Archive

Posts Tagged ‘finite fields’

Grothendieck’s functions-sheaves correspondence

21 November 2013 3 comments

Today begins a shift in perspective for this series of posts. Whereas before we took as our basic objects of study the (abelian) étale covers of a smooth projective curve X, from now on we will study (rank 1) \ell-adic local systems on X, which can be considered as a “linearized” version of the problem. More precisely, since understanding the structure of the fundamental group \pi_1(X,\overline{\eta}) (here \overline{\eta} is the geometric generic point, not that it really matters) is tantamount to understanding all étale covers of X, the Tannakian philosophy says that we should consider the monoidal category of finite-dimensional representations of \pi_1(X,\overline{\eta}), from which the group can be recovered. Just as in topology, representations of the fundamental group are equivalent to local systems.

In fact, the other side of the Artin reciprocity map also has a useful interpretation in terms of \ell-adic local systems: characters of the Picard group \text{Pic}(X) (more precisely, its profinite completion) correspond to “multiplicative” rank 1 local systems on the Picard scheme \text{Pic}_X. This situation is very special to finite fields! A multiplicative local system on \text{Pic}_X can be pulled back along the Abel-Jacobi map X \to \text{Pic}_X, given by x \mapsto \mathscr{O}(x) on points, and global unramified class field theory says that this pullback establishes a bijection between multiplicative local systems on \text{Pic}_X and rank 1 local systems on X. This reformulation of class field theory is true over any field whatsoever, and has a beautiful geometric proof due to Deligne which we will hopefully get to next time.

Fix a prime \ell (traditionally one assumes \ell is not equal to p, the characteristic of our finite ground field \mathbb{F}_q, but for us \ell = p is fine). Let \overline{\mathbb{Q}}_{\ell} be an algebraic closure of the \ell-adic numbers \mathbb{Q}_{\ell}. The totally disconnected topology on \overline{\mathbb{Q}}_{\ell} makes it better suited to our purposes than the complex numbers, although it is worth mentioning that they are isomorphic as discrete fields. For the most part we will only care that \overline{\mathbb{Q}}_{\ell} is an algebraically closed field of characteristic zero.

Let S be a connected scheme and \overline{s} : \text{Spec } \Omega \to S a geometric point.

Definition An \ell-adic local system \mathscr{F} on S is a finite-dimensional continuous \overline{\mathbb{Q}}_{\ell}-representation of \pi_1(S,\overline{s}). The dimension of the representation is called the rank of \mathscr{F}.

As our terminology and notation suggest, \ell-adic local systems can be thought of as locally constant sheaves of \overline{\mathbb{Q}}_{\ell}-vector spaces. This is not literally true, however: in order to get nontrivial local systems one must first consider locally constant étale sheaves over finite coefficient rings, pass to pro-systems of these sheaves, then localize (or kill torsion) to obtain a \overline{\mathbb{Q}}_{\ell}-linear category, which one can prove is monoidally equivalent to representations of the fundamental group. It seems clear that there is no such procedure for complex coefficients. Although these beasts are not literally sheaves, most of our sheafy intuition and technique applies, whence the power of this approach.

We will choose our notation accordingly: for instance, if f : T \to S and \overline{t} \in f^{-1}(\overline{s}), we will write f^*\mathscr{F} for the local system on T obtained by restricting \mathscr{F} along the homomorphism \pi_1(T,\overline{t}) \to \pi_1(S,\overline{s}) induced by f. The underlying \overline{\mathbb{Q}}_{\ell}-vector space of the representation \mathscr{F} is denoted by \mathscr{F}_{\overline{s}}.

Now we come to the namesake of this post, for which we should assume that S is defined over \mathbb{F}_q. This construction takes an \ell-adic local system \mathscr{F} on S and produces from it a function t_{\mathscr{F}} : S(\mathbb{F}_q) \to \overline{\mathbb{Q}}_{\ell}. Given x : \text{Spec } \mathbb{F}_q \to S,  there is an isomorphism \pi_1(S,\overline{x}) \cong \pi_1(S,\overline{s}) well-defined up to conjugation, so that after making this choice we can form the pullback x^*\mathscr{F}, a local system on \text{Spec } \mathbb{F}_q. Then we set

t_{\mathscr{F}}(x) = \text{tr}(\text{Frob};x^*\mathscr{F}),

the trace of the action of the Frobenius a \mapsto a^q, which does not depend on our choice because the trace is conjugation-invariant.

