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## The man who knew elliptic integrals, prime number theorems, and black holes

I went to see The Man Who Knew Infinity yesterday. I have nothing much to say about the movie as a movie that wasn’t already said in Scott Aaronson‘s review, except that I learned a few fun facts during the Q&A session with writer/director Matthew Brown afterwards. Namely, it’s a little surprising the movie was able to get high-profile stars like Dev Patel and Jeremy Irons on board given that it was made on a relatively low budget. Apparently, Dev Patel signed on because he really wanted to popularize the story of Ramanujan, and Jeremy Irons signed on because he was hooked after being given a copy of Hardy’s A Mathematician’s Apology.

(Disclaimer: this blog does not endorse any of the opinions Hardy expresses in the Apology, e.g. the one about mathematics being a young man’s game, the one about pure math being better than applied math, or the one about exposition being an unfit activity for a real mathematician. The opinion of this blog is that the Apology should be read mostly for insight into Hardy’s psychology rather than for guidance about how to do mathematics.)

Anyway, since this is a movie about Ramanujan, let’s talk about some of the math that appears in the movie. It’s what he would have wanted, probably.

## Separable algebras

Let $k$ be a commutative ring and let $A$ be a $k$-algebra. In this post we’ll investigate a condition on $A$ which generalizes the condition that $A$ is a finite separable field extension (in the case that $k$ is a field). It can be stated in many equivalent ways, as follows. Below, “bimodule” always means “bimodule over $k$.”

Definition-Theorem: The following conditions on $A$ are all equivalent, and all define what it means for $A$ to be a separable $k$-algebra:

1. $A$ is projective as an $(A, A)$-bimodule (equivalently, as a left $A \otimes_k A^{op}$-module).
2. The multiplication map $A \otimes_k A^{op} \ni (a, b) \xrightarrow{m} ab \in A$ has a section as an $(A, A)$-bimodule map.
3. $A$ admits a separability idempotent: an element $p \in A \otimes_k A^{op}$ such that $m(p) = 1$ and $ap = pa$ for all $a \in A$ (which implies that $p^2 = p$).

(Edit, 3/27/16: Previously this definition included a condition involving Hochschild cohomology, but it’s debatable whether what I had in mind is the correct definition of Hochschild cohomology unless $k$ is a field or $A$ is projective over $k$. It’s been removed since it plays no role in the post anyway.)

When $k$ is a field, this condition turns out to be a natural strengthening of the condition that $A$ is semisimple. In general, loosely speaking, a separable $k$-algebra is like a “bundle of semisimple algebras” over $\text{Spec } k$.

## Forms and Galois cohomology

Yesterday we gave a brief and abstract description of Galois descent, the punchline of which was that Galois descent could abstractly be described as a natural equivalence

$\displaystyle C(k) \cong C(L)^G$

where $f : k \to L$ is a Galois extension, $G = \text{Aut}(L)$ is the Galois group of $L$ (thinking of $L$ as an object of the category of field extensions of $k$ at all times), $C(k)$ is a category of “objects over $k$,” and $C(l)$ is a category of “objects over $L$.”

In fact this description is probably only correct if $k \to L$ is a finite Galois extension; if $k \to L$ is infinite it should probably be modified by requiring that every function of $G$ that occurs (e.g. in the definition of homotopy fixed points) is continuous with respect to the natural profinite topology on $G$. To avoid this difficulty we’ll stick to the case that $k \to L$ is a finite extension.

Today we’ll recover from this abstract description the somewhat more concrete punchline that $k$-forms $c_k \in C(k)$ of an object $c_L \in C(L)$ can be classified by Galois cohomology $H^1(BG, \text{Aut}(c_L))$, and we’ll give some examples.

## Stating Galois descent

After a relaxing and enjoyable break, we’re finally in a position to state what it means for structures to satisfy Galois descent.

