Iwasawa theory for chromatic localizations

I’m visiting MIT for a semester, which has already been a real delight — it’s been a long time since I’ve been around so many people all interested in Morava $E$-theory and willing to sit in a room to discuss it. That really doesn’t happen anywhere else on the planet. There are a lot of things to try proving about $E$-theory, lots considered low-hanging fruit and a few things not so simple. I want to tell you about one of the more fantastical ones that’s not so simple to even state clearly, nevermind prove.

This story starts with the calculation of the homotopy of the $K(1)$-local sphere. The chromatic spectral sequence for this is concentrated in two lines: $H^{0, *}(\mathbb{S}_1; E_1)$ is a $\mathbb{Z}_p$ concentrated in degree $0$, and $H^{1, 1+*}(\mathbb{S}_1; E_1)$ is a scattering of groups, collectively called the $\alpha$-family, which altogether take the following form:

What’s of course implicit in even talking about the $K(1)$-local sphere is that we’re also working locally against some (here odd) prime $p$. It would be nice to be able to stitch these answers together, to have some integral data which reduces to this when requested.

Such objects exist (and are even sort of unsurprising if you’ve spent enough time around $K$-theory): the Bernoulli numbers $B_{2s}/(2s)$ have the property that their denominators have exactly the right $p$-local reduction. So, another way to write the answer above (ignoring the non-patterned group at $0$) is

That’s pretty cool.

Now here’s a different thing to notice about this answer: another way to write $\mathbb{Z}_p / (ps)$ is as $\mathbb{Z}_p / p^{1-\nu_p s}$, where $\nu_p$ is the $p$-adic valuation. This means two things: one is that you can replace the integer $s$ by a $p$-adic integer instead, and the computation will still give an answer, and two, the answer that you get is “continuous in $s$”, in the sense that it’s (up to isomorphism type) locally constant.

This spurred a(n unfortunately short-lived) line of investigation in the early 90s, as this lines up with yet another computation. The $K(1)$-local category comes with a monoidal structure defined so that $K(1)$-localization and $K(1)$-homology itself are both monoidal functors. Since all the functors we care about preserve the monoidal structure, it’s fruitful to ask questions about its invariants: for instance, what is the group of “invertible” elements, i.e., which $K(1)$-local spectra $X$ have pair spectra $X^{-1}$ such that $X \hat\wedge X^{-1} \simeq \mathbb{S}^0_{K(1)}$?

It turns out that the answer is: a lot. The standard spheres $\mathbb{S}^n$ always embed into this group (called the Picard group), which can also be shown to be a pro-$p$-group, and so it’s guaranteed to be some enormous thing. You can ask this question in the $K(n)$-local category for any $n$, but in the case that $n = 1$, this actually turns out to be it — the $K(1)$-local Picard group is the $p$-adic completion of the standard spheres. (The construction of such a $p$-adic sphere is neat and simple, but I think it’s orthogonal to the rest of what I want to say, so I’ll omit it.)

And so, putting all these thoughts next to each other, the question naturally arises: if the integral homotopy groups are graded over the integers (i.e., the standard spheres), should $K(1)$-local homotopy groups be indexed by the $K(1)$-local Picard group? Our answer for $\pi_* \mathbb{S}^0_{K(1)}$ certainly suggests that this is at least not an outright insane question. In fact, if you build these p-adic spheres, you see that their homotopy groups even match those suggested in the answer above.

So, what is this p-adic function that we’re sampling? We get a hint by noticing another place where Bernoulli numbers appear: they arise as the negative odd integer values of the Riemann zeta function,

So, one thing we’re doing is taking a dense sequence of zeta values in the $p$-adic integers and interpolating — this sounds a lot like it deserves to be called a $p$-adic zeta function.

…And it sort of is. Both fortunately and unfortunately, number theorists are light years ahead of us, and they’ve decided that “$p$-adic zeta function” ought to mean something slightly different — namely, that it should live in the land of Galois theory. The name for this set-up is “Iwasawa theory,” and while I understand very little of it, I’m prepared to say a very few words: start with the Galois group $G = \mathrm{Gal}(\mathbb{Q}(\zeta_{p^\infty}) : \mathbb{Q})$. This comes with a natural character $\kappa: G \to \mathbb{Z}_p^\times$ determined by $\zeta_{p^n}^{\kappa(\sigma)} = \sigma \cdot \zeta_{p^n}$ for any $\sigma$ in the Galois group $G$. As a general source of inputs to $\kappa$, we can consider the profinitely continuous group-ring $\Lambda = \mathbb{Z}_p[\![G]\!]$; an element of $\Lambda$ we think of as a holomorphic function, and the pushforward $\kappa^{2r}_*: \Lambda \to \mathbb{Z}_p$ we think of as evaluation at $r$. Then, the $p$-adic zeta function is a certain element of the fraction field of $\Lambda$ (i.e., a “meromorphic function”), which interpolates the complex zeta function at our chosen points and which has dramatic constraints on its poles.