We will be interested in the case  where \mathscr{F} has rank 1, and then it is easier to describe t_{\mathscr{F}}. Namely, x determines a canonical map \widehat{\mathbb{Z}} = \pi_1(\mathbb{F}_q) \to \pi_1(S,\overline{s})^{\text{ab}}, which we compose with \mathscr{F} (now thought of a one-dimensional representation) to obtain a homomorphism \widehat{\mathbb{Z}} \to \overline{\mathbb{Q}}_{\ell}^{\times}. Evaluate this map at 1 to obtain t_{\mathscr{F}}(x).

Sometimes interesting classes of functions and sheaves match up under this correspondence. Let G be a commutative algebraic group over \mathbb{F}_q: then one such class of functions is the group of characters of G(\mathbb{F}_q), meaning one-dimensional representations G(\mathbb{F}_q) \to \overline{\mathbb{Q}}_{\ell}^{\times}. What sort of local system \mathscr{F} on G has the property that t_{\mathscr{F}} is a character?

Definition A rank 1 local system \mathscr{F} on G is called a character sheaf provided that \mu^*\mathscr{F} \cong \mathscr{F} \boxtimes \mathscr{F}, where \mu : G \times G \to G is the multiplication map.

(Notation for those who haven’t seen it: if \mathscr{F} is a sheaf on S and \mathscr{G} is a sheaf on T, then their external tensor product is \mathscr{F} \boxtimes \mathscr{G} = p_S^*\mathscr{F} \otimes p_T^*\mathscr{G} where p_S : S \times T \to S and p_T : S \times T \to T are the projections.)

Sometimes character sheaves are called multiplicative local systems. The latter terminology is arguably better, since “character sheaf” has other meanings. This is analogous to how “character” can refer not only to a one-dimensional representation but also to the trace function associated to a higher-dimensional representation.

Before we prove the main result of this post, we need a lemma.

Lemma Let \mathscr{F} be an \ell-adic local system on S. Then there is a canonical isomorphism \mathscr{F} \to \text{Fr}_S^*\mathscr{F}.

Proof. We reduce immediately to the case that \mathscr{F} is a locally constant étale sheaf of finite sets, so there is some finite étale map f : T \to S whose sheaf of local sections is \mathscr{F}. Write \text{Fr}_S^*T for the fiber product of \text{Fr}_S : S \to S and f, so it suffices to produce an isomorphism T \to \text{Fr}_S^*T of S-schemes. Using the relation f \circ \text{Fr}_T = \text{Fr}_S \circ f, we obtain the desired map g : T \to \text{Fr}_S^*T from \text{Fr}_T and f. Since f and \text{Fr}_S^*T \to S are finite étale, so is g, and similarly g is radicial (i.e. universally injective) because \text{Fr}_T is. But a map which is both étale and radicial must be an open embedding, and g is also finite, hence an isomorphism.

\Box

Now we come to the really interesting part. As usual G is a commutative algebraic group over \mathbb{F}_q, which we now assume to be smooth and connected (the smoothness hypothesis is really not necessary).

Proposition Under the above assumptions, \mathscr{F} \to t_{\mathscr{F}} is a bijection from character sheaves on G to characters of G(\mathbb{F}_q).

Proof. That t_{\mathscr{F}} is a character follows from the easy identities t_{f^*\mathscr{F}} = f^*t_{\mathscr{F}} and t_{\mathscr{F} \otimes \mathscr{G}} = t_{\mathscr{F}} \cdot t_{\mathscr{G}}. Suppose that we are given a character \chi : G(\mathbb{F}_q) \to \overline{\mathbb{Q}}_{\ell}^{\times}. The Lang isogeny L : G \to G is a pointed finite Galois covering with group G(\mathbb{F}_q), hence gives rise to a map \pi_1(G,\overline{1}) \to G(\mathbb{F}_q), which we can restrict \chi along to obtain a rank 1 local system \mathscr{F}(\chi) on G. The same identities show that \mathscr{F}(\chi) is a character sheaf if one argues in the opposite direction.

It remains to show that these constructions are mutually inverse, and first we’ll check that t_{\mathscr{F}(\chi)} = \chi. Given x \in G(\mathbb{F}_q), we obtain a canonical map \widehat{\mathbb{Z}} = \pi_1(\mathbb{F}_q) \to \pi_1(G,\overline{1})^{\text{ab}}, whose value on 1 = \text{Frob} we will call \text{Frob}_x. By definition t_{\mathscr{F}(\chi)}(x) is the value of \mathscr{F}(\chi) on \text{Frob}_x, but \mathscr{F}(\chi) factors through \chi : G(\mathbb{F}_q) \to \overline{\mathbb{Q}}_{\ell}^{\times} by construction, so it suffices to show that the map \pi_1(G,\overline{1}) \to G(\mathbb{F}_q) induced by L sends \text{Frob}_x to x. This means that when \text{Frob}_x acts on the fiber L^{-1}(\overline{1}) = G(\mathbb{F}_q), it just translates by x. Fixing \overline{y} \in L^{-1}(\overline{x}), we obtain an identification L^{-1}(\overline{1}) \cong L^{-1}(\overline{x}) since L^{-1}(\overline{x}) = \overline{y}G(\mathbb{F}_q). Tracing through definitions, our claim follows from the calculation

\text{Frob}_x \cdot \overline{y} = \text{Fr}_G(\overline{y}) = \overline{x} \overline{y}.