Fix a field $k$. The gadgets we want to study assign to each separable extension $k \to L$ a category $C(L)$ of “objects over $L$,” to each morphism $f : L_1 \to L_2$ of extensions an “extension of scalars” functor $f_{\ast} : C(L_1) \to C(L_2)$, and to each composable pair $L_1 \xrightarrow{f} L_2 \xrightarrow{g} L_3$ of morphisms of extensions a natural isomorphism

$\displaystyle \eta(f, g) : f_{\ast} g_{\ast} \cong (fg)_{\ast}$

of functors $C(L_1) \to C(L_3)$ (where again we’re taking compositions in diagrammatic order) satisfying the usual cocycle condition that the two natural isomorphisms $f_{\ast} g_{\ast} h_{\ast} \cong (fgh)_{\ast}$ we can write down from this data agree. We’ll also want unit isomorphisms $\varepsilon : \text{id}_{C(L)} \cong (\text{id}_L)_{\ast}$ satisfying the same compatibility as before. This is just spelling out the definition of a 2-functor from the category of separable extensions of $k$ to the 2-category $\text{Cat}$, and in particular each $C(L)$ naturally acquires an action of $\text{Aut}(L)$ (where we mean automorphisms of extensions of $k$, hence if $L$ is Galois this is the Galois group) in precisely the sense we described earlier.

We’ll call such an object a Galois prestack (of categories, over $k$) for short. The basic example is the Galois prestack of vector spaces $\text{Mod}(-)$, which sends an extension $L$ to the category $\text{Mod}(L)$ of $L$-vector spaces and sends a morphism $f : L_1 \to L_2$ to the extension of scalars functor

$\displaystyle \text{Mod}(L_1) \ni V \mapsto V \otimes_{L_1} L_2 \in \text{Mod}(L_2)$.

Every example we consider will in some sense be an elaboration on this example in that it will ultimately be built out of vector spaces with extra structure, e.g. the Galois prestacks of commutative algebras, associative algebras, Lie algebras, and even schemes. In these examples, fields are not really the natural level of generality, and to make contact with algebraic geometry we should replace them with commutative rings, but for now we’ll ignore this.

In order to state the definition, we need to know that if $f : k \to L$ is an extension, then the functor $f_{\ast} : C(k) \to C(L)$ naturally factors through the category $C(L)^G$ of homotopy fixed points for the action of $G = \text{Aut}(L)$ on $C(L)$. We’ll elaborate on why this is in a moment.

Definition: A Galois prestack satisfies Galois descent, or is a Galois stack, if for every Galois extension $k \to L$ the natural functor $C(k) \to C(L)^G$ (where $G = \text{Aut}(L) = \text{Gal}(L/k)$) is an equivalence of categories.

In words, this condition says that the category of objects over $k$ is equivalent to the category of objects over $L$ equipped with homotopy fixed point structure for the action of the Galois group (or Galois descent data).

(Edit, 11/18/15:) This definition is slightly incorrect in the case of infinite Galois extensions; see the next post and its comments for some discussion.

## Fixed points of group actions on categories

Previously we described what it means for a group $G$ to act on a category $C$ (although we needed to slightly correct our initial definition). Today, as the next step in our attempt to understand Galois descent, we’ll describe what the fixed points of such a group action are.

John Baez likes to describe (vertical) categorification as replacing equalities with isomorphisms, which we saw on full display in the previous post: we replaced the equality $F(g) F(h) = F(gh)$ with isomorphisms $\eta(g, h) : F(g) F(h) \cong F(gh)$, and as a result we found 2-cocycles lurking in this story.

I prefer to describe categorification as replacing properties with structures, in the nLab sense. That is, the real import of what we just did is to replace the property (of a function between groups, say) that $F(g) F(h) = F(gh)$ with the structure of a family of isomorphisms between $F(g) F(h)$ and $F(gh)$. The use of the term “structure” emphasizes, as we also saw in the previous post, that unlike properties, structures need not be unique.

