# What is the algebro-geometric or measure-theoretic “content” of Dhillon and Mináč's motivic Artin symbols over an arbitrary ground field?

1. Short version. In this text, Dhillon and Mináč define motivic Artin symbols. Having fixed a ground field $k$ and a smooth projective curve $Y$ over $k$ equipped with the action of a finite group $G$, they associate a virtual Chow motive $\text{Ar}(C,n)=\text{Ar}(Y,G,C,n)$ to each conjugacy class $C\subset G$ and each positive integer $n$. In fact, it is not hard to see that their motivic Artin symbols $\text{Ar}(C,n)$ are well-defined independently of $\text{char}_{\ \!}k$ when $\text{dim}_{k\ \!}Y=1$, and are well-defined independently of $\text{char}_{\ \!}k$ and $\text{dim}_{k\ \!}Y$ as soon as the Chow motive $h(X)$ is finite-dimensional in the sense of Kimura (see this MathOverflow question).
Dhillon and Mináč show [Proposition 6.2] that when $k=\mathbb{F}_{\!q}$ and $Y$ is a smooth projective curve over $\mathbb{F}_{\!q}$, each motivic Artin symbol $\text{Ar}(C,n)$ is a global, motivic incarnation of a family classical Artin symbols $(\!\frac{X/Y}{\mathfrak{p}}\!)$ at closed points $\mathfrak{p}\in X=Y/G$. Specifically: $$\sum_{i=0}^{\infty}(-1)^{i}\text{Tr}\Big(\text{Fr}_{q}\big|H^{i}\big(\text{Ar}(C,n)_{\overline{\mathbb{F}_{\!q}}},\mathbb{Q}_{\ell}\big)\Big) \ =\ n\cdot\#\Big\{\mathfrak{p}\in X: \begin{array}{l} Y\!\to\!X\text{ is unramified over }\mathfrak{p}, \\ \text{deg}_{\ \!}\mathfrak{p}=n,\text{ and }(\!\tfrac{X/Y}{\mathfrak{p}}\!)\in C \end{array} \Big\},$$ where $H^{i}\big((-)_{\overline{\mathbb{F}_{\!q}}},\mathbb{Q}_{\ell}\big)$ denotes the $i^{\text{th}}$ $\ell$-adic cohomological realization functor on Chow motives, for $\ell\neq\text{char}_{\ \!}\mathbb{F}_{\!q}$. We get a more geometric interpretation by recalling that $\big(\!\frac{X/Y}{\mathfrak{p}}\!\big)\in C$ precisely when $Y_{\mathfrak{p}}$ has decomposition type $C$, i.e., when the decomposition structure of $Y_{\mathfrak{p}}$ matches the cycle decomposition of elements in $C$. Then the above alternating sum becomes $$=\ n\cdot\#\big\{\mathfrak{p}\in X:\text{deg}_{\ \!}\mathfrak{p}=n\text{ and }Y_{\mathfrak{p}}\text{ has decomposition type }C\big\}.$$ Because the alternating sum of traces of Frobenius returns the count of $\mathbb{F}_{\!q}$-valued points on any projective $\overline{\mathbb{F}_{\!q}}$-variety, we can imagine that the above formulae count the $\mathbb{F}_{\!q}$-valued points in the virtual motive $\text{Ar}(C,n)$, and maybe even offer some sort of moduli-theoretic description of $\text{Ar}(C,n)$.
This interpretation of the motivic Artin symbols falls apart over arbitrary ground fields $k$: absence of Frobenius makes the alternating sum meaningless, and it can happen that the decomposition type of the fiber $Y_{\mathfrak{p}}$ over every closed point $\mathfrak{p}\in X$ is that of the conjugacy class $\{1\}\subset G$, for instance when $k=\mathbb{C}$). The latter makes any "measure of factorization type" uninteresting. Thus my question is:

Question 1.1. What is the general algebro-geometric content/meaning of the motivic Artin symbols $\text{Ar}(C,n)$ for schemes of arbitrary dimension over an arbitrary ground field $k$ (especially when $\text{char}_{\ \!}k=0$)? Is there some description comparable to the above identities?

See below for more specific questions.

Remark 1.2. I'm especially curious about the characteristic-0 case. I suspect that the $\text{dim}_{k\ \!}Y\ge 2$ case over $k=\mathbb{F}_{\!q}$ is closely related to Bary-Soroker's work on geometric embedding problems. I am certainly interested in this case, but what I'm really interested in are the cases where $\text{char}_{\ \!}k=0$. Even the $\text{dim}_{k\ \!}Y=1$ case over $k=\mathbb{C}$ would be of great interest to me.

Remark 1.3. Dhillon and Mináč provide a partial answer for curves over global fields. In the special case that $Y$ is a curve over a global field $k$, [Dhillon and Mináč, Proposition 6.2] includes the statement that at almost all places $\nu$ of $k$, the base change of $\text{Ar}(C,n)$ to the finite residue field $\kappa(\nu)$ exists and satisfies the above point-counting identity. However, this does not account for infinite places of $k$, it does not account for more general fields $k$, and it does not account for higher-dimensional $Y$.

Remark 1.4. I want something more "moduli-theoretic" than the $K_{0}$-decomposition. If $h(X)$ is Kimura finite-dimensional, then in theory the definition of $\text{Ar}(C,n)$ gives us an algorithm for writing down a presentation of $\text{Ar}(C,n)$ as a formal $\mathbb{Q}$-linear combination of classes in $K_{0}(\mathcal{M}^{\text{rat}}_{k})$ (see below). In some sense, the expression for each $\text{Ar}(C,n)$ can be taken as an algebro-geometric description of $\text{Ar}(C,n)$. However, the complexity of these expressions grows rapidly with $n$, and I see nothing in the expressions suggesting anything like a geometric version of the above identities.

