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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.