Accordingly, it’s not surprising that being a fixed point of a group action on a category is also a structure and not a property. Suppose $F : G \to \text{Aut}(C)$ is a group action as in the previous post, and $c \in C$ is an object. The structure of a fixed point, or more precisely a homotopy fixed point, is the data of a family of isomorphisms

$\displaystyle \alpha(g) : c \cong F(g) c$

which satisfy the compatibility condition that the two composites

$\displaystyle c \xrightarrow{\alpha(g)} F(g) c \xrightarrow{F(g)(\alpha(h))} F(g) F(h) c \xrightarrow{\eta(g, h)(c)} F(gh) c$

and

$\displaystyle c \xrightarrow{\alpha(gh)} F(gh) c$

are equal, as well as the unit condition that

$\displaystyle \alpha(e) = \varepsilon(c) : c \to F(e) c$

where $\varepsilon$ is the unit isomorphism $\text{id}_C \cong F(e)$. This is, in a sense we’ll make precise below, a 1-cocycle condition, but this time with nontrivial (local) coefficients.

Curiously, when the action $F$ is trivial (meaning both that $F(g) = \text{id}_C$ and that $\eta(g, h) = e \in Z(C)^{\times}$), this reduces to the definition of a group action of $G$ on $c \in C$ in the usual sense. In general, we can think of homotopy fixed point structure as a “twisted” version of a group action on $c \in C$ where the twist is provided by the group action on $C$.

## The puzzle of Galois descent

Suppose we have a system $f_1, f_2, \dots f_n \in k[x_1, x_2, \dots x_m]$ of polynomial equations over a perfect (to keep things simple) field $k$, and we’d like to consider solutions of it over various field extensions $L$ of $k$. Write $V(L)$ for the set of all solutions to this system over $L$.

As it happens, knowing $V(L)$ for any algebraic extension $L$ of $k$ is equivalent to knowing $V(\bar{k})$, where $\bar{k}$ denotes the algebraic closure of $k$, together with the action of the absolute Galois group $G = \text{Gal}(\bar{k}/k)$. After picking an embedding of $L$ into $\bar{k}$, the infinite Galois correspondence says that $L$ is precisely the set of fixed points of the closed subgroup $H$ of $G$ which stabilizes $L$, and it’s not hard to see that this extends to $V(L)$; that is, $G$ naturally acts on $V(\bar{k})$, and we have a natural identification

$\displaystyle V(L) \cong V(\bar{k})^H$.

Now let’s categorify this situation. Before we considered, for each algebraic extension $L$ of $k$, a set $V(L)$. There are many situations in mathematics in which it’s natural to consider instead a category $F(L)$, such that a morphism $L_1 \to L_2$ induces a functor $F(L_1) \to F(L_2)$, and so forth. The basic example is the case that $F(L) = \text{Mod}(L)$ is the category of $L$-vector spaces, and for $f : L_1 \to L_2$ a morphism the corresponding functor is given by extension of scalars

$\displaystyle \text{Mod}(L_1) \ni V \mapsto V \otimes_{L_1} L_2 \in \text{Mod}(L_2)$.

This leads to many other examples coming from equipping vector spaces with extra structure: for example, $F(L)$ might be

• the category of representations of some finite group $G$ over $L$,
• the category of commutative (or associative, or Lie) algebras over $L$, or
• the category of schemes over $L$.

It would be great if understanding all of these categories was in some sense as simple as understanding the category $F(\bar{k})$, which generally tends to be simpler, and the action of the absolute Galois group $G$ on it, whatever that means. For example, the representation theory of finite groups over algebraically closed fields of characteristic zero is well understood, as is, say, the classification of semisimple Lie algebras. The general problem of trying to extract an understanding of $F(L)$ from an understanding of $F(\bar{k})$ is the problem of Galois descent.

