John Baez Dirichlet species and the Hasse-Weil zeta function

This is a draft of a paper by John Baez and James Dolan. For a popularized treatment, start with week300 of This Week’s Finds in Mathematical Physics.

Introduction

In this paper we begin categorifying the theory of zeta functions. As a precursor, we must understand how to get a Dirichlet series from a species. This is a very nice counterpart to the usual recipe for computing a formal power series from a species, namely its generating function (Joyal).

Briefly, a species is any type of structure that one can put on finite sets: for example, a coloring, or ordering, or tree structure. Suppose $F$ is some such structure. If we denote the set of $F$-structures on the $n$-element set by $F(n)$, the generating function of $F$ is the formal power series

$$\sum_{n \ge 0} \frac{|F(n)|}{n!} x^n \, .$$

On the other hand, the Dirichlet series associated to $F$ is

$$\sum_{n \ge 1} \frac{|F(n)|}{n!} n^{-s} \, .$$

Far from being ad hoc tricks, the generating function and the Dirichlet series fit into a nice unified theory, which we shall explain here.

The most famous of all Dirichlet series is the Riemann zeta function:

$$\zeta(s) = \sum_{n \ge 1} n^{-s} \, .$$

This comes from a species we call the Riemann species, $Z$. A $Z$-structure on a finite set is a way of making it into a semisimple commutative ring — that is, a product of finite fields.

The Riemann zeta function has many generalizations, notably the Hasse-Weil zeta function. This sort of zeta function is usually defined for any projective variety defined over the integers. It is easier to work in still more generality, starting from any functor

$$S : \Comm\Ring \to \Set$$

that preserves products and takes finite fields to finite sets. Any such functor gives a species which we call the Hasse–Weil species, $Z_S$. A $Z_S$-structure on a finite set is defined to be a way of making it into a finite semisimple commutative ring, say $k$, and then picking an element of $S(k)$. The Dirichlet series of the Hasse-Weil species is

$$\sum_{n \ge 1} \frac{|Z_S(n)|}{n!} n^{-s} \, .$$

We shall prove that this equals the usual Hasse-Weil zeta function, which is defined as a product over primes:

$$\prod_p exp \left( \sum_{n \gt 0} \frac{|S(\mathbb{F}_{p^n})|}{n} p^{-n s} \right)$$

where $\mathbb{F}_{p^n}$ is the field with $p^n$ elements.

To illustrate how this works, consider a classic example of a Hasse-Weil zeta function: the Dedekind zeta function of an algebraic number field. Given such a field, let $R$ be its ring of algebraic integers.

Starting from any such ring we obtain a functor from commutative rings to sets, which sends any commutative ring $k$ to the set of homomorphisms from $R$ to $k$. As described above, this functor gives a Hasse–Weil species. Let us call this species $Z_R$. Unravelling the constructions we have described, it is easy to see that a $Z_R$-structure on a finite set is a way of making that set into a semisimple commutative ring and choosing a homomorphism from $R$ to that ring. By definition, the Dirichlet series of this species is

$$\sum_{n \ge 1} \frac{|Z_R(n)|}{n!} n^{-s} \,.$$

But in fact, this equals the usual Dedekind zeta function, namely

$$\sum_{I} {|R/I|}^{-s}$$

where $I$ ranges over all ideals of $R$, and $|R/I|$ is the cardinality of the quotient ring.

Two Examples

Before plunging into the general theory, let us consider a couple of examples of Hasse–Weil species and their zeta functions. We will tackle these in a brute-force way and make only slight progress. Everything will become easier later, after we have introduced more technology. Still, this first attempt is amusing and perhaps instructive.

First take $R = \mathbb{Z}$. In this case, a $Z_R$-structure on a finite set is a way of making that set into a commutative semisimple ring and choosing a homomorphism from $\mathbb{Z}$ to that ring. But there is always exactly one such homomorphism, so $Z_R$-structure is just a way of making a finite set into a commutative semisimple ring. In other words, $Z_R$ is the Riemann species. We call this species $Z$ for short.

Let us count the $Z$-structures on an $n$-element set for a few small values of $n$. To do this, we can start by classifying finite semisimple commutative rings, with the help of two facts:

• By the Artin–Wedderburn theorem, a finite semisimple commutative ring is the same as a finite product of finite fields.

• There is one field with $q$ elements, denoted $\mathbb{F}_q$, when $q$ is a power of a prime number, and none otherwise.

This lets us classify the semisimple commutative rings with $n$ elements. For example:

• There is none when $n = 0$.

• There is one when $n = 1$: the ring with one element. (This is the empty product of finite fields.)

• There is one when $n = 2$: $\mathbb{F}_2$.

• There is one when $n = 3$: $\mathbb{F}_3$.

• There are two when $n = 4$: $\mathbb{F}_4$ and $\mathbb{F}_2 \times \mathbb{F}_2$.

• There is one when $n = 5$: $\mathbb{F}_5$.

