## 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 , from now on we will study (rank ) -adic local systems on , which can be considered as a “linearized” version of the problem. More precisely, since understanding the structure of the fundamental group (here is the geometric generic point, not that it really matters) is tantamount to understanding all étale covers of , the Tannakian philosophy says that we should consider the monoidal category of finite-dimensional representations of , 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 -adic local systems: characters of the Picard group (more precisely, its profinite completion) correspond to “multiplicative” rank local systems on the Picard scheme . This situation is very special to finite fields! A multiplicative local system on can be pulled back along the Abel-Jacobi map , given by on points, and global unramified class field theory says that this pullback establishes a bijection between multiplicative local systems on and rank local systems on . 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 (traditionally one assumes is not equal to , the characteristic of our finite ground field , but for us is fine). Let be an algebraic closure of the -adic numbers . The totally disconnected topology on 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 is an algebraically closed field of characteristic zero.

Let be a connected scheme and a geometric point.

**Definition** An *-adic local system* on is a finite-dimensional continuous -representation of . The dimension of the representation is called the *rank* of .

As our terminology and notation suggest, -adic local systems can be thought of as locally constant sheaves of -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 -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 and , we will write for the local system on obtained by restricting along the homomorphism induced by . The underlying -vector space of the representation is denoted by .

Now we come to the namesake of this post, for which we should assume that is defined over . This construction takes an -adic local system on and produces from it a function . Given , there is an isomorphism well-defined up to conjugation, so that after making this choice we can form the pullback , a local system on . Then we set

,

the trace of the action of the Frobenius , which does not depend on our choice because the trace is conjugation-invariant.

We will be interested in the case where has rank , and then it is easier to describe . Namely, determines a canonical map , which we compose with (now thought of a one-dimensional representation) to obtain a homomorphism . Evaluate this map at to obtain .

Sometimes interesting classes of functions and sheaves match up under this correspondence. Let be a commutative algebraic group over : then one such class of functions is the group of characters of , meaning one-dimensional representations . What sort of local system on has the property that is a character?

**Definition** A rank local system on is called a *character sheaf* provided that , where is the multiplication map.

(Notation for those who haven’t seen it: if is a sheaf on and is a sheaf on , then their *external tensor product* is where and 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 be an -adic local system on . Then there is a canonical isomorphism .

*Proof**.* We reduce immediately to the case that is a locally constant étale sheaf of finite sets, so there is some finite étale map whose sheaf of local sections is . Write for the fiber product of and , so it suffices to produce an isomorphism of -schemes. Using the relation , we obtain the desired map from and . Since and are finite étale, so is , and similarly is radicial (i.e. universally injective) because is. But a map which is both étale and radicial must be an open embedding, and is also finite, hence an isomorphism.

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

**Proposition** Under the above assumptions, is a bijection from character sheaves on to characters of .

*Proof.* That is a character follows from the easy identities and . Suppose that we are given a character . The Lang isogeny is a pointed finite Galois covering with group , hence gives rise to a map , which we can restrict along to obtain a rank local system on . The same identities show that 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 . Given , we obtain a canonical map , whose value on we will call . By definition is the value of on , but factors through by construction, so it suffices to show that the map induced by sends to . This means that when acts on the fiber , it just translates by . Fixing , we obtain an identification since . Tracing through definitions, our claim follows from the calculation

.

Finally, we must prove that . We claim that is trivial, or equivalently that factors through the homomorphism determined by . Since this map sends the claim implies that is determined by its values on for all , and we just proved that , from which it follows that . As for the claim, observe that

.

By the lemma , so we just have to check that is trivial. The latter sheaf is the pullback of along , which is the trivial homomorphism, and since we are done.

Next time we will consider , 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 . 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 arise as pullbacks of the Lang isogeny of the Jacobian along an Abel-Jacobi map.

Recall that any scheme over has the *Frobenius endomorphism* , which is defined to be the identity on the underlying set of and sends for local sections . It is not hard to see that is natural in the sense that if is a morphism of -schemes, we have . A fancy way of saying this is that is an element of the Bernstein center of the category of -schemes, i.e. an endomorphism of the identity functor on . In the sequel we will omit the subscript and simply write .

Let be a group scheme over . We want to study the difference between the Frobenius endomorphism and the identity.

**Definition** The *Lang map* is the endomorphism of the underlying -scheme of given as the composition

,

where and are the inversion and multiplication maps of , respectively.

Thus, if for some -scheme , we can write . From this formula and the aforementioned fact that commutes with all morphisms, one deduces that if is commutative then is a group endomorphism.

Taking , we see that the fiber of over is precisely .

**Proposition** Suppose is smooth (in particular, of finite type) over . Then the Lang map is finite étale.

*Proof.* To prove that is étale, it suffices to show that the differential of is an isomorphism on the tangent space at each point of . Since we have , whence is étale at . Observe that intertwines two actions of on itself: right translations and . Since the action by right translations is transitive, it follows that 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 is finite.

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

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

,

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

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

**Corollary** If is connected then any -torsor is trivial.

*Proof*. Let us prove the equivalent statement .The -torsor clearly has a point . Now one can find such that , and we claim that if (which exists according to the previous corollary) then . Indeed,

.

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

**Corollary** Let be a smooth, projective, and geometrically connected curve over . Then admits a degree zero-cycle.

*Proof.* Let denote the Picard functor introduced previously and its fppf-sheafification, which we know to be representable by general results. Moreover, is smooth and connected (these facts are drawn from Chapter 6 of *Néron Models* by Bosch, Lütkebohmert, and Raynaud). Now is a -torsor, so by the last corollary we have . But because (see the post on the Picard scheme for the exact sequence that implies this), so has a degree line bundle, or equivalently a degree zero-cycle.

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