2. More specific questions. My central problem is that after the iterative procedure involved in the definition of the $\text{Ar}(C,n)$ and the power series manipulations in the proof of [Dhillon and Mináč, Proposition 6.2], I have no algebro-geometric understanding of $\text{Ar}(C,n)$.
Here are more detailed questions, ignoring the issue of well-definedness of the $\text{Ar}(C,n)$:

Question 2.1. (Measure-theoretic description?). If $\text{char}_{\ \!}k=0$, then we have a ring homomorphism \begin{equation}\label{equation: hom 1} K(\text{Var}_{k})\longrightarrow K_{0}(\mathcal{M}^{\text{rat}}_{k}) \end{equation} from the Grothendieck ring of varieties to the Grothendieck ring of the category of Chow motives $\mathcal{M}^{\text{rat}}_{k}$, taking each class $[X]\in K(\text{Var}_{k})$ of a smooth projective $k$-variety $X$ to its corresponding Chow motive $h(X)$ (see the answers to this MathOverflow question). In the case that $\text{char}_{\ \!}k>0$, the formula above relates the images of the motivic Artin symbols under the alternating-trace-of-Frobenius ring homomorphism \begin{equation}\label{equation: hom 2} K_{0}(\mathcal{M}^{\text{rat}}_{k})_{\mathbb{Q}}\longrightarrow\mathbb{Q} \end{equation} to point counts. Point counts are the values of a particular ring homomorphism \begin{equation}\label{equation: hom 3} \mu:K(\text{Var}_{k})\longrightarrow\mathbb{Z}. \end{equation} My brain tries to draw a commutative triangle from this; I want to say "$\frac{1}{n}\text{Ar}(C,n)$ is a virtual moduli space of something, and the alternating trace formula is telling me a bit about what it's the virtual moduli space of by showing me its counting measure." This leads to the following question:
In the $\text{char}_{\ \!}k=0$ case, is there any sense in which we can describe the content of the motivic Artin symbols $\text{Ar}(C,n)$ in terms closely related to the values of some measure on $K(\text{Var}_{k})$? Specifically, are there natural ring homomorphisms $K_{0}(\mathcal{M}^{\text{rat}}_{k})_{\mathbb{Q}}\to\mathbb{Q}$ and $K(\text{Var}_{k})\to\mathbb{Z}$ such that we can describe the image of each $\text{Ar}(C,n)$ under $K_{0}(\mathcal{M}^{\text{rat}}_{k})_{\mathbb{Q}}\to\mathbb{Q}$ in terms of easier-to-understand values of $K(\text{Var}_{k})\to\mathbb{Z}$?
For a concrete example when $k=\mathbb{C}$, the Euler charateristic on $K(\text{Var}_{\mathbb{C}})$ factors through a morphism $K(\mathcal{M}^{\text{rat}}_{\mathbb{C}})_{\mathbb{Q}}\to\mathbb{Q}$. Is there an intuitive description of the values of the motivic Atrin symbols $\text{Ar}(C,n)$ under this morphism, something that illuminates the meaning of $\text{Ar}(C,n)$?

Question 2.2. (Direct algebro-geometric constructions?). Are there more direct algebro-geometric constructions of the motivic Artin symbols $\text{Ar}(C,n)$ in arbitrary characteristic? Specifically, are there constructions that better clarify the geometric meaning or content of each $\text{Ar}(C,n)$?

Question 2.3. (Monodromy over a punctured disk in place of Frobenius?). One case that seems closer to Dhillon and Mináč's setting is that of varieties over $k=\mathbb{C}(\!(s)\!)$, with monodromy taking the place of Frobenius. A vague guess is that in this case, the motivic Artin symbols might contain information about monodromy on Hodge structures. Is anyone aware of a way to see something along these lines?

Question 2.4. (The case of trivial G). Let $G=\{1\}$, so that $Y=X$. Then the only conjugacy class is $C=\{1\}$, and $L(X,\{1\},t)$ (see [Dhillon and Mináč, §2.3]) becomes the motivic L-function $$L(X,\{1\},t) \ =\ \sum_{n=0}^{\infty}h(\text{Sym}^{n}X)\ \!t^{n}.$$ If $k=\mathbb{F}_{\!q}$, the motivic Artin symbols $\text{Ar}(\{1\},n)$ become carriers of straightforward degree-wise point counts, as one can see by comparing $$\text{log}_{\ \!}L(X,\{1\},t) \ =\ \sum_{n>0}\text{Ar}(\{1\},n)\ \!\frac{t^{n}}{n} \ \ \ \ \ \ \ \ \ \mbox{and}\ \ \ \ \ \ \ \zeta(X,t) \ =\ \text{exp}\Big(\sum_{n>0}N_{n}\frac{t^{n}}{n}\Big).$$ We can ask Question 1.1 and Questions 2.1 through 2.3 above in this possibly simpler case:
With $G=\{1\}$ and $Y=X$, what is the geometric meaning of the coefficients $\text{Ar}(\{1\},n)$ that appear in $\text{log}_{\ \!}L(X,\{1\},t)$? Are there examples in characteristic $0$, for instance over $k=\mathbb{C}$, where their measures give us moduli-theoretic interpretations? Are there simple descriptions of the virtual motives $\frac{1}{n}\text{Ar}(\{1\},n)$ in terms of more familiar algebro-geometric spaces?

• You're asking five questions, all of them concerning one specific, relatively recent, paper. Have you tried emailing the authors? – David Loeffler Apr 26 '17 at 9:02
• In fact, I just mentioned to Dhillon that there is an MO question about his work, and he seemed surprised that you didn't simply email him directly. – Donu Arapura Apr 26 '17 at 17:53