On L-functions and algebraic K-theory

Hi all! In this post, I’ll attempt to make some progress towards demystifying the relationship between L-functions and algebraic K-theory, with reference to a very simple example. I’ll say something about L-functions a little lower down, but the only thing I’ll say about K-theory in general is that it’s a way to get a spectrum from, say, a scheme.

The deep connection between these two objects is supposed to be exemplified by the following very approximate and possibly incorrect statement.

Ur-conjecture. [Lichtenbaum, Bloch-Kato, Beilinson, Scholbach…] Let $X$ be a geometric object, probably a scheme of a particular kind, and let $\zeta_X$ be the zeta function associated to $X$. Then for $n$ a sufficiently large positive integer,

where $R(n)$ is some factor. Obviously this is completely meaningless until you know what $R(n)$ is, but people do have expressions for it (a relevant buzzword here is “Borel regulator”), about which I’ll say no more. The point is that there are formulae for values of L-functions at integers (and I’m sorry, but I’ll use “L-function” and “zeta function” interchangeably) involving torsion orders of K-groups.

A conjecture of this form has been proven for $X$ the ring of integers of a totally real abelian field by piling Iwasawa theory and other manoeuverings onto the cases of the Quillen-Lichtenbaum conjecture proven by Voevodsky. It’s a humongous proof. In general, these conjectures are wide open and should be regarded with reverence.

## L-functions

Before I really get going, though, I’m going to say a word about L-functions, which by convention are meromorphic functions on $\mathbb{C}$ with certain properties. I’m going to list these salient properties in bold, and illustrate them using the historically first L-function, which is of course the Riemann zeta $\zeta(s)$. We’ll later find that each of these properties has a purely homotopy-theoretic analogue on the K-theory spectrum.

Dirichlet series. To an L-function $L$ is associated a Dirichlet series expansion

with $a_n \in \mathbb{C}$. $\zeta(s)$ has $a_n = 1$ for all $n$.

Abscissa of convergence. There is some $\sigma \in \mathbb{R}$, called the abscissa of convergence, such that the Dirichlet series converges to $L(s)$ for $\mathbf{Re}(s) > \sigma$. The abscissa of convergence of $\zeta$ is 1.

Euler product expansion. There is an infinite product, typically over closed points of a scheme, of nice simple functions that also converges to $L(s)$ on some respectable proportion of $\mathbb{C}$. The Euler product expansion of $\zeta$ is

Analytic continuation. This is implicit in the fact that I said an L-function was a meromorphic function on $\mathbb{C}$ in the first place, but if you take the more practical view that an L-function is defined by its Dirichlet series or Euler product, there’s some content here, and analytic continuation statements are usually hard to prove.

Functional equation. An L-function has a (possibly twisted) reflective symmetry: there is some real number $\tau$, frequently equal to the abscissa of convergence, such that

For the Riemann zeta, we have (lifted from Wikipedia) $\tau = 1$ and

This just comes from the Archimedean L-factor (we’ll say nothing about that here) but there can be other things too. Analytic continuation and functional equation are usually proved together.

You probably expect to be able to formulate a Riemann hypothesis for a general L-function, too, but we won’t get into that here.

## The simplest L-function

I said the Riemann zeta was the historically first L-function, but it’s not the simplest. The simplest is the zeta function of a finite field $\mathbb{F}_q$, which is a single Euler factor:

The number theorists probably use different notation for this, but I am a carefree rogue who does as he wills.

$\Xi_q$ is manifestly meromorphic; it is its own Euler product; its Dirichlet series is given by expanding the above expression as a geometric series, with abscissa of convergence $0$; and its functional equation is

This is the only example of an L-function which one can comfortably manipulate with one’s hands without substantial analytic ingenuity. It’s this basic example that we’ll spend much of the rest of this blog post studying.

## The K-theory of a finite field

In a 1972 paper, Quillen proved one of my all-time favourite theorems by describing the homotopy type of the K-theory spectrum of a finite field.

Theorem. [Quillen] The connected component $K(\mathbb{F}_q)_0$ of the $0$-space of the K-theory spectrum of $\mathbb{F}_q$ fits into a fibre sequence

where $\psi^q$ is the $q$th Adams operation.

This is the only example of a (global) K-theory spectrum which one can comfortably manipulate with one’s hands without substantial topological ingenuity. By Bott periodicity, we know the homotopy groups of BU:

We also know that $\psi^q$ acts by multiplication by $q^m$ on $\pi_{2m} BU$. That means we can read off the positive-degree homotopy groups of $K(\mathbb{F}_q)$:

Motivated by the ur-conjecture stated above, let’s define a function $f_q : \mathbb{N} \to \mathbb{C}$ by

Suppose, as if possessed by a ghost, you wrote this function down without ever having heard of an L-function. You would still notice that

is the restriction to $\mathbb{N}$ of an obvious complex-analytic function: $-\Xi_q$. This is the easiest case of the conjecture.