• There is one when $n = 6$: $\mathbb{F}_2 \times \mathbb{F}_3$.

• There is one when $n = 7$: $\mathbb{F}_7$.

• There are three when $n = 8$: $\mathbb{F}_8$, $\mathbb{F}_2 \times \mathbb{F}_4$ and $\mathbb{F}_2 \times \mathbb{F}_2 \times \mathbb{F}_2$.

Next, how many ways are there to make an $n$-element set into a ring isomorphic to one on our list? For starters: how many ways are there to make an $n$-element set into a ring isomorphic to some fixed $n$-element ring, say $k$? Each bijection between the set and the ring $k$ gives a way to do this, but not all of them give different ways: two bijections differing by an automorphism of $k$ give the same way. So, the answer is $n!/|Aut(k)|$.

To go further, we need to know that if $q = p^m$ for some prime $p$, then

$$Aut(\mathbb{F}_q) \cong \mathbb{Z}/m \, ,$$

a cyclic group generated by the Frobenius automorphism

$$\array{ F : &\mathbb{F}_q &\to& \mathbb{F}_q \\ & x &\mapsto & x^p \, . }$$

More generally, if we have a finite product of finite fields, its automorphisms all come from automorphisms of the factors together with permutations of like factors. So, for example, $\mathbb{F}_2 \times \mathbb{F}_4$ has 2 automorphisms (coming from automorphisms of the second factor), while $\mathbb{F}_2 \times \mathbb{F}_2 \times \mathbb{F}_2$ has 6 (coming from permutations of the factors).

Now we can count how many ways there are to make an $n$-element set into a commutative semisimple ring:

• For $n = 0$ there are $0$ ways.

• For $n = 1$ there is $1!/1 = 1!$ ways.

• For $n = 2$ there are $2!/1 = 2!$ ways.

• For $n = 3$ there are $3!/1 = 3!$ ways.

• For $n = 4$ there are $4!/2 + 4!/2 = 4!$ ways.

• For $n = 5$ there are $5!/1 = 5!$ ways.

• For $n = 6$ there are $6!/1 = 6!$ ways.

• For $n = 7$ there are $7!/1 = 7!$ ways.

• For $n = 8$ there are $8!/3 + 8!/2 + 8!/6 = 8!$ ways.

From this evidence, one might boldly guess there are always $n!$ ways. In fact, though it is hard to see why from our work so far, this guess is correct for all $n$. So, the Dirichlet function of the Riemann species is the Riemann zeta function:

$$\sum_{n \ge 1} \frac{n!}{n!} n^{-s} = \zeta(s) \, ,$$

Next take $R = \mathbb{Z}[i]$ to be the Gaussian integers. Then a $Z_R$-structure on a finite set is a way of making that set into a commutative semisimple ring and choosing a square root of $-1$ in this ring. Let us count the $Z_R$-structures on an $n$-element set for a few small values of $n$.

For starters, choosing a square root of $-1$ in a product of finite fields is the same as choosing a square root of $-1$ in each factor. So, how many square roots of $-1$ does $\mathbb{F}_q$ have, where $q = p^m$ is a prime power? The answer depends on the prime $p$, and number theorists have names for 3 cases:

• Split: if $p \equiv 1 \, mod \, 4$, then there are 2 square roots of $-1$ in $\mathbb{F}_{p^m}$ for all $m \ge 1$.

• Inert: if $p \equiv 3 \, mod \, 4$, then there is no square root of $-1$ in $\mathbb{F}_{p^m}$ when $m$ is odd and 2 when $m$ is even.

• Ramified: if $p = 2$, then there is 1 square root of $-1$ in $\mathbb{F}_{p^m}$.

Combining this information with the tables above, we can count the ways are to make an $n$-element set into a commutative semisimple ring equipped with a square root of $-1$:

• For $n = 0$ there are $0 = 0 \times 1!$ ways.

• For $n = 1$ there is $1 \times 1!$ ways.

• For $n = 2$ there are $1 \times 2!$ ways.

• For $n = 3$ there are $0 \times 3!$ ways.

• For $n = 4$ there are $0 \times 4!/2 + 1 \times 4!/2 = 1 \times 4!$ ways.

• For $n = 5$ there are $2 \times 5!$ ways.

• For $n = 6$ there are $0 \times 6!$ ways.

• For $n = 7$ there are $0 \times 7!$ ways.

• For $n = 8$ there are $1 \times 8!/3 + 1 \times 8!/2 + 1 \times 8!/6 = 1 \times 8!$ ways.