Finally, we must prove that \mathscr{F}(t_{\mathscr{F}}) = \mathscr{F}. We claim that L^*\mathscr{F} is trivial, or equivalently that \mathscr{F} factors through the homomorphism \pi_1(G,\overline{1}) \to G(\mathbb{F}_q) determined by L. Since this map sends \text{Frob}_x \mapsto x the claim implies that \mathscr{F} is determined by its values on \text{Frob}_x for all x \in G(\mathbb{F}_q), and we just proved that t_{\mathscr{F}(t_{\mathscr{\chi}})} = t_{\mathscr{F}}, from which it follows that \mathscr{F}(t_{\mathscr{F}}) = \mathscr{F}. As for the claim, observe that

L^*\mathscr{F} = (i,\text{Fr}_G)^*\mu^*\mathscr{F} = (i,\text{Fr}_G)^*(\mathscr{F} \boxtimes \mathscr{F}) = i^*\mathscr{F} \otimes \text{Fr}_G^*\mathscr{F}.

By the lemma \text{Fr}_G^*\mathscr{F} = \mathscr{F}, so we just have to check that i^*\mathscr{F} \otimes \mathscr{F} is trivial. The latter sheaf is the pullback of \mathscr{F} along \mu \circ (i,\text{id}_G) : G \to G, which is the trivial homomorphism, and since t_{\mathscr{F}}(1) = 1 we are done.

\Box

Next time we will consider G = \text{Pic}_X, a disconnected group for which the proposition is almost true.

Advertisements

The Lang isogeny

2 September 2013 Leave a comment

Sorry for the long wait: I had almost abandoned the project when I spoke to someone at a conference who claimed he was actually reading these posts. This took me by surprise and has inspired me to continue writing.

Today we specialize to the case of a finite ground field \mathbb{F}_q. Our goal is to prove Lang’s theorem on the surjectivity of his eponymous map and give some applications. This map plays a vital role in geometric class field theory: all geometrically connected étale covers of a smooth projective curve over \mathbb{F}_q arise as pullbacks of the Lang isogeny of the Jacobian along an Abel-Jacobi map.

Recall that any scheme S over \mathbb{F}_q has the Frobenius endomorphism \text{Fr}_S, which is defined to be the identity on the underlying set of S and sends f \mapsto f^q for local sections f \in \mathscr{O}_S. It is not hard to see that \text{Fr} is natural in the sense that if f : S \to T is a morphism of \mathbb{F}_q-schemes, we have f \circ \text{Fr}_S = \text{Fr}_T \circ f. A fancy way of saying this is that \text{Fr} is an element of the Bernstein center of the category \text{Sch}_{\mathbb{F}_q} of \mathbb{F}_q-schemes, i.e. an endomorphism of the identity functor on \text{Sch}_{\mathbb{F}_q}. In the sequel we will omit the subscript and simply write \text{Fr}.

Let G be a group scheme over \mathbb{F}_q. We want to study the difference between the Frobenius endomorphism and the identity.

Definition The Lang map L is the endomorphism of the underlying \mathbb{F}_q-scheme of G given as the composition

L : G \stackrel{(i,\text{Fr})}{\longrightarrow}G \times G \stackrel{m}{\longrightarrow} G,

where i and m are the inversion and multiplication maps of G, respectively.

Thus, if g \in G(S) for some \mathbb{F}_q-scheme S, we can write L(g) = g^{-1}\text{Fr}_G(g). From this formula and the aforementioned fact that \text{Fr} commutes with all morphisms, one deduces that if G is commutative then L is a group endomorphism.

Taking S = \overline{\mathbb{F}}_q, we see that the fiber of L over \overline{1} \in G(\overline{\mathbb{F}}_q) is precisely G(\mathbb{F}_q).

Proposition Suppose G is smooth (in particular, of finite type) over \mathbb{F}_q. Then the Lang map L : G \to G is finite étale.