Now what of the values of $\Xi_q$ at negative integers? We might hope that they have something to do with the negative homotopy groups of $K(\mathbb{F}_q)$, which is, after all, a spectrum. But as things stand, these negative homotopy groups are zero. This rift is signaling that the arithmetic incarnation of the bare K-theory spectrum is not the L-function; rather, it’s the Dirichlet series of the L-function, or the part of the L-function to the right of the abscissa of convergence. The topological object that we should really associate to the full L-function is the K(1)-localisation of the K-theory spectrum. To do this, we need to fix an ambient prime $p$, and let’s make sure it’s different from the characteristic of $\mathbb{F}_q$.

## The K(1)-localisation of the K-theory of a finite field

I’m going to self-consciously write out all $L_{K(1)}$s in full. It’ll take up space, but hopefully it’ll aid clarity of notation.

Here’s a slight variant of Quillen’s calculation; if you like, it arises by applying the Bousfield-Kuhn functor to the statement of Quillen’s theorem above.

Proposition. The K(1)-localised K-theory spectrum $L_{K(1)} K(\mathbb{F}_q)$ fits into a fibre sequence

Thus the homotopy groups of $L_{K(1)} K(\mathbb{F}_q)$, positive and negative, are given by

Until further notice, when we talk about the “value” of an L-function at a point, we’ll mean up to multiplication by a rational number which is a p-adic unit. With this caveat, we now have the identity

for all $n \in \mathbb{Z}$, except at $n = 0$, where things are untidy because $\Xi_q$ has a pole and both homotopy groups in the formula are infinite, and a regulator gets involved. That’s nice.

We need to make a couple more observations about the homotopy groups of the localised K-theory spectrum.

First, the natural map

is an equivalence in degrees greater than $0$. (It’s also an equivalence in degree $0$, but I’d like to propose that we regard that as an accident.) This $0$ is the very same $0$ that occurs as the abscissa of convergence of $\Xi_q$, and it also shows up as the étale cohomological dimension of $\mathbb{F}_q$ minus 1 - a number I’d like to refer to as the Quillen-Lichtenbaum constant of $\mathbb{F}_q$.

Second, the functional equation for $\Xi_q(n)$ is reflected in the homotopy groups of $L_{K(1)} K(\mathbb{F}_q)$ as follows. For any integer $m$, the $p$-adic valuation of $q^m - 1$ is the same as that of $q^{-m} - 1$. Thus for any odd integer $n$, we have an isomorphism

Can this apparent duality be given a topological explanation?

We’re going to do this by invoking a total of four or five different dualities, one of which is deep and will be totally blackboxed, so get ready.

Let’s start tackling this by rearranging the above isomorphism slightly. Both of the groups there arise as quotients of $\mathbb{Z}_p$, so they come with a choice of generator: the image of $1$. That means that either group can be identified with its Pontryagin dual, so let’s do it on the left:

There, now it looks more like a sensible duality. And the Pontryagin dual in there is a strong hint that Brown-Comenetz duality is involved. Briefly, the Brown-Comenetz dualising spectrum is a spectrum $I$ defined by the natural isomorphism

for spectra $E$. To save pixels, we’ll write $IE$ for $\mathbf{Map}(E, I)$. Now our isomorphism reads

But height 1 Gross-Hopkins duality states that for a K(1)-local spectrum E,

Here $M$ is the first monochromatic slice functor, which for a K(1)-local spectrum is just the fibre of the map to its rationalisation. $M$ isn’t doing anything too destructive here, so let’s omit it for now and figure out its role in a moment. We’ve arrived at

at least for odd $n \neq -1$. So we’d be home if $K(\mathbb{F}_q)$ were to be K(1)-locally Spanier-Whitehead self-dual. And it is! I learned this in Dustin Clausen’s thesis defence, but one can prove it by showing that

exhibits $KU^\wedge_p$ as K(1)-locally self-dual up to a shift of 1, compatibly with Adams operations, and plugging this into the fiber sequence for $L_{K(1)}K(\mathbb{F}_q)$. And thus the duality is explained.

Let’s circle back for a moment and say a quick word about the role of the monochromatic slice functor $M$. Since $L_{K(1)} K(\mathbb{F}_q)$ doesn’t have many torsion-free homotopy groups, the only effect of $M$ is to convert the $\mathbb{Z}_p$s in degrees $0$ and $-1$ into $\mathbb{Q}_p/ \mathbb{Z}_p$s in degrees $-1$ and $-2$. But that was exactly what was needed to get the original duality statement to work in those degrees, once we put in the Pontryagin dual. Thus the duality is not merely explained but embellished with a pristine little bow.