So, when $R = \mathbb{Z}[i]$, we obtain the zeta function

$$\zeta_R(s) = 1^{-s} + 2^{-s} + 4^{-s} + 2 \cdot 5^{-s} + 8^{-s} + \cdots$$

Let us see how this compares to the usual Dedekind zeta function. Recall that this is defined as

$$\sum_{I} {|R/I|}^{-s}$$

where $I$ ranges over all ideals of $R$, and $|R/I|$ is the cardinality of the quotient ring. However, every ideal in a ring of algebraic integers can be uniquely factored into prime ideals, so a standard calculation shows

$$\sum_{I} {|R/I|}^{-s} = \prod_{P} \frac{1}{1- {|R/P|}^{-s}}$$

where $P$ ranges over prime ideals of $R$. The prime ideals of $R = \mathbb{Z}[i]$ come in three flavors, corresponding to the three kinds of primes mentioned above:

• Split: if $p \equiv 1 \, mod \, 4$, then the ideal of $\mathbb{Z}[i]$ generated by $p$ is a product of two different prime ideals $P, P'$, with $|R/P| = |R/P| = p$.

• Inert: if $p \equiv 3 \, mod \, 4$, then the ideal $P \subseteq \mathbb{Z}[i]$ generated by $p$ is a prime ideal with $|R/P| = p^2$.

• Ramified: if $p = 2$, then the ideal of $\mathbb{Z}[i]$ generated by $p$ is the square of a prime ideal $P$ with $|R/P| = 2$.

These account for all the prime ideals in $\mathbb{Z}[i]$, so the Dedekind zeta function is

$$\array{ \prod_{P} \frac{1}{1- {|R/P|}^{-s}} &=& \frac{1}{(1 - 2^{-s})} \; \left(\prod_{p \equiv 1 \, mod \, 4} \frac{1}{(1 - p^{-s})^2} \right) \; \left(\prod_{p \equiv 3 \, mod \, 4} \frac{1}{(1 - p^{-2s})} \right) \\ &=& \frac{1}{(1 - 2^{-s})} \frac{1}{(1 - 3^{-2s})} \frac{1}{(1 - 5^{-s})^2} \frac{1}{(1 - 7^{-2s})} \cdots \\ &=& (1 + 2^{-s} + 4^{-s} + 8^{-s} + \cdots)(1 + 9^{-s} + \cdots)(1 + 2 \cdot 5^{-s} + \cdots)(1 + 49^{-s} + \cdots) \cdot \cdots \\ &=& 1 + 2^{-s} + 4^{-s} + 2 \cdot 5^{-s} + 8^{-s} + \cdots }$$

So, the zeta function arising from the Hasse–Weil species seems to match the usual Dedekind zeta function. And indeed this is true — though again, the reason is not obvious from our work so far. We need a little theory to see what is really going on.

Dirichlet series

Dirichlet series from tame species

We call the groupoid of finite sets and bijections $core(\Fin\Set)$, since it is the core of the category $\Fin\Set$, whose objects are finite sets and whose morphisms are functions. The category of species or structure types is the functor category

$$[core(\Fin\Set), \Set ]$$

An object in here, say $F: core(\Fin\Set) \to \Set$, describes a type of structure that you can put on a finite set. For any finite set $n$, $F(n)$ is the collection of structures of that type that you can put on the set $n$.

We will mainly be interested in species satisfying a certain finiteness property. So, we define the category of tame species to be the functor category

$$[core(\FinSet), \FinSet]$$

Like the category of species itself, the category of tame species has quite a few interesting monoidal structures. Two of these come from addition and multiplication in the target category $\Fin\Set$:

• the pointwise coproduct, given by

  $$(F + G)(n) = F(n) + G(n)$$

  This is usually called addition of species.

• the pointwise product, given by

  $$(F \times G)(n) = F(n) \times G(n)$$

  This is sometimes called the Hadamard product.

Two more arise via Day convolution from $+$ and $\times$ in $core(\Fin\Set)$. Note that while this groupoid does not have coproducts or products, it inherits operations which we may call $+$ and $\times$ from $\Fin\Set$, which does. These in turn give the category of tame species the following two monoidal structures:

• The operation

  $$(F \cdot_{C} G)(n) = \sum_{k + k' = n} F(k) \times G(k'),$$

  also called the Cauchy product. Be careful of the notation: here $n$ is a finite set, and we are summing over all ways of writing this set as a disjoint union of two subsets $k$ and $k'$. This is just a lowbrow (and very convenient) way of saying that we sum over equivalence classes of coproduct cocones

  $$\array{ && k &&&& k' \\ & && \searrow & & \swarrow && \\ &&&& n &&&& }$$

  where we consider two such cocones, say

  $$\array{ && k_1 &&&& k'_1 \\ & && \searrow & & \swarrow && \\ &&&& n &&&& }$$

  and

  $$\array{ && k_2 &&&& k'_2 \\ & && \searrow & & \swarrow && \\ &&&& n &&&& }$$

  equivalent if there are isomorphisms $k_1 \to k_2$, $k'_1 \to k'_2$ making the diagram built from these isomorphisms and the above two diagrams commute. For example, if $n$ is $35$-element set, $k$ is a $10$-element set and $k'$ is a $25$-element set, there are

  $$\frac{35!}{10! \, 25!}$$

  equivalence classes of coproduct cocones. This is a highbrow way of saying that this binomial coefficient counts the ways of writing a 35-element set as a disjoint union of a 10-element set and a 25-element set.