Proof. To prove that L is étale, it suffices to show that the differential of \overline{L} : \overline{G} \to \overline{G} is an isomorphism on the tangent space at each point of \overline{G} = G \times_{\mathbb{F}_q} \overline{\mathbb{F}}_q. Since d\text{Fr} = 0 we have dL_{\overline{1}} = -\text{id}, whence L is étale at \overline{1}. Observe that L intertwines two actions of G on itself: right translations and g \cdot h = h^{-1}g\text{Fr}(h). Since the action by right translations is transitive, it follows that L is étale everywhere.

Now any étale morphism is finite over some nonempty open set in the target, as can be seen by writing it locally as a standard étale morphism. Another routine application of equivariance shows that L is finite.

\Box

This justifies the terminology “Lang isogeny,” at least when G is commutative. We record below a couple of important consequences of this result.

Corollary (Lang) If G is connected then the Lang map is surjective. In particular, L fits into a short exact sequence of pointed sets

1 \to G(\mathbb{F}_q) \to G(\overline{\mathbb{F}}_q) \stackrel{L}{\to} G(\overline{\mathbb{F}}_q) \to 1,

which is a short exact sequence of groups in case G is commutative.

Proof. Since L is finite étale it is both closed and open, so for connected G it is surjective. The rest is a combination of previous remarks.

\Box

Corollary If G is connected then any G-torsor X is trivial.

Proof. Let us prove the equivalent statement X(\mathbb{F}_q) \neq \varnothing.The \overline{G}-torsor \overline{X} clearly has a point x \in \overline{X}(\overline{\mathbb{F}}_q) = X(\overline{\mathbb{F}}_q). Now one can find g \in G(\overline{\mathbb{F}}_q) such that g \cdot \text{Fr}(x) = x, and we claim that if h \in L^{-1}(g) (which exists according to the previous corollary) then h \cdot x \in X(\mathbb{F}_q). Indeed,

\text{Fr}(h \cdot x) = hL(h) \cdot \text{Fr}(x) = hg \cdot \text{Fr}(x) = h \cdot x.

\Box

Finally, we give an application that was promised in a previous post.

Corollary Let X be a smooth, projective, and geometrically connected curve over \mathbb{F}_q. Then X admits a degree 1 zero-cycle.

Proof. Let \text{Pic}_X denote the Picard functor introduced previously and \widetilde{\text{Pic}}_X its fppf-sheafification, which we know to be representable by general results. Moreover, \widetilde{\text{Pic}}_X^0 = \text{ker}(\widetilde{\text{Pic}}_X \stackrel{\text{deg}}{\to} \mathbb{Z}) is smooth and connected (these facts are drawn from Chapter 6 of Néron Models by Bosch, Lütkebohmert, and Raynaud). Now \widetilde{\text{Pic}}_X^1 is a \widetilde{\text{Pic}}_X-torsor, so by the last corollary we have \widetilde{\text{Pic}}_X^1(\mathbb{F}_q) \neq \varnothing. But \widetilde{\text{Pic}}_X^1(\mathbb{F}_q) = \text{Pic}^1(X) because \text{Br}(\mathbb{F}_q) = 0 (see the post on the Picard scheme for the exact sequence that implies this), so X has a degree 1 line bundle, or equivalently a degree 1 zero-cycle.

\Box

Next time we’ll discuss \ell-adic local systems and Grothendieck’s functions-sheaves correspondence. The Lang isogeny will be important in the passage from functions to sheaves.

Geometric class field theory

15 February 2013 Leave a comment

Today, after a long absence from the blogosphere, I’m starting a series of posts on geometric class field theory. My goal is to make the presentation so geometrical that it is easily comprehensible to readers with backgrounds in algebraic geometry but not number theory. Of course, the story is enriched by the analogy with number fields, and I will frequently draw attention to this analogy, but it will be unnecessary for both the statements and the proofs of the main results.

The main character is a smooth, projective, and (geometrically) connected curve X over a field k, which we will generally assume is either a finite field \mathbb{F}_q or the complex numbers \mathbb{C}. Very broadly speaking, the goal is to understand all “covers” of X, by which we mean finite separable maps Y \to X where Y is another curve over k, but this is far too ambitious for us. We will focus our attention on abelian covers, which are the connected Galois covers whose automorphism group is abelian (recall that a connected cover is called Galois if its automorphism group acts transitively on the geometric generic fiber, or equivalently has cardinality equal to the degree of the cover). Then there is a correspondence involving moduli of line bundles on X, as we will explain at length. When k = \mathbb{F}_q, abelian covers correspond to finite-index subgroups of the Picard group of X (with level structure in the ramified case).

This is very much like the number theorist’s goal of understanding (abelian) extensions of a number field. Indeed, we are doing the same for the field of rational functions on X. The case k = \mathbb{F}_q was developed classically along the same lines as arithmetic class field theory, and to my knowledge it was Deligne who first gave a purely geometric proof in the sixties.