## The dictionary, and some questions

As promised, we’ve given homotopy-theoretic interpretations of (almost) all the fundamental analytic properties of L-functions. Here’s a table to summarise:

Analysis   Topology

L-function   K(1)-localised K-theory spectrum
Dirichlet series   Bare K-theory spectrum
Abscissa of convergence   Quillen-Lichtenbaum constant
Euler product   ?? (But see below)
Analytic continuation   K(1)-localisation
Functional equation   Gross-Hopkins + Spanier-Whitehead duality

(I apologise for the poor formatting - I’m not used to Markdown, and the latex support doesn’t seem to run to table environments).

Some questions I don’t know how to answer at this point:

• Let X be a Dedekind domain, L its field of fractions, and $k(x)$ the residue field at a closed point $x \in X$. Then there’s a cofiber sequence

The left-hand arrow looks an awful lot like it’s trying to express K(X) as an Euler product over x of K(k(x)), but I don’t know how to make that precise. K(L) is a nuisance.

• Now let X be a smooth proper variety over $\mathbb{F}_q$. There’s a spectral sequence with $E_2$ term given by étale cohomology of X with suitably Tate-twisted $\mathbb{Z}_p$ coefficients which abuts to the K(1)-local K-theory of X. Can we use this to relate the formula for the zeta function of X in terms of the characteristic polynomial of Frobenius on étale cohomology to a K-theoretic description of the special values?

• What’s the analogue of $K(\mathbb{F}_q)$ for an archimedean L-factor? This might be related to work of Connes and Consani on cyclic homology with adelic coefficients.

This section is tangential to the rest of the post, but I think it suggests some interesting avenues of investigation. Up until now we’ve been discussing L-functions defined on $\mathbb{C}$, but L-functions defined on $\mathbb{Z}_p$ are also of interest to number theorists. We’ll construct the p-adic analogue of an Euler factor, and see that it throws up some unexpected topology.

In this section, we’ll assume that $p$ is odd. Things are probably substantially more awkward when $p = 2$.

Let’s return (up to sign) to our function $f_q$, this time regarded as a function $\mathbb{N} \to \mathbb{Q}_p$:

We’d like to extend $f_q$ to a continuous function $E_q: \mathbb{Z}_p \to \mathbb{Q}_p$. p-adic continuity sounds weak compared to complex analyticity, but it’s actually kind of hard to be p-adically continuous, and since $\mathbb{N}$ is dense in $\mathbb{Z}_p$, we’re going to have at most one choice of extension.

In fact, we don’t have any choices, a fact which won’t surprise anyone who’s studied p-adic L-functions before. The proof of this is a byproduct of the construction of the real p-adic L-function. Let

Observe that $S$ is already dense in $\mathbb{Z}_p$, so if we can find a continuous function $\mathbb{Z}_p \to \mathbb{Q}_p$ that agrees with $f_q$ on $S$, that’s the best we can do, whether the two functions agree on the whole of $\mathbb{N}$ or not.

Here’s the crux: There’s a continuous function

known as the Iwasawa logarithm, and a continuous function

known as the p-adic exponential. They have most of the properties you’d expect of a pair of functions called $\exp$ and $\log$, except that the identity

only holds when $x$ is congruent to $1$ mod $p$. In general,

where the integer $r$ and the $(p-1)$st root of unity $\omega$ are uniquely determined by the requirement that $p^r \omega x \equiv 1 \mod p$.

So we can follow our hearts and define

but we have to live with the fact that $E_q(n) = f_q(n)$ only when $n \in S$. For general $n \in \mathbb{N}$,

for a suitable $(p-1)$st root of unity $\omega$.

Fine. Where’s the topology? What spectrum is $E_q$ telling us about? At integers outside $S$, the values of $E_q$ don’t match up with the homotopy groups of $L_{K(1)} K(\mathbb{F}_q)$ anymore. Instead,

for all integers $n \neq 0$, where M is a K(1)-local spectrum defined as the fiber of

$\psi^{\omega q} - 1 : KU^\wedge_p \to KU^\wedge_p$.

So, a couple more questions:

• What’s special about M? Why is it any better than $L_{K(1)}K(\mathbb{F}_q)$? What other spectra are “p-adically continuous” in this sense?

• Constructing p-adic L-functions more serious than $E_q$ usually requires some fairly heavy p-adic analysis. Can we partly bypass this by proving p-adic continuity results for the Picard-graded homotopy groups of certain K(1)-local spectra and using the orders of these homotopy groups? I believe some such continuity results already exist, in the folklore if not in the literature.

Thanks for reading this many words! Let me finish up by remarking that some of the stuff in this post carries through to K-theory spectra of rings of integers, and using work of Dwyer and Mitchell as a bridge, one can obtain results on the special values of Kubota-Leopoldt-type p-adic zeta functions. If I can think of enough interesting stuff to say about that, it might be the subject of a future post.

I think the Chromotopy platform doesn’t currently support blog comments, but if you have any comments or questons, please email me at sglasman at math dot mit dot edu.

Update: Dustin Clausen emailed me with some helpful comments about the Spanier-Whitehead self-duality of $L_{K(1)} K(\mathbb{F}_q)$, resulting in some alterations in the text.