• The operation

  $$(F \cdot_{D} G)(n) = \sum_{k \times k' = n} F(k) \times G(k')$$

  which we hereby dub the Dirichlet product. (This seems to be what Maia and Méndez called the ‘arithmetic product’.) Here we must be even more careful to understand the notation, which is a bit misleading. As before, $n, k$ and $k'$ denote finite sets, not just numbers. And, dually to above case, we are really summing over equivalence cases of product cones

  $$\array{ &&&& n \\& && \swarrow && \searrow \\ && k &&&& k' }$$

  with the equivalence relation defined in a precisely dual way to that above. More concretely, we are summing over ways of organizing the elements of $n$ into a $k \times k'$ rectangle, but where we count two ways of doing this as equivalent if they differ by permuting the rows and/or permuting the columns. So, for example, if $n$ is $35$-element set, $k$ is a $5$-element set and $k'$ is a $7$-element set, there are

  $$\frac{35!}{5! \, 7!}$$

  equivalence classes of product cones. We see here a funny mutant version of a binomial coefficient.

We are mainly interested here in addition and the Dirichlet product, though they fit into a bigger story. Why these two operations? First of all, they make the category of tame species into a rig category. Second, any tame species

$$F: core(\FinSet) \to \FinSet$$

has a Dirichlet series

$$\overline{F}(s) = \sum_{n \ge 1} \frac{|F(n)|}{n!} n^{-s}$$

and it is easy to check that

$$\overline{F+G} = \overline{F} + \overline{G}$$

and more interestingly,

$$\overline{F \cdot_{D} G} = \overline{F} \, \overline{G}$$

Checking the second equation here uses some simple facts about the ‘mutant binomial coefficients’ mentioned above. We believe the appearance of Dirichlet series in number theory is largely due to this fact.

All this should be compared to the much more familiar story involving generating functions of species. Namely, any tame species $F$ has an (exponential) generating function

$$|F|(z) = \sum_{n \ge 0} \frac{|F(n)|}{n!} z^n$$

and it is easy to check that

$$|F+G| = |F| + |G|$$

and

$$|F \cdot_{C} G| = |F| |G|$$

(Here checking the second equation uses some simple facts about binomial coefficients.)

In short, the Dirichlet series well-adapted to studying the Dirichlet product of species just as generating functions are well-adapted to the Cauchy product. But, they are just two ways of presenting the same information. The Dirichlet series $\overline{F}$ is obtained from the generating function $|F|$ by the the change of basis

$$z^n \mapsto n^{-s} \, .$$

In number theory a closely related change of basis is called the Mellin transform:

$$(e^{-t})^n \mapsto n^{-s} \Gamma(s)$$

and this explains the frequent appearance of Mellin transforms in number theory — though there are some wrinkles here that need to be ironed out. But, regardless of these wrinkles, the basic point is that as long as a species $F$ obeys $|F(0)| = 0$, we can freely go back and forth between its generating function and its Dirichlet series.

Dirichlet series from tame stuff types

Species, or structure types, are a special case of stuff types, and we can generalize all our remarks so far to stuff types. Here we just sketch this very briefly; for some more details see (BaezDolan, Morton). A stuff type is a weak 2-functor

$$F: core(\FinSet) \to \Gpd$$

which describes a type of stuff that you can put on a finite set. By the Grothendieck construction, we can also think of a stuff type as a functor

$$p: X \to core(\FinSet)$$

from some fixed groupoid to the groupoid of finite sets. The idea is that $p$ is a ‘forgetful functor’ sending any ‘set equipped with $F$-stuff’ to its underlying set.

There is a concept of groupoid cardinality generalizing the concept of cardinality for finite sets. Namely, for any groupoid $X$, we compute its cardinality $|X|$ as a sum over isomorphism classes of objects. For each isomorphism class, say $[x]$, we take a representative object $x$ and compute the reciprocal of the order of its automorphism group. Then, we sum these up:

$$|X| = \sum_{[x]} \frac{1}{|Aut(x)|}$$

Of course the sum could diverge; we say a groupoid is tame if every object has a finite automorphism group and the sum converges. We say a stuff type

$$F: core(\FinSet) \to \Gpd$$

is tame if $F(n)$ is tame for every finite set $n$. A tame stuff type $F$ has a Dirichlet series

$$\overline{F}(s) = \sum_{n \ge 1} \frac{|F(n)|}{n!} n^{-s}$$

However, this formula simplifies if we treat our stuff type as a functor $p: X \to core(\Fin\Set)$. Then we have

$$\overline{F}(s) = \sum_{n \ge 1} |p^{-1}(n)| n^{-s} \, .$$

Here $p^{-1}(n)$ is the full inverse image of the $n$-element set: the groupoid of all objects $x \in X$ such that $p(x)$ is an $n$-element set, and all morphisms in $X$ between such objects. Now the factor of $1/n!$ is built into the groupoid cardinality of $p^{-1}(n)$.