Here is a more precise outline of the plan. Our short-term goal will be to prove the main theorem of class field theory in the unramified setting. After that, we will move on to local class field theory, which we will approach using a geometric version of Lubin-Tate theory. The natural next step is to return to and prove the the general ramified case of global class field theory. Along the way we will explain how, in the case k = \mathbb{F}_q, the basic correspondence can be realized using moduli of shtukas on X, and how this relates to Drinfeld modules and explicit class field theory. Finally, in the distant future we might say some words about the higher rank case, which is the geometric Langlands correspondence for \text{GL}_n, and especially Drinfeld’s proof of the case n = 2 in positive characteristic.

So that this post is not entirely devoid of content, let’s go ahead and state the main theorem of unramified global geometric class field theory when k = \mathbb{F}_q (the case k = \mathbb{C} is slightly harder to formulate, but we’ll get to it). Next time we’ll give (some) definitions and explain how our statement relates to more classical formulations, and probably move on to the proof two posts from now.

We will denote by \pi_1(X) the étale fundamental group of X based at the geometric generic point and \pi_1(X)^{\text{ab}} its abelianization (as a profinite group). The structure morphism X \to \text{Spec } \mathbb{F}_q induces a homomorphism \pi_1(X) \to \widehat{\mathbb{Z}}, and we write W_X (respectively W_X^{\text{ab}}) for the Weil group of X, i.e. the preimage of \mathbb{Z} in \pi_1(X) (respectively \pi_1(X)^{\text{ab}}). It is not hard to see that the Weil group is a dense discrete subgroup of \pi_1(X). Any closed point x : \text{Spec } \mathbb{F}_{q^d} \to X induces a map \mathbb{Z} \to W_X, well-defined up to conjugation, and the image of 1 is a conjugacy class in W_X called the (arithmetic) Frobenius at x, which we denote by \text{Fr}_x. In particular, \text{Fr}_x maps to a single element of W_X^{\text{ab}}, which we also denote by \text{Fr}_x.

The other object which appears in the theorem is the Picard group \text{Pic}_X(\mathbb{F}_q) of isomorphism classes of line bundles on X under tensor product. As the notation suggests, the Picard group is the group of rational points of the Picard group scheme \text{Pic}_X, which will be relevant later. For now, just observe that \text{Pic}_X(\mathbb{F}_q) is generated by the line bundles \mathcal{O}(x) as x varies through the closed points of X. Now we can state the theorem.

Theorem (Unramified global class field theory) There is a unique map \text{Pic}_X(\mathbb{F}_q) \to \pi_1(X)^{\text{ab}} which sends \mathcal{O}(x) \mapsto \text{Fr}_x for each closed point x \in X. This map induces an isomorphism \text{Pic}_X(\mathbb{F}_q) \cong W_X^{\text{ab}}.

Note that the isomorphism \text{Pic}_X(\mathbb{F}_q) \cong W_X^{\text{ab}} intertwines the degree map \text{Pic}_X(\mathbb{F}_q) \to \mathbb{Z} with the natural map W_X^{\text{ab}} \to \mathbb{Z}. This is because if x \in X is a degree d point, then \mathcal{O}(x) is a degree d line bundle and \text{Fr}_x induces the automorphism a \mapsto a^{q^d} on \overline{\mathbb{F}}_q.

The uniqueness in the theorem is obvious, since the line bundles \mathcal{O}(x) generate the Picard group. But the existence of this map is already a highly nontrivial statement: this says that if \sum n_ix_i is a principal divisor on X, then \prod \text{Fr}_{x_i}^{n_i} is trivial in \pi_1(X)^{\text{ab}}. This is an example of a reciprocity law in the sense of arithmetic class field theory.

Conjugacy classes in the finite general and special linear groups

21 April 2012 Leave a comment

Now that I’m finally done with school for the summer, I’d like to get back into the routine of blogging regularly. If you were following last summer: I never completed my project of understanding the Weil representation, so I probably won’t be continuing that series of posts. I may be helping some people complete that project this summer, in which case I can hopefully link to some further information eventually.

This week I’m going to give a detailed description of the conjugacy classes in \text{GL}_2(\mathbb{F}_q) and \text{SL}_2(\mathbb{F}_q), where \mathbb{F}_q is the finite field with q elements. This is relevant to representation theory because the conjugacy classes in a finite group correspond bijectively to irreducible representations, and in particular we will find out how many irreducible representations these groups have. A quick Google search reveals that it is easy to find the final answers, but somewhat harder to find a careful explanation, which is what I will attempt now.

