Etale descent obstruction and anabelian geometry of curves over finite fields

Let $C$ and $D$ be smooth, proper and geometrically integral curves over a finite field ${\mathbb F}$. We consider the arithmetic of the curve $C\times_{\mathbb F}K$ over the global function field $K:={\mathbb F}(D)$ (which we still denote by $C$ by abuse of notation). We denote by ${\mathbb A}_{K}$ the ring of adèles of $K$ and consider the set $C({\mathbb A}_{K})$ of adelic points of $C$, which is also the product $\prodop\displaylimits_{v}C(K_{v})$, where $v$ runs through the places of $K$, with its natural product topology. Let $\overline{C(K)}$ denote the topological closure of $C(K)$ inside $C({\mathbb A}_{K})$.

The definition of the set $C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}}$ of adelic points surviving descent by all torsors under finite étale group schemes over $K$ is recalled in Section 2.2. We also consider the set $C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}\text{-}\operatorname{Br}}$ of adelic points surviving the étale-Brauer obstruction; see [PoonenRatPoints, Section 8.5.2]. These are closed subsets of $C({\mathbb A}_{K})$ containing $\overline{C(K)}$. A special case of [PV, Conjecture C] implies that $\overline{C(K)}=C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}\text{-}\operatorname{Br}}$. For any of the other containments in the sequence

there are examples showing that, in general, they can be proper. The first will be proper when $C(K)$ is infinite, which occurs whenever there is a nonconstant morphism $\phi\in\operatorname{Mor}_{\mathbb F}(D,C)=C(K)$, as it may be composed with the Frobenius endomorphism of $C$. Examples where the third inclusion is proper are given in [CV, Proposition 4.5] and are accounted for by a descent obstruction coming from torsors under finite abelian group schemes that are not étale.

Despite this, it is still expected that the information obtained from $C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}}$ should determine the set of rational points, as we now describe. For a place $v$, let ${\mathbb F}_{v}$ denote the residue field of the integer ring $\mathcal{O}_{v}\subset K_{v}$. We define $C({\mathbb A}_{K,{\mathbb F}}):=\prodop\displaylimits_{v}C({\mathbb F}_{v})$, which is a closed subset of $C({\mathbb A}_{K})=\prodop\displaylimits_{v}C(K_{v})$ admitting a continuous retraction $r\colon C({\mathbb A}_{K})\to C({\mathbb A}_{K,{\mathbb F}})$ (see Section 2.1). Define $C({\mathbb A}_{K,{\mathbb F}})^{\operatorname{\textup{\'{e}t}}}=C({\mathbb A}_{K,{\mathbb F}})\cap C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}}$. Then $r(\overline{C(K}))$ is a closed subset of $C({\mathbb A}_{K,{\mathbb F}})^{\operatorname{\textup{\'{e}t}}}$. We conjecture the following.

Conjecture 1.1 is a nonabelian analogue of a conjecture in the number field case by Poonen, see [Poonen], in a setup first studied in [Scharaschkin]. It is equivalent, by [CV, Theorem 1.2], to the conjecture that $\overline{C(K)}=C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}\text{-}\operatorname{Br}}$. When $C$ has genus $1$, Conjecture 1.1 follows from the Tate conjecture for abelian varieties over finite fields. It is also known when the genera of $C$ and $D$ satisfy $g(D)<g(C)$ by [CV, Theorem 1.5], and in some other cases where $C(K)=C({\mathbb F})$; see [CVV, Theorem 2.14]. The goal of this paper is to provide further evidence for this conjecture, by relating it to anabelian geometry.

Fix geometric points $\overline{x}\in C({\overline{{\mathbb F}}})$ and $\overline{y}\in D({\overline{{\mathbb F}}})$, where ${\overline{{\mathbb F}}}$ denotes an algebraic closure of ${\mathbb F}$, and let $\pi_{1}(C):=\pi_{1}(C,{\overline{x}})$ and $\pi_{1}(D):=\pi_{1}(D,{\overline{y}})$ be the étale fundamental groups of $C$ and $D$ with these base points. Any morphism of curves $D\to C$ induces a morphism of étale fundamental groups $\pi_{1}(D)\to\pi_{1}(C)$ up to conjugation by an element of the geometric fundamental group $\pi_{1}({\overline{C}}):=\pi_{1}(C\times_{\mathbb F}{\overline{{\mathbb F}}},{\overline{x}})$. Grothendieck’s anabelian philosophy suggests that, when $C$ has genus at least $2$, all open homomorphisms between the étale fundamental groups should arise in this way from a nonconstant morphism of schemes; see [ST2009, ST2011]. In Section 3 we define a notion of well-behaved morphisms between fundamental groups of curves (see Definition 3.1). We expect all open homomorphisms are well behaved, but we have not been able to prove this.