The operations of addition and Dirichlet product — and indeed all the operations listed for species — extend to stuff types, and it is easy to check that for tame stuff types

$$\overline{F+G} = \overline{F} + \overline{G}$$

and

$$\overline{F \cdot_{D} G} = \overline{F} \, \overline{G}$$

To illustrate usefulness of stuff types, let us recompute the first 8 terms of the Riemann zeta function. This time let us treat the Riemann species as a stuff type, namely the forgetful functor

$$p: core(\FinSSCommRing) \to core(\FinSet)$$

where $\Fin\SS\Comm\Ring$ is the category of finite semisimple commutative rings. The Dirichlet series of the Riemann species is then

$$\overline{F}(s) = \sum_{n \ge 1} |p^{-1}(n)| n^{-s}$$

where $p^{-1}(n)$ is the groupoid of $n$-element semisimple commutative rings. As we shall later see, we always get $|p^{-1}(n)| = 1$. For example:

• There is one isomorphism class of semisimple commutative rings with $n$ elements when $n = 1$: the empty product of finite fields, which is the ring with one element. This has just one automorphism, so

  $$|p^{-1}(1)| = \frac{1}{1} = 1 \, .$$

• There is one isomorphism class when $n = 2$: $\mathbb{F}_2$. Thus

  $$|p^{-1}(2)| = \frac{1}{|Aut(\mathbb{F}_2)|} = \frac{1}{1} = 1 \, .$$

• There is one when $n = 3$: $\mathbb{F}_2$. Thus

  $$|p^{-1}(3)| = \frac{1}{|Aut(\mathbb{F}_3)|} = \frac{1}{1} = 1 \, .$$

• There are two when $n = 4$: $\mathbb{F}_4$ and $\mathbb{F}_2 \times \mathbb{F}_2$. Thus:

  $$|p^{-1}(4)| = \frac{1}{|Aut(\mathbb{F}_4)|} + \frac{1}{|Aut(\mathbb{F}_2 \times \mathbb{F}_2)|} = \frac{1}{2} + \frac{1}{2} = 1 \, .$$

• There is one when $n = 5$: $\mathbb{F}_5$. Thus

  $$|p^{-1}(5)| = \frac{1}{|Aut(\mathbb{F}_5)|} = \frac{1}{1} = 1 \, .$$

• There is one when $n = 6$: $\mathbb{F}_2 \times \mathbb{F}_3$. Thus

  $$|p^{-1}(6)| = \frac{1}{|Aut(\mathbb{F}_2 \times \mathbb{F}_3)|} = \frac{1}{1} = 1 \, .$$

• There is one when $n = 7$: $\mathbb{F}_7$. Thus

  $$|p^{-1}(7)| = \frac{1}{|Aut(\mathbb{F}_7)|} = \frac{1}{1} = 1 \, .$$

• There are three when $n = 8$: $\mathbb{F}_8$, $\mathbb{F}_2 \times \mathbb{F}_4$ and $\mathbb{F}_2 \times \mathbb{F}_2 \times \mathbb{F}_2$. Thus

  $$|p^{-1}(8)| = \frac{1}{|Aut(\mathbb{F}_8)|} + \frac{1}{|Aut(\mathbb{F}_2 \times Aut(\mathbb{F}_4))|} + \frac{1}{|Aut(\mathbb{F}_2 \times \mathbb{F}_2 \times \mathbb{F}_2)|}= \frac{1}{3} + \frac{1}{2} + \frac{1}{6} = 1 \, .$$

The calculations here are fundamentally the same as those we did before, but they are cleaner: no cancellation of factorials is needed.

Dirichlet exponentiation

Given a tame species $F$ with $F(0) = 0$, there is a tame species $exp_D(F)$ called the Dirichlet exponential such that an $exp_D(F)$-structure on a finite set is the same as a way of writing that set as an unordered product of finite sets and putting an $F$-structure on each factor. This has the property that

$$\overline{exp_D(F)} = exp(\overline{F})$$

where on the right we take the exponential of the Dirichlet series of $F$, obtaining another Dirichlet series.

This is reminiscent of the ‘usual’ exponential of a tame species $F$, which we dub the Cauchy exponential $exp_C(F)$. This is a tame species such that an $exp_C(F)$-structure on a finite set is the same as a way of writing that set as an unordered coproduct of finite sets and putting an $F$-structure on each summand. The Cauchy exponential has the property that

$$|exp_C(F)| = exp(|F|) \, .$$

Like the Cauchy exponential, the Dirichlet exponential generalizes from species to stuff types.