We might very optimistically hope that the story here is directly analogous to the story above. This suggests the following puzzle.

Puzzle: In what sense could the statement $F(L) \cong F(\bar{k})^H$ be true for the examples given above?

## Topological Diophantine equations

The problem of finding solutions to Diophantine equations can be recast in the following abstract form. Let $R$ be a commutative ring, which in the most classical case might be a number field like $\mathbb{Q}$ or the ring of integers in a number field like $\mathbb{Z}$. Suppose we want to find solutions, over $R$, of a system of polynomial equations

$\displaystyle f_1 = \dots = f_m = 0, f_i \in R[x_1, \dots x_n]$.

Then it’s not hard to see that this problem is equivalent to the problem of finding $R$-algebra homomorphisms from $S = R[x_1, \dots x_n]/(f_1, \dots f_m)$ to $R$. This is equivalent to the problem of finding left inverses to the morphism

$\displaystyle R \to S$

of commutative rings making $S$ an $R$-algebra, or more geometrically equivalent to the problem of finding right inverses, or sections, of the corresponding map

$\displaystyle \text{Spec } S \to \text{Spec } R$

of affine schemes. Allowing $\text{Spec } S$ to be a more general scheme over $\text{Spec } R$ can also capture more general Diophantine problems.

The problem of finding sections of a morphism – call it the section problem – is a problem that can be stated in any category, and the goal of this post is to say some things about the corresponding problem for spaces. That is, rather than try to find sections of a map between affine schemes, we’ll try to find sections of a map $f : E \to B$ between spaces; this amounts, very roughly speaking, to solving a “topological Diophantine equation.” The notation here is meant to evoke a particularly interesting special case, namely that of fiber bundles.

We’ll try to justify the section problem for spaces both as an interesting problem in and of itself, capable of encoding many other nontrivial problems in topology, and as a possible source of intuition about Diophantine equations. In particular we’ll discuss what might qualify as topological analogues of the Hasse principle and the Brauer-Manin obstruction.

## The p-group fixed point theorem

The goal of this post is to collect a list of applications of the following theorem, which is perhaps the simplest example of a fixed point theorem.

Theorem: Let $G$ be a finite $p$-group acting on a finite set $X$. Let $X^G$ denote the subset of $X$ consisting of those elements fixed by $G$. Then $|X^G| \equiv |X| \bmod p$; in particular, if $p \nmid |X|$ then $G$ has a fixed point.

Although this theorem is an elementary exercise, it has a surprising number of fundamental corollaries.

## Small factors in random polynomials over a finite field

Previously I mentioned very briefly Granville’s The Anatomy of Integers and Permutations, which explores an analogy between prime factorizations of integers and cycle decompositions of permutations. Today’s post is a record of the observation that this analogy factors through an analogy to prime factorizations of polynomials over finite fields in the following sense.

Theorem: Let $q$ be a prime power, let $n$ be a positive integer, and consider the distribution of irreducible factors of degree $1, 2, ... k$ in a random monic polynomial of degree $n$ over $\mathbb{F}_q$. Then, as $q \to \infty$, this distribution is asymptotically the distribution of cycles of length $1, 2, ... k$ in a random permutation of $n$ elements.

One can even name what this random permutation ought to be: namely, it is the Frobenius map $x \mapsto x^q$ acting on the roots of a random polynomial $f$, whose cycles of length $k$ are precisely the factors of degree $k$ of $f$.

Combined with our previous result, we conclude that as $q, n \to \infty$ (with $q$ tending to infinity sufficiently quickly relative to $n$), the distribution of irreducible factors of degree $1, 2, ... k$ is asymptotically independent Poisson with parameters $1, \frac{1}{2}, ... \frac{1}{k}$.

Today’s post is a record of a very small observation from my time at PROMYS this summer. Below, by $\text{Spec } R$ I mean a commutative ring $R$ regarded as an object in the opposite category $\text{CRing}^{op}$.