First, the general linear group: for any A \in \text{GL}_2(\mathbb{F}_q), consider the \mathbb{F}_q[t]-module \mathbb{F}_q^2 where t acts by A. Two matrices are conjugate if and only if the corresponding modules are isomorphic, and it is easy to analyze these isomorphism classes using the structure theorem for principal ideal domains. Note that since we are counting invertible matrices, we need only consider polynomials with nonzero constant term.

  • The nonzero scalar matrices are precisely the center of \text{GL}_2(\mathbb{F}_q), so these account for q-1 conjugacy classes with one element each.
  • For each \lambda,\mu \in \mathbb{F}_q^{\times} such that \lambda \neq \mu, there is the semisimple conjugacy class of matrices with minimal polynomial (x-\lambda)(x-\mu): the centralizer of such a matrix is a split maximal torus \mathbb{F}_q^{\times} \times \mathbb{F}_q^{\times} \subset \text{GL}_2(\mathbb{F}_q), so each of these \frac12 (q-1)(q-2) conjugacy classes has q^2+q elements.
  •  For each \lambda \in \mathbb{F}_q^{\times} there is a conjugacy class of matrices with minimal polynomial (x-\lambda)^2 which are not semisimple, and hence conjugate to a Jordan block. If we write a Jordan block as \lambda I + N, where N is the nilpotent matrix defined by Ne_1 = 0 and Ne_2 = e_1, it is easy to see that the centralizer consists of matrices of the form a I + bN where a,b \in \mathbb{F}_q and a \neq 0. Thus each of these q-1 conjugacy classes has q^2 - 1 elements.
  •  Finally, there are the matrices which have no eigenvalue in \mathbb{F}_q, and therefore have a conjugate pair of eigenvalues \alpha,\alpha^q \in \mathbb{F}_{q^2} \setminus \mathbb{F}_q. Such matrices are semisimple because \mathbb{F}_q is perfect, so their conjugacy class is determined by their eigenvalues, and in particular we see that there are \frac12 (q^2 - q) conjugacy classes of these matrices. If A \in \text{GL}_2(\mathbb{F}_q) has eigenvalue \alpha \in \mathbb{F}_{q^2} \setminus \mathbb{F}_q, then the subalgebra \mathbb{F}_q[A] \subset \text{Mat}_2(\mathbb{F}_q) is isomorphic to \mathbb{F}_{q^2}. If we use the basis 1,\alpha to identify \mathbb{F}_{q^2} with \mathbb{F}_q \oplus \mathbb{F}_q, we get an isomorphism \text{Mat}_2(\mathbb{F}_q) \cong \text{End}_{\mathbb{F}_q}(\mathbb{F}_{q^2}), and here \mathbb{F}_q[A] corresponds to \text{End}_{\mathbb{F}_{q^2}}(\mathbb{F}_{q^2}) \cong \mathbb{F}_{q^2}. The centralizer of this subalgebra is \mathbb{F}_{q^2}^{\times}, so we see that the centralizer of A in \text{GL}_2(\mathbb{F}_q) is isomorphic to the non-split torus \mathbb{F}_{q^2}^{\times} and in particular the conjugacy class of A has q^2 - q elements.

Note that the total number of conjugacy classes of \text{GL}_2(\mathbb{F}_q) is

q-1 + \frac12 (q-1)(q-2) + q-1 + \frac12 (q^2 - q) = q^2 - 1.

As for \text{SL}_2(\mathbb{F}_q), we first find the \text{GL}_2(\mathbb{F}_q)-conjugacy classes in \text{SL}_2(\mathbb{F}_q) and then determine how they split into \text{SL}_2(\mathbb{F}_q)-conjugacy classes. Unfortunately, we must now keep track of whether q is even or odd.