Multiplicative species and stuff types

In number theory, an arithmetic function is a function

$$f: \mathbb{N}^+ \to \mathbb{C}$$

An arithmetic function is said to be multiplicative if

$$f(m n) = f(m) f(n)$$

whenever $m$ and $n$ are relatively prime. Since $1$ is relatively prime to everything, this forces $f(1) = 1$. A multiplicative function is determined by its values on prime powers, so we may think of it as a function of isomorphism classes of finite fields.

Any arithmetic function $f$ gives a Dirichlet series

$$\hat{f}(s) = \sum_{n \ge 1} f(n) n^{-s}$$

and $f$ is multiplicative if and only if this Dirichlet series has an Euler factorization:

$$\sum_{n \ge 1} f(n) n^{-s} = \prod_{p} (1 + f(p) p^{-s} + f(p^2) p^{-2s} + \cdots )$$

where the product is taken over all primes. The most famous case is when $f(n) = 1$ for all $n$; then we get Euler’s formula for the Riemann zeta function:

$$\sum_{n \ge 1} n^{-s} = \prod_{p} \frac{1}{1 - p^{-s}}$$

All this is ripe for categorification.

Definition: A species

$$F : core(FinSet) \to Set$$

is multiplicative if

$$F = \prod^D_p F_p$$

where $p$ ranges over prime numbers and $F_p$ is a species with the property that $F_p(n)$ is the empty set unless the cardinality of $n$ is a power of $p$, and the one-element set when $n$ has one element.
Here $\displaystyle{\prod^D}$ stands for the Dirichlet product. (Note that while this is an infinite product, there is no real problem, since any natural number is a product of finitely many primes.)

It is easy to check that if a tame species $F$ is multiplicative, so is the arithmetic function

$$n \mapsto |F(n)|$$

or for that matter

$$n \mapsto |F(n)|/n!$$

So, the Dirichlet series of $F$ has an Euler factorization.

The concept of ‘multiplicative species’ easily generalizes to stuff types, and these too have Dirichlet series that admit Euler factorizations.

The Riemann species

The Riemann species and the Riemann zeta function

Definition: The Riemann species is the species $Z$ that assigns to any finite set the collection of ways of making that set into a product of finite fields.

Theorem: The Dirichlet series of the Riemann species is the Riemann zeta function:

$$|Z| = \zeta \, .$$

In other words, the number of ways to make an $n$-element set into a product of finite fields is $n!$. To prove this we use some lemmas:

Lemma: The Riemann species is multiplicative:

$$Z \cong \prod^D_p Z_p$$

where $Z_p$ is the species that assigns to any finite set the collection of ways of making that set into a product of finite fields of characteristic $p$.

Proof: Use the the definition of ‘multiplicative species’ and the fact that any finite field is a field of characteristic $p$ for some unique prime $p$.

Lemma: The species $Z_p$ can be written as a Dirichlet exponential

$$Z_p \cong exp_D(F_p)$$

where $F_p$ is the species that assigns to any finite set the collection of ways of making that set into a finite field of characteristic $p$.

Proof: Use the definition of ‘Dirichlet exponential’.

Lemma: The species $F_p$ has the following Dirichlet series:

$$F_p(s) = p^{-s} + \frac{p^{-2s}}{2} + \frac{p^{-3s}}{3} + \cdots = ln \left(\frac{1}{1 - p^{-s}}\right)$$

Proof: A finite field of characteristic $p$ has cardinality $p^n$ for some $n \ge 1$. All fields of this cardinality are isomorphic, and the automorphism group of any one of these is a cyclic group with $n$ elements, generated by the Frobenius automorphism

$$x \mapsto x^p \, .$$

So, to count number of different ways to make an $p^n$-element set into a finite field, we can just count the group of all permutations of that set, modulo the subgroup that fixes a given finite field structure, obtaining

$$\frac{(p^n)!}{n}$$

To get the corresponding coefficient of the Dirichlet series for the species $F_p$, we just divide by $(p^n)!$. So, the Dirichlet series is

$$|F_p|(s) = \sum_{n \ge 1} \frac{1}{n} p^{-n s} \, .$$

This is just another way of writing the expression in the statement of the lemma.

Lemma: The species $Z_p$ has the following Dirichlet series:

$$|Z_p|(s) = \frac{1}{1 - p^{-s}} \, .$$

Proof: Since $Z_p$ is the Dirichlet exponential of $F_p$ we have

$$|Z_p|(s) = exp(|F_p|(s))\, ,$$

so this result follows from the previous Lemma.

These lemmas, together with the Euler product formula for the Riemann zeta function, add up to prove the Theorem:

$$\array{ |Z|(s) &=& \prod_p |Z_p|(s) \\ &=& \prod_p \frac{1}{1 - p^{-s}} \\ &=& \zeta(s) \, . }$$

Hasse–Weil species

These days, algebraic geometry is often formulated in terms of schemes. The most popular definition takes a bit of getting used to, but for the purposes of these notes we can work with a simpler and more general concept, namely a functor

$$S: \Comm\Ring \to Set \, .$$

For example:

• We can start with a polynomial equation with integer coefficients, say

  $$x^3 + y^3 = z^3$$

  and let $S(k)$ be the set of solutions of this equation when the variables take values in the commutative ring $k$. Any homomorphism $k \to k'$ gives a map $S(k) \to S(k')$, and it is easy to check that $S$ is a functor. We call $S(k)$ the set of $k$-points of $S$.