Our main result is the following theorem, which relates the set $C({\mathbb A}_{K,{\mathbb F}})^{\operatorname{\textup{\'{e}t}}}$ appearing in Conjecture 1.1 to an object of interest in anabelian geometry.

This theorem is a strengthening of an analogous result for curves over number fields, which shows that an adelic point surviving étale descent gives rise to a section of the fundamental exact sequence; see [Harari-Stix, Stoll]. Combining Theorem 1.2 with the results in [CV], we prove the following.

In addition to establishing new instances of the conjecture, this result allows us to relate it in the case $g(D)=g(C)$ to a recent conjecture of Sutherland and the second author, see [SV], which we now recall. We embed $C$ into its Jacobian $J_{C}$ by a choice of divisor of degree $1$ (which always exists by the Lang–Weil estimates since $C$ is defined over a finite field). The Hilbert class field is defined as follows. Let $\Phi\colon J_{C}\to J_{C}$ denote the ${\mathbb F}$-Frobenius map. Define $H(C):=(I-\Phi)^{*}(C)\subset J_{C}$, where $I$ denotes the identity map on $J$. Then $H(C)$ is an unramified abelian cover of $C$ with Galois group $J_{C}({\mathbb F})$, well defined up to a twist that corresponds to a choice of divisor of degree $1$ embedding $C$ into $J_{C}$. Define $H_{0}(C):=C$, $H_{1}(C):=H(C)$, and successively define $H_{n+1}(C):=H_{n}(H(C))$ for integers $n\geq 1$.

The set of places of the global field $K={\mathbb F}(D)$ is in bijection with the set $D^{1}$ of closed points of $D$. Given $v\in D^{1}$, we use $K_{v}$, $\mathcal{O}_{v}$ and ${\mathbb F}_{v}$ to denote the corresponding completion, ring of integers and residue field, respectively. Fix a separable closure $K^{\operatorname{s}}$ of $K$, and let ${\overline{{\mathbb F}}}$ denote the algebraic closure of ${\mathbb F}$ inside $K^{\operatorname{s}}$. For each $v\in D^{1}$, fix a separable closure $K_{v}^{\operatorname{s}}$ of $K_{v}$ and an embedding $K^{\operatorname{s}}\hookrightarrow K_{v}^{\operatorname{s}}$. This determines an embedding ${\mathbb F}_{v}\subset{\overline{{\mathbb F}}}$ and an inclusion $\theta_{v}\colon\operatorname{Gal}(K_{v})\to\operatorname{Gal}(K)$. The embedding ${\mathbb F}_{v}\subset{\overline{{\mathbb F}}}$ fixes a geometric point $\overline{v}\in D({\overline{{\mathbb F}}})$ in the support of the closed point $v\in D$. The inclusions ${\mathbb F}\subset{\mathbb F}_{v}\subset\mathcal{O}_{v}\subset K_{v}$ endow $\mathcal{O}_{v},K_{v}$ and the adele ring ${\mathbb A}_{K}$ with the structure of an ${\mathbb F}$-algebra. We define the locally constant adele ring ${\mathbb A}_{K,{\mathbb F}}:=\prodop\displaylimits_{v\in D^{1}}{\mathbb F}_{v}$. This is an ${\mathbb F}$-subalgebra of the adele ring ${\mathbb A}_{K}$.

The constant curve $C\times_{\operatorname{Spec}({\mathbb F})}\operatorname{Spec}(K)$ spreads out to a smooth proper model $C\times_{\operatorname{Spec}({\mathbb F})}D$ over $D$. For any $v\in D^{1}$, this gives a reduction map $r_{v}\colon C(K_{v})\to C({\mathbb F}_{v})$. Since $C$ is proper, $C({\mathbb A}_{K})=\prodop\displaylimits C(K_{v})$ and the reduction maps give rise to a continuous projection $r\colon C({\mathbb A}_{K})\to C({\mathbb A}_{K,{\mathbb F}})$ sending $(x_{v})$ to $(r_{v}(x_{v}))$.

