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Posts Tagged ‘algebraic groups’

## Grothendieck’s functions-sheaves correspondence

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.

## The Lang isogeny

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.

Categories: algebraic geometry