• More generally, we can specify a functor $S_R: \Comm\Ring \to Set$ by giving a commutative ring $R$ and letting $S_R(k)$ be the set of homomorphisms $f: R \to k$. The previous example is the special case where we take $R = \mathbb{Z}[x,y,z]/\langle x^3 + y^3 = z^3 \rangle$.

• More generally, any scheme determines a functor from $\Comm\Ring$ to $Set$, often called its ‘functor of points’. The previous example is a special case of this: a so-called affine scheme.

Definition: Given a functor $S: \Comm\Ring \to Set$, we define its Hasse–Weil species $Z_S$ as follows: a $Z_S$-structure on a finite set is a way to make that set into a finite commutative semisimple ring, say $k$ and then choose an element of $S(k)$.

Remember that ‘finite commutative semisimple ring’ is just an erudite term for ‘finite product of finite fields’.

Definition: We say a functor $S: \Comm\Ring \to Set$ is tame if its Hasse–Weil species is tame. Equivalently, $S$ is tame if $S(k)$ is finite whenever $k$ is a finite commutative semisimple ring.

Whenever $S$ is the functor of points of some scheme of finite type defined over the integers, it is tame (Serre). We can think of such a scheme as a projective algebraic variety defined over the integers.

Definition: Given a tame functor $S: \Comm\Ring \to Set$ we define its zeta function $\zeta_S$ to be the Dirichlet series of the Hasse–Weil species $Z_S$:

$$\zeta_S(s) = \sum_{n \ge 1} \frac{|Z_S(n)|}{n!} \, n^{-s} \, .$$

Definition: A functor $S: \Comm\Ring \to Set$ is multiplicative if it preserves products.

In fact whenever $S$ is the functor of points of some scheme defined over the integers, it is multiplicative. This is easiest to see for affine schemes.

Proposition: If $S: \Comm\Ring \to Set$ is multiplicative, the Hasse-Weil species $Z_S$ is multiplicative:

$$Z_S \cong \prod_p^D Z_{S,p}$$

where a $Z_{S,p}$-structure on a finite set is a way to make that set into a ring $k$ that is a product of finite fields of characteristic $p$ and then choose an element of $S(k)$.

Proof: The proof is analogous to the case of the Riemann species.

Proposition: If $S: \Comm\Ring \to Set$ is multiplicative, each species $Z_{S,p}$ can be written as the Dirichlet exponential

$$Z_{S,p} \cong exp_p^D \, (F_{S,p})$$

where an $F_{S,p}$-structure on a finite set is a way to make that set into a field $k$ of characteristic $p$ and then choose an element of $S(k)$.

Proof: The proof is analogous to the case of the Riemann species.

Here we have factored the Hasse–Weil species into Euler factors $Z_{S,p}$ and then written each factor as an exponential. We can also reverse the order, writing the Hasse-Weil species as an exponential of some species and then writing this species as a sum. In other words, we have a commuting square of isomorphisms:

(1)$$\array{ Z_S & \rightarrow & \prod_p^D Z_{S,p} \\ \downarrow & & \downarrow \\ exp_D(F_S) &\rightarrow & exp_D (\sum_p F_{S,p}) \cong \prod_p^D \exp_D(F_{S,p}) }$$

Proposition: If $S \Comm\Ring \to Set$ is multiplicative, the species $Z_S$ can be written as the Dirichlet exponential

$$Z_S \cong \exp_D(F_S)$$

where an $F_S$-structure on a finite set is a way to make that set into a field $k$ and then choose an element of $S(k)$.

Proof: We use the assumption that $Z_S$ is multiplicative and the fact that every finite semisimple commutative algebra can be written as a product of finite fields in a unique way.

It is then easy to derive the commutative square (1) with the help of the fact that for any sequence of species $G_i$, we have a natural isomorphism

$$\exp_D(\sum_i G_i) \cong \prod^D_i \exp_D(G_i) \, .$$

Let us illustrate these concepts with two easy examples:

Example: Affine $d$-space, $\mathbb{A}^d$. This is the affine scheme corresponding to the commutative ring $R = \mathbb{Z}[x_1, \dots, x_d]$. Here

$$S(k) = k^d$$

for any commutative ring $k$. The corresponding Hasse–Weil species, which we denote as $Z_{\mathbb{A}^d}$, works as follows. A $Z_{\mathbb{A}^d}$-structure on a finite set is a way to make that set into a semisimple commutative ring, say $k$, and then choose a $d$-tuple of elements of it. We have already seen from our study of the Riemann species that there are $n!$ ways of making an $n$-element set into a product of finite fields. So, the number of $Z_{\mathbb{A}^d}$-structures on an $n$-element set is $n^d n!$, and the corresponding zeta function is

$$\array{ \zeta_{\mathbb{A}^d}(s) &=& \sum_{n \ge 1} n^d n^{-s} \\ &=& \zeta(s-d) \, , }$$

just a translate of the Riemann zeta function.