  • The center of \text{SL}_2(\mathbb{F}_q) is trivial if q is even or \{ \pm I \} if q is odd. Hence this accounts for one conjugacy class if q is even or two if q is odd, with one element each in either case.
  • For each \lambda \in \mathbb{F}_q^{\times} with \lambda \neq \pm 1, there is the semisimple conjugacy class of matrices with minimal polynomial (x-\lambda)(x-\lambda^{-1}). If q is even then there are \frac12 (q-2) of these conjugacy classes, and if q is odd then there are \frac12 (q-3). We already saw that the stabilizer of such a matrix in \text{GL}_2(\mathbb{F}_q) is a split maximal torus, so each conjugacy class has q^2+1 elements.
  • There are matrices with minimal polynomial (x \pm 1)^2 which are not semisimple, and hence conjugate to a Jordan block. If q is even then there is only one such\text{GL}_2(\mathbb{F}_q)-conjugacy class, and if q is odd then there are two. We saw that the stabilizer in \text{GL}_2(\mathbb{F}_q) of such a matrix has q(q-1) elements, so these \text{GL}_2(\mathbb{F}_q)-conjugacy classes contain q^2-1 matrices each.
  • The conjugacy classes of matrices which have no eigenvalue in \mathbb{F}_q are parameterized by conjugate pairs \alpha,\alpha^q \in \mathbb{F}_{q^2} \setminus \mathbb{F}_q where \alpha^{q+1} = 1. The latter equation has q+1 solutions in the cyclic group \mathbb{F}_{q^2}^{\times}, and if q is even only one of those solutions comes from \mathbb{F}_q, while if q is odd then two do. Thus there are \frac12 q such conjugacy classes if q is even and \frac12 (q-1) if q is odd. As we saw, the stabilizers in \text{GL}_2(\mathbb{F}_q) of these matrices are non-split maximal tori, so each of these conjugacy classes has q^2-q elements.

So we have described the  \text{GL}_2(\mathbb{F}_q)-conjugacy classes in  \text{SL}_2(\mathbb{F}_q), but it remains to see how these split as \text{SL}_2(\mathbb{F}_q)-conjugacy classes. We will show momentarily that semisimple \text{GL}_2(\mathbb{F}_q)-conjugacy classes in \text{SL}_2(\mathbb{F}_q) do not split further as \text{SL}_2(\mathbb{F}_q)-conjugacy classes, and here the only non-semisimple matrices are conjugate to one of the Jordan blocks \pm I + N (where N is the nilpotent matrix mentioned earlier). Let’s write G = \text{GL}_2(\mathbb{F}_q) and H = \text{SL}_2(\mathbb{F}_q) for the moment to improve the notation. Now if X is the G-conjugacy class in H of A \in H, then X \cong G/C_G(A) as G-sets and in particular as H-sets. In particular we get a bijection X/H \cong G/HC_G(A) \cong \mathbb{F}_q^{\times}/\text{det} C_G(A). We saw earlier that if A is a Jordan block then C_G(A) consists of matrices of the form aI + bN with a,b \in \mathbb{F}_q and a \neq 0, so \text{det} C_G(A) = \mathbb{F}_q^{\times 2} is the subgroup of squares. Thus if q is odd then the two G-conjugacy classes of \pm I + N split into two H-conjugacy classes with \frac12 (q^2-1) elements each, and if q is even then the G-conjugacy class of I + N does not split further as an H-conjugacy class. We see now that if q is odd then \text{SL}_2(\mathbb{F}_q) has

2 + \frac12 (q-3) + 4 + \frac12 (q-1) = q+4

conjugacy classes, and if q is even then the number is

1 + \frac12 (q-2) + 1 + \frac12 q = q+1.

It remains to show that if X \subset H is a semisimple G-conjugacy class, then X does not split further as an H-conjugacy class. This is true for G = \text{GL}_n(F) and H = \text{SL}_n(F) where n \geq 1 is arbitrary and F is any field with the property that the norm map N_{E/F} : E^{\times} \to F^{\times} is surjective for any finite extension E/F. Even more generally, suppose R is a finite-dimensional commutative semisimple algebra over such a field F, and M a finite-dimensional R-module. Then we have the determinant map \text{Aut}_F(M) \to F^{\times}, and we claim the subgroup \text{Aut}_R(M) surjects onto F^{\times}. Now R \cong E_1 \times \cdots \times E_r by the semisimplicity hypothesis, where each E_i is a finite field extension of F, so M \cong M_1 \times \cdots \times M_r where each M_i is an E_i-vector space and R acts diagonally. Thus the automorphism group splits as well:

\text{Aut}_R(M) \cong \text{Aut}_{E_1}(M_1) \times \cdots \times \text{Aut}_{E_r}(M_r).

It is enough to show that \text{det} : \text{Aut}_{E_i}(M_i) \to F^{\times} is surjective for some 1 \leq i \leq r, so we have reduced to the case that R = E is a finite field extension of F. But now we can see from the definitions that the determinant \text{Aut}_E(M) \to F^{\times} factors into the determinant \text{Aut}_E(M) \to E^{\times} followed by the norm E^{\times} \to F^{\times}, and the latter is surjective by assumption. Applying this to the case when A \in H is semisimple, R = F[A] \subset \text{Mat}_n(F), and M = F^n, we have \text{Aut}_R(M) = C_G(A) and the claim follows.

The affine line over a finite field

13 May 2011 4 comments

Exercise II.2.11 in Hartshorne’s book asks for a description of Spec \mathbb{F}_p[x], where \mathbb{F}_p denotes the field with p elements, including the number of points with a given residue field. I’m going to discuss the affine line over an arbitrary finite field \mathbb{F}_q, since this is no more difficult than the case when q = p is prime.