This is a lot to swallow, and it’s not clear how to connect all this silly Galois theory up to homotopy theory conceptually — but Neil Strickland has plunged ahead and twisted up our computation our the homotopy groups of the $K(1)$-local sphere to use all these same letters. It’s an amusing exercise to expand out, and Neil’s note about this is a little terse, so I want to take the time to write it down now:

We’ll deal just with the “even” part of the $K(1)$-local Picard group, which is isomorphic to $\mathbb{Z}_p^\times$. We interpret this as $\mathbb{Z}_p^\times \cong \operatorname{End}(\mathbb{Z}_p^\times)$, which sends a $p$-adic integer $b$ to the function $\beta: x \mapsto x^b$. Our replacement for $\Lambda$ is simply $\Lambda = \mathbb{Z}_p[\![\mathrm{Pic}_1^*]\!]$, and we construct a single $\Lambda$-module $\pi \mathbb{S}^0_{K(1)}$ and an auxiliary module $M_\lambda$ for each $\lambda \in \mathrm{Pic}_1$, such that

Here’s how we do it: take $\pi \mathbb{S}^0_{K(1)}$ to be $\mathbb{Z}_p$ with the trivial $\mathrm{Pic}_1^*$-action, and take $M_\lambda$ to be $\mathbb{Z}_p$ with the action $\beta \cdot x = \beta(\lambda)x = \lambda^b x$.

With all this in hand, we make the following calculation, taking $x \otimes y$ to be a pure tensor built out of generators for $\pi \mathbb{S}^0_{K(1)}$ and $M_\lambda$, and $b$ to be a topological generator:

and hence $0 = (1 - \lambda^b)(x \otimes y)$. We are then presented with options: in the case $\lambda = 1 + ps$, this relation produces the quotient $\mathbb{Z}_p / (ps)$, and in every other case this coefficient is a unit and so the whole thing collapses. So, (up to mistakes in my exposition — I see that my grading is off by a sign, at least) we recover the answer that we want, phrased in terms friendly to Iwasawa theory. Pretty cool!

The reason this was abandoned is that a few people thought really hard about the $K(2)$-local case, couldn’t figure it out, and then got distracted by much more attractive and more modern developments in homotopy theory. However, Mark Behrens published a result a few years ago that indicates there may yet be hope for a $K(2)$-local statement. Mark’s result concerns detecting the $\beta$ family, which are a bunch of analogous elements in $H^{2, *}$ at chromatic height $2$, using (spectra related to) $\operatorname{TMF}$. If $\operatorname{TMF}$ picks up these elements, then their existence and nonexistence should be reflected and recorded by statements about modular forms. This turns out to be exactly the case: the element $\beta_{i/j,k}$ exists exactly when a modular form $f_{i/j,k}$ exists, satisfying some minimality property and a variety of complicated and strange conditions (even for seasoned number theorists!) about the weights of $f$ and its reductions.

Now, what might this mean in the context of the above program? Well, remember that in the end this all comes down to picking off the denominator of some Bernoulli numbers, which is supposed to tell you how many times you can divide your favorite $\alpha$-element by $p$. To phrase this operation algebraically, you’d want to follow the composite

which exactly detects the fractional part of the $p$-adic rational special value of the zeta function. Analogously, the divisibility properties of modular forms are encoded in the theory of $L$-functions. Moreover, there’s even an analogous quotient: $L$-functions are valued in modular forms, and $p$-adic modular forms are built out of integral modular forms essentially by inverting the Eisenstein series. Hence, one could hypothesize the existence of a fantastical $p$-adic $L$-function whose composite

records the divisibilities in the $\beta$-family.

Now, that’s already a lot to ask for — but, ideally, this would all fit into an Iwasawa theory for $\mathrm{Pic}_2$, which would wrap up the homotopy groups of $\mathbb{S}^0_{K(2)}$ too. You’ll notice that the module $\pi \mathbb{S}^0_{K(1)}$ was extraordinarily simple. This is partly because the actual homotopy groups are not so awful, but it’s also reflective of a broader phenomenon in number theory: computing special values of modular forms / $L$-functions / whatever is potentially very difficult, but the objects themselves are often easier to deal with “in the aggregate.” By the same token, one could hope for a sort of $L$-function which “embodies” the $K(2)$-local homotopy groups of spheres in a way that gave interesting structural information about them without requiring us to unpack the $\Lambda$-module into the individual groups.

This would be a really extraordinary victory for stable homotopy theory, I feel.

Two things worth reading in connection to this topic are: Steve Mitchell’s chapter $K(1)$-local homotopy theory, Iwasawa theory, and algebraic $K$-theory of the $K$-theory Handbook, as well as his student Rebekah Hahn’s PhD thesis Iwasawa theory for $K(1)$-local spectra. They, in turn, cite inspirational articles by Bousfield and Ravenel, which are also worth considering but don’t do much investigation along these lines. Finally, certainly Mike Hopkins’s name should be all over this document; many (or maybe all) of these ideas trace back to him.