Any locally constant adelic point $(x_{v})\in C({\mathbb A}_{K,{\mathbb F}})$ determines a unique Galois equivariant map of sets $\psi\colon D({\overline{{\mathbb F}}})\to C({\overline{{\mathbb F}}})$ with the property that $\phi(\overline{v})=x_{v}$. This induces a bijection $C({\mathbb A}_{K,{\mathbb F}})\leftrightarrow\operatorname{Map}_{G_{\mathbb F}}(D({\overline{{\mathbb F}}}),C({\overline{{\mathbb F}}}))$. Moreover, a locally constant adelic point on $C$ determines, and is uniquely determined by, a map $f\colon D^{1}\to C^{1}$ together with an embedding ${\mathbb F}_{f(v)}\subset{\mathbb F}_{v}$ for each $v\in D^{1}$ (see [CV, Lemma 2.1]).

The set $C(K)/F$ is finite by the theorem of de Franchis [Lang, Chapter 8, pp. 223-224]. Over a finite field ${\mathbb F}$, there is a simpler proof. The degree of a separable map $D\to C$ is bounded by Riemann–Hurwitz. Looking at coordinates of an embedding of $C$, it now suffices to show that there are only finitely many functions on $D/{\mathbb F}$ of degree bounded by some $m$. The zeros and poles of such a function have degree at most $m$ over ${\mathbb F}$, so there are only finitely choices for the divisor of such a function. Finally, the function itself is determined up to a scalar in ${\mathbb F}^{*}$ by its divisor, but ${\mathbb F}^{*}$ is finite by hypothesis.

Let $f\colon C^{\prime}\to C$ be a torsor under a finite étale group scheme $G/K$. We use $\operatorname{H}^{1}(K,G)$ to denote the étale cohomology set parameterizing isomorphism classes of $G$-torsors over $K$ (and similarly with $K$ replaced by $K_{v},{\mathcal{O}}_{v},{\mathbb F}_{v}$, etc.). The distinguished element of this pointed set is represented by the trivial torsor.

Following the terminology in [Stoll], we say an adelic point $(x_{v})\in C({\mathbb A}_{K})$ survives $f$ if the element of $\prodop\displaylimits_{v}\operatorname{H}^{1}(K_{v},G)$ given by evaluating $f$ at $(x_{v})$ lies in the image of the diagonal map

Equivalently, $(x_{v})$ survives $f$ if and only if $(x_{v})$ lifts to an adelic point on some twist of $f$ by a cocycle representing a class in $\operatorname{H}^{1}(K,G)$. We use $C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}}$ to denote the set of adelic points surviving all $C$-torsors under étale group schemes over $K$. Then $C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}}$ is a closed subset of $C({\mathbb A}_{K})$ containing $C(K)$. We define $C({\mathbb A}_{K,{\mathbb F}})^{\operatorname{\textup{\'{e}t}}}=C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}}\cap C({\mathbb A}_{K,{\mathbb F}})$. By [CV, Proposition 4.6], an adelic point lies in $C({\mathbb A}_{K})^{\operatorname{\textup{\'{e}t}}}$ if and only if its image under the reduction map $r\colon C({\mathbb A}_{K})\to C({\mathbb A}_{K,{\mathbb F}})$ lies in $C({\mathbb A}_{K,{\mathbb F}})^{\operatorname{\textup{\'{e}t}}}$.

The following lemma is a special case of a well-known statement in étale cohomology over a henselian ring (cf. [Milne, Remark 3.11(a) on p. 116]).

An element of $\operatorname{H}^{1}(K_{v},G)$ is called unramified if it lies in the image of the map $\operatorname{H}^{1}(\mathcal{O}_{v},G)\to\operatorname{H}^{1}(K_{v},G)$ induced by the inclusion $\mathcal{O}_{v}\subset K_{v}$. Thus, the lemma identifies $\operatorname{H}^{1}({\mathbb F}_{v},G)$ with the set of unramified elements in $\operatorname{H}^{1}(K_{v},G)$.