As is the situation over any field, the closed points of the affine line correspond to monic irreducible polynomials, and there is a unique non-closed point, namely the zero ideal, which plays the role of generic point. The nonempty open sets are just the cofinite subsets, and the \mathbb{F}_q-algebra over such an open set is obtained by inverting the irreducible polynomials in \mathbb{F}_q[x] corresponding to the removed points. In particular, every open subset of Spec \mathbb{F}_q[x] is affine. The local ring at a closed point (f) consists of all rational functions whose denominator does not divide f, and if f has degree n the residue field is \mathbb{F}_{q^n}. The local ring at (0) is the function field \mathbb{F}_q(x), which in some respects looks like a number field of positive characteristic: the completions of the various local rings play the role of p-adic number rings.

Now for the interesting part: one can count the number of points with a given residue field by using Möbius inversion from elementary number theory. First we must define the Möbius \mu-function \mathbb{N} \to \{ -1,0,1 \}, which is given by \mu(1) = 1, \mu(n) = (-1)^r if n is squarefree with r prime factors, and \mu(n) = 0 if n is not squarefree. Then Möbius inversion can be stated as follows: if f,g : \mathbb{N} \to \mathbb{C} satisfy

g(n) = \sum_{d | n} f(d),

then one can solve for f by the formula

f(n) = \sum_{d | n} \mu (\frac{n}{d}) f(d).

Let’s apply this to our situation: let h(n) be the number of points of Spec \mathbb{F}_q[x] with residue field \mathbb{F}_{q^n}, which is the same as the number of monic irreducible polynomials in \mathbb{F}_q[x] of degree n. Recall that the polynomial x^{q^n} - x is the product of all irreducible monic polynomials of degree dividing n, so upon counting degrees we obtain the formula

q^n = \sum_{d | n} d \cdot h(d).

Now apply Möbius inversion with f(n) = n \cdot h(n) and g(n) = q^n, so we get

h(n) = \frac{1}{n} \sum_{d | n} \mu (\frac{n}{d}) q^d.

This relates to the Hasse-Weil zeta function for Spec \mathbb{F}_q[x] in the following way. Recall that an \mathbb{F}_{q^n}-point of a variety X over \mathbb{F}_q is a morphism Spec \mathbb{F}_{q^n} \to X of \mathbb{F}_q-schemes, which corresponds to a point p \in X and an \mathbb{F}_q-homomorphism k(p) \to \mathbb{F}_{q^n}, where k(p) denotes the residue field of X at p. Clearly such a homomorphism exists if and only if p is closed with a residue field whose degree over \mathbb{F}_q divides n, so if we write X(\mathbb{F}_{q^n}) for the set of \mathbb{F}_{q^n}-points of X and deg(p) = [k(p) : \mathbb{F}_{q}] we see that

X(\mathbb{F}_{q^n}) = \coprod_{\text{deg}(p) | n} Hom(k(p),\mathbb{F}_{q^n}).

When counting the elements of Hom(k(p),\mathbb{F}_{q^n}) we must take into account the action of the Frobenius: this set carries a transitive action by Gal(\mathbb{F}_{q^n} / \mathbb{F}_q), which is canonically identified with \mathbb{Z}/n\mathbb{Z}, and if deg(p) = d the stabilizer of any point is just Gal(\mathbb{F}_{q^n} / \mathbb{F}_{q^d}) \cong d\mathbb{Z}/n\mathbb{Z}. Thus by the point-stabilizer theorem we have |\text{Hom}(k(p),\mathbb{F}_{q^n})| = d and consequently

|X(\mathbb{F}_{q^n})| = \sum_{d | n} d \cdot | \{ p \in X \ | \ \text{deg}(p) = d \}|.

If we write N_n = |X(\mathbb{F}_{q^n})| then the Hasse-Weil zeta function of X is the element of \mathbb{Q}[[t]] defined by

Z(X,t) = \text{exp}(\sum_{n \geq 1} \frac{N_n}{n} t^n).

In the case X = \text{Spec } \mathbb{F}_q[x], by the previous observation we have N_n = q^n, so

Z(X,t) = \text{exp}(\sum_{n \geq 1} \frac{q^n}{n} t^n) = \text{exp}(-\text{log}(1-qt)) = \frac{1}{1-qt}.

So in our situation the zeta function is actually rational! The miracle is that this is true for any variety over \mathbb{F}_q whatsoever. Several proofs are known, all of which are quite hard, although it is possible to give a relatively elementary proof for curves.