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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.

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