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 $F$. Any morphism from $D$ to $C$ induces a morphism of their \'etale fundamental groups. The anabelian philosophy proposed by Grothendieck suggests that, when $C$ has genus at least $2$, all open homomorphisms between the \'etale fundamental groups should arise in this way from a nonconstant morphism of curves. We relate this expectation to the arithmetic of the curve $C_K$ over the global function field $K = F(D)$. Specifically, we show that there is a bijection between the set of conjugacy classes of well-behaved morphism of fundamental groups and locally constant adelic points of $C_K$ that survive \'etale descent. We use this to provide further evidence for the anabelian conjecture by relating it to another recent conjecture by Sutherland and the second author.


Introduction
For a smooth, proper and geometrically integral curve X over a global field k, it is well known that the Hasse principle can fail.That is, X may contain points over every completion of k, yet fail to have any k-rational point.All known examples of this phenomenon can be explained by a finite descent obstruction.This means that there is a torsor f : Y → X under a finite group scheme over k such that no twist of Y contains points over every completion.Since any k-rational point must lift to some twist of f , this yields an obstruction to the existence of k-rational points on X.A central question in the arithmetic of curves over global fields is to determine whether this is the only obstruction to the existence of k-rational points.
This problem is expected to be very hard in general.For curves of genus 1, it is equivalent to standard conjectures concerning the Tate-Shafarevich groups of elliptic curves.For curves of genus at least 2 over number fields, it is known to follow from Grothendieck's section conjecture, but there are essentially no general results.For a discussion of the finite descent obstruction over number fields, we refer to [Sto07], which, despite being published over a decade ago, still conveys the state of the art.
The situation is much more promising when X is defined over a global function field, i.e., when k = F (D) is the function field of a smooth, proper and geometrically connected curve D over a finite field F .Building on work of Poonen-Voloch, see [PV10], and Rössler, see [Rös13], [CV23,Appendix], the authors have recently completed a proof that finite descent is the only obstruction for all nonisotrivial curves of genus at least 2; see [CV23].It thus remains to consider the situation for isotrivial curves.Recall that X is called constant if it is isomorphic to the base change of a curve defined over F , and that X is called isotrivial if it becomes constant after base change to a finite extension of k.
We formulate a precise version of the conjecture that finite descent is the only obstruction to the existence of k-rational points on a constant curve over a global function field (Conjecture 1.1) and prove the equivalence of this conjecture with an analogue of Grothendieck's section conjecture for curves over finite fields (see Theorem 3.8).This enables us to use techniques from anabelian geometry which we combine with results of [CV22] to establish new instances of these conjectures.We prove that finite descent is the only obstruction to the existence of k-rational points for a constant curve X ≃ C × F k such that the Jacobian of C is not an isogeny factor of the Jacobian of D (see Theorem 1.3).

Main results and conjectures
Let C and D be smooth, proper and geometrically integral curves over a finite field F .We consider the arithmetic of the curve C × F K over the global function field K := F (D) (which we still denote by C by abuse of notation).We denote by A K the ring of adèles of K and consider the set C(A K ) of adelic points of C, which is also the product v C(K v ), where v runs through the places of K, with its natural product topology.Let C(K) denote the topological closure of C(K) inside C(A K ).
The definition of the set C(A K ) ét 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(A K ) ét -Br of adelic points surviving the étale-Brauer obstruction; see [Poo17, Section 8.5.2].These are closed subsets of C(A K ) containing C(K).A special case of [PV10, Conjecture C] implies that C(K) = C(A K ) ét -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 φ ∈ Mor 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 [CV22, 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(A K ) ét should determine the set of rational points, as we now describe.For a place v, let F v denote the residue field of the integer ring Conjecture 1.1 is a nonabelian analogue of a conjecture in the number field case by Poonen, see [Poo06], in a setup first studied in [Sch99].It is equivalent, by [CV22, Theorem 1.2], to the conjecture that C(K) = C(A K ) ét -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 [CV22, Theorem 1.5], and in some other cases where C(K) = C(F ); see [CVV18, 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 x ∈ C(F ) and y ∈ D(F ), where F denotes an algebraic closure of F , and let π 1 (C) := π 1 (C, x) and π 1 (D) := π 1 (D, y) be the étale fundamental groups of C and D with these base points.Any morphism of curves D → C induces a morphism of étale fundamental groups π 1 (D) → π 1 (C) up to conjugation by an element of the geometric fundamental group π 1 (C) := π 1 (C × F F , 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 [ST09,ST11].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(A K,F ) ét appearing in Conjecture 1.1 to an object of interest in anabelian geometry.

locally constant adelic points surviving étale descent.
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 [HS12,Sto07].Combining Theorem 1.2 with the results in [CV22], we prove the following.
Theorem 1.3.If the Jacobian J C of C is not an isogeny factor of J D , then Conjecture 1.1 holds for C and D.
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 [SV19], 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 Φ : J C → J C denote the F -Frobenius map.Define H(C) := (I − Φ) * (C) ⊂ 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 (F ), well defined up to a twist that corresponds to a choice of divisor of degree 1 embedding C into

Notation
The set of places of the global field K = F (D) is in bijection with the set D 1 of closed points of D. Given v ∈ D 1 , we use K v , O v and F v to denote the corresponding completion, ring of integers and residue field, respectively.Fix a separable closure K s of K, and let F denote the algebraic closure of F inside K s .For each v ∈ D 1 , fix a separable closure K s v of K v and an embedding K s → K s v .This determines an embedding and the adele ring A K with the structure of an F -algebra.We define the locally constant adele ring A K,F := v∈D 1 F v .This is an F -subalgebra of the adele ring A K .
The constant curve C × Spec(F ) Spec(K) spreads out to a smooth proper model C × Spec(F ) D over D. For any v ∈ D 1 , this gives a reduction map r v : and the reduction maps give rise to a continuous projection r : Any locally constant adelic point (x v ) ∈ C(A K,F ) determines a unique Galois equivariant map of sets ψ : . Moreover, a locally constant adelic point on C determines, and is uniquely determined by, a map f : , where C(K)/F denotes the set of K-rational points up to Frobenius twist; i.e., P ∼ Q if and only if there are m, n ≥ 0 such that F m P = F n Q.
Proof.The first statement follow from the fact (e.g., [GW10, Exercise 5.17]) that a morphism defined on a geometrically reduced variety is determined by what it does to geometric points.For the second statement, see [Sti02,Proposition 2.3].□ The set C(K)/F is finite by the theorem of de Franchis [Lan83, Chapter 8, pp.223-224].Over a finite field F , there is a simpler proof.The degree of a separable map D → 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/F of degree bounded by some m.The zeros and poles of such a function have degree at most m over 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 F * by its divisor, but F * is finite by hypothesis.

Etale descent obstruction
Let f : C ′ → C be a torsor under a finite étale group scheme G/K.We use 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 , O v , F v , etc.).The distinguished element of this pointed set is represented by the trivial torsor.
Following the terminology in [Sto07], we say an adelic point 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 H 1 (K, G).We use C(A K ) ét to denote the set of adelic points surviving all C-torsors under étale group schemes over K. Then C(A K ) ét is a closed subset of C(A K ) containing C(K).We define . By [CV22, Proposition 4.6], an adelic point lies in C(A K ) ét if and only if its image under the reduction map r : The following lemma is a special