Example: A ring of algebraic integers. This is the affine scheme corresponding to a ring $R$ consisting of the algebraic integers for some algebraic number field. We define a $Z_R$-structure on a finite set to be a way to make that set into a semisimple commutative ring, say $k$, and then choose a homomorphism from $R$ to $k$. We write the corresponding zeta function as $\zeta_R$:

$$\zeta_R(s) = \sum_{n \ge 1} \frac{|Z_R(n)|}{n!} \, n^{-s} \, .$$

We claim that $\zeta_R(s)$ equals the usual Dedekind zeta function of our algebraic number field, namely

$$\sum_{I} {|R/I|}^{-s}$$

where $I$ ranges over all ideals of $R$, and $|R/I|$ is the cardinality of the quotient ring.

On the one hand, it is known that the Dedekind zeta function has an Euler factorization into local zeta functions:

(2)$$\sum_{I} {|R/I|}^{-s} = \prod_p exp \left( \sum_{n \gt 0} \frac{|hom(R,\mathbb{F}_{p^n})|}{n} p^{-n s} \right)$$

where $|hom(R,\mathbb{F}_{p^n})|$ is the number of homomorphisms from $R$ to $\mathbb{F}_{p^n}$.

On the other hand, we have seen that

$$Z_R \cong \prod^D_p exp_D(F_{p,R})$$

where an $F_{p,R}$-structure on a finite set is a way of making it into a finite field. It follows that

(3)$$\zeta_R(s) = \prod_p exp \left(\sum_{n \ge 1} \frac{|F_{R,p}(p^n)|}{p^n!} p^{-n s} \right) \, .$$

Comparing (2) and (3), we see that to prove our claim it suffices to show that

$$\frac{p^n!}{n} |hom(R,\mathbb{F}_{p^n})| = |F_{R,p}(p^n)| \, .$$

By definition $|F_{R,p}(p^n)|$ is the number of ways to make an $p^n$-element set into a field and choose a homomorphism from $R$ to this field. But there are $p^n! / n$ ways to make a $p^n$-element set into a field, since there are $p^n!$ bijections between the set and $\mathbb{F}_{p^n}$, and this field has $n$ automorphisms. So, there are $\frac{p^n!}{n} |hom(R,\mathbb{F}_{p^n})|$ ways to make a $p^n$-element set into a field and then choose a homomorphism from $R$ to this field.

The Hasse–Weil Zeta Function

We are now ready to prove the main theorem. All the real work has been done, since all the essential ideas appear already for the Dedekind zeta function — our second example above. Indeed, the original definition of the Dedekind zeta function is, from a modern point of view, just a clever repackaging of the Hasse–Weil zeta function. Everything becomes simpler if we work with the Hasse-Weile zeta function itself. We can generalize the usual definition of this function as follows:

Definition: Suppose a functor $S: \Comm\Ring \to Set$ is tame and multiplicative. Define its Hasse–Weil zeta function to equal

$$\prod_p exp \left( \sum_{n \gt 0} \frac{|S(\mathbb{F}_{p^n})|}{n} p^{-n s} \right) \, .$$

Theorem: Suppose a functor $S: \Comm\Ring \to Set$ is tame and multiplicative. Then the Dirichlet series of its Hasse–Weil species

$$\zeta_S(s) = \sum_{n \ge 1} \frac{|Z_S(n)|}{n!} \, n^{-s}$$

is equal to its Hasse–Weil zeta function.

Proof: By (1), in this situation $Z_S$ can be written as a Dirichlet product of Dirichlet exponentials

$$Z_S \cong \prod_p^D \exp_D(F_{S,p})$$

so taking Dirichlet series we have

$$\zeta_S(s) = \prod_p \exp \left( \sum_{n \gt 0} \frac{|F_{S,p}(n)|}{p^n!} p^{-n s} \right) \, .$$

It therefore suffices to show that

$$\frac{p^n!}{n} |S(\mathbb{F}_{p^n})| = |F_{R,p}(p^n)| \, .$$

By definition $|F_{S,p}(p^n)|$ is the number of ways to make an $p^n$-element set into a field, say $k$, and choose an element of $S(k)$. But as we have seen, there are $p^n! / n$ ways to make a $p^n$-element set into a field. So, there are $\frac{p^n!}{n} |S(\mathbb{F}_{p^n})|$ ways to make a $p^n$-element set into a field $k$ and then choose an element of $S(k) \cong S(\mathbb{F}_{p^n})$.

Acknowledgements

We thank Matthew Emerton and other denizens of the n-Category Café for help with Hasse-Weil zeta functions.

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