## 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.