John Baez
Zeta functions of Z-sets

Preface

This is work in progress by John Baez and James Dolan. based in part on material from week218 of This Week’s Finds.

In Zeta functions, we discussed a way to categorify the Hasse-Weil zeta function of a scheme. Another class of mathematical objects that have zeta functions are dynamical systems. The simplest example is the Artin–Mazur zeta function of a discrete dynamical system. A discrete dynamical system is simply a set $X$ equipped with a bijection $f:X\to X$, and its Artin–Mazur zeta function is

$${\zeta }_X(z)=\exp (\sum _{n=1}^{\mathrm{\infty }}\mid \mathrm{fix}(f^n)\mid \frac{z^n}{n}),$$\zeta_X(z)= exp\left(\sum_{n=1}^\infty |fix(f^n)| \frac {z^n}{n} \right)\, ,

where $\mathrm{fix}(f^n)$ is the set of fixed points of the $n$th iterate of $f$. Mathematically, we can also think of a discrete dynamical system as a $\mathbb{Z}$-set: a set equipped with an action of the additive group $\mathbb{Z}$. Any $\mathbb{Z}$-set $X$ gives a species $Z_X$ for which a $Z_X$-structure on a finite set is a way of making it into a $\mathbb{Z}$-set and equipping it with a $\mathbb{Z}$-set map to $X$. We shall show that the Dirichlet series of this species is equal to the Artin–Mazur zeta function ${\zeta }_X$.

We shall also explain how any Hasse–Weil species gives rise to a sequence of $\mathbb{Z}$-sets, one for each prime. Each of these gives a species as above, and under some mild conditions we can multiply these (using the ‘Dirichlet product’ of species) to recover the Hasse–Weil species. As we shall see, this explains why the zeta functions of such Hasse–Weil species have Euler factorizations.

The zeta function of a Z-set

Suppose we have a $\mathbb{Z}$-set $S$, or in other words, a set equipped with a bijection

$$f:S\to S.$$f: S \to S \, .

We can form three closely related sequences as follows:

• the number of fixed points of $f^n$;

• the number of orbits of cardinality $n$;

• the number of invariant multisubsets of weight $n$.

The last is perhaps the least familiar, but it is the most closely connected to zeta functions. In applications to number theory, if we start with a function

$$S:CommRing\to \mathrm{Set}$$S: \Comm\Ring \to Set

as discussed above. Then for each prime $p$, we obtain a set $S({\overline{𝔽}}_p)$, where ${\overline{𝔽}}_p$ is the algebraic closure of ${𝔽}_p$. This algebraic closure is equipped with an automorphism

$$F:{\overline{𝔽}}_p\to {\overline{𝔽}}_p$$F: \overline{\mathbb{F}}_p \to \overline{\mathbb{F}}_p

called the Frobenius automorphism. Since $S$ is a functor, this makes $S({\overline{𝔽}}_p)$ into a $\mathbb{Z}$-set where

$$f:S({\overline{𝔽}}_p)\to S({\overline{𝔽}}_p)$$f : S(\overline{\mathbb{F}}_p) \to S(\overline{\mathbb{F}}_p)

is given by

$$f=S(F).$$f = S(F) \, .

It then turns out that….

• fixed points of $f^n$ correspond to homomorphisms $\psi :A\to {𝔽}_{p^n}$. The reason is that ${𝔽}_{p^n}$ can be seen as precisely the fixed points of $F^n$.

• orbits of cardinality $n$ correspond to prime ideals $I$ of order $n$ in the ring $A$ localized at $p$ (???)

• invariant multisubsets of weight $n$ correspond to ideals of order $n$ in $A$ localized at $p$ (???)

Next:

• Let $\mathrm{fix}(f^n)$ be the set of fixed points of $f^n:S\to S$, and let $\mid \mathrm{fix}(f^n)\mid$ be the cardinality of this set.

• Let ${\mathrm{orb}}_S(n)$ be the number of $f$-orbits of $S$ having cardinality $n$.

• Let $z_S(n)$ be the number of invariant multi-subsets of $S$ of weight $n$.

Here a multi-subset of a set $S$ is a function $\psi :S\to \mathbb{N}$, and its weight is ${\sum }_{x\in S}\psi (x)$, which could be infinite. Any subset $X\subseteq S$ gives a multi-subset where $\psi$ is the characteristic function of $X$, and then the weight of this multi-subset is the cardinality of $X$.

As we shall see, if one of the above sequences consist of finite numbers, they all do. We say a $\mathbb{Z}$-set $S$ is tame if all these sequences consist of finite numbers.

Some of the above sequences arise naturally from the generating functions of certain structure types. For example, we can “cyclically order” a finite set by drawing it as a little circle of dots with arrows pointing clockwise from each dot to the next. (For convenience we decree that there are no ways to cyclically order the empty set.) A cyclically ordered set is automatically a $\mathbb{Z}$-set in an obvious way. So, here is a type of structure we can put on a finite set: cyclically ordering it and equipping the resulting $\mathbb{Z}$-set with a morphism to a fixed $\mathbb{Z}$-set $S$. And, we have:

Proposition: The power series

$$\sum _{n>0}\mid \mathrm{fix}(f^n)\mid \frac{x^n}{n}$$\sum_{n \gt 0} |fix(f^n)| \frac{x^n}{n}

is the generating function for the structure type “being cyclically ordered and equipped with a morphism to the $\mathbb{Z}$-set $S$”.

Proof: Given an $n$-element set, there are $(n-1)!$ ways to cyclically order it if $n>0$, and none if $n=0$. After having chosen a cyclic ordering, to specify a $\mathbb{Z}$-equivariant map to $S$ we simply map a chosen point to any element of $\mathrm{fix}(f^n)$. So there are $\mid \mathrm{fix}(f^n)\mid (n-1)!$ to do this, and the generating function of “being cyclically ordered and equipped with a morphism to the $\mathbb{Z}$-set $S$” is as described.

Proposition: The power series

$$\exp (\sum _{n>0}\mid \mathrm{fix}(f^n)\mid \frac{x^n}{n})$$exp \left( \sum_{n \gt 0} |fix(f^n)| \frac{x^n}{n} \right)

is the generating function for the structure type “being a $\mathbb{Z}$-set over $S$”.

Proof: For any structure type $F$ there is a structure type “being partitioned into nonempty parts, each equipped with an $F$-structure”, which is called $\exp (F)$ and satisfies

$$\mid \exp (F)\mid =\exp (\mid F\mid )$$|exp(F)| = exp(|F|)

So, by the previous proposition,

$$\exp (\sum _{n>0}\mid \mathrm{fix}(f^n)\mid \frac{x^n}{n})$$exp \left( \sum_{n \gt 0} |fix(f^n)| \frac{x^n}{n} \right)

is the generating function for “being partitioned into nonempty parts, each equipped with a cyclic ordering and a morphism to the $\mathbb{Z}$-set $S$”. But this is just a long way of saying: “being made into a $\mathbb{Z}$-set and equipped with a morphism to the $\mathbb{Z}$-set $S$ — or in category-theoretic jargon, “being a $\mathbb{Z}$-set over $S$”.