }\pi ^{n}\cdot g=0\text {.} \end{aligned}$$

Since $\pi \in F_{0}$, it follows that G is a topological P-torsion group, for some prime P of ${\mathcal {O}}_{F_{0}}$ (and more concretely, a prime such that $P\cap {\mathbb {Z}}=(p)$, and $P\cdot F=(\pi )F$, i.e. after going to our local field F, the prime becomes principal and is generated by the image of $\pi$). By Proposition 3.4 we conclude that G is a vector-free adelic block. As G was arbitrary, we learn that all objects in ${\mathsf {LCA}}_{F,{\mathrm{top}}}$ are topological P-torsion modules for this concrete P. We will not repeat the proof, but a mild variant of our version of the Braconnier–Vilenkin Theorem (Theorem 3.6) then produces the claimed equivalence of categories (The only necessary changes in the proof are to remove the contributions from all infinite places and all other finite places except for P, since the above argument rules out that any of these occur in ${\mathsf {LCA}}_{F,{\mathrm{top}}}$. The rest of the proof carries over verbatim).$\square$

Corollary A.5

Let ${\mathsf {A}}$ be any stable $\infty$-category and $K:{\mathrm{Cat}}_{\infty }^{{\mathrm{ex}}} \rightarrow {\mathsf {A}}$ be a localizing invariant (in the sense of [3]). Then there is an equivalence $K({\mathsf {LCA}} _{F,{\mathrm{top}}})\overset{\sim }{\longrightarrow }K(F)$.

1.3 A.3. Grothendieck–Witt theory

The following conjecture seems plausible:

Conjecture 2

If $({\mathsf {C}}_{i})_{i\in I}$ (for some index set I) are exact categories with duality, then there is a canonical equivalence of spectra

$$\begin{aligned} GW\left( {\textstyle \prod \nolimits _{i}} {\mathsf {C}}_{i}\right) \overset{\sim }{\longrightarrow } {\textstyle \prod \nolimits _{i}} GW\left( {\mathsf {C}}_{i}\right) \text {.} \end{aligned}$$

This would be a natural analogue of [8].

Theorem A.6

Let F be a number field. If the Conjecture 2 is true, then there are canonical isomorphisms

$$\begin{aligned} GW_{n}({\mathsf {LCA}}_{F,ab})\cong \left\{ \left. (\alpha _{P})_{P}\in \prod _{P}GW_{n}({\widehat{F}}_{P})\right| \begin{array}[c]{l} \alpha _{P}\in GW_{n}(\widehat{{\mathcal {O}}}_{P})\text { for all but finitely}\\ \text {many among the finite places} \end{array} \right\} \text {.} \end{aligned}$$

Proof

The proof of Theorem 4.10 generalizes since all underlying exact equivalences preserve duality.$\square$

We should point out that our proof of Theorem 5.4 relies on several subcategories which are not closed under duality, and in particular it teaches us nothing about anything relying on duals.

Question 2

What is $GW({\mathsf {LCA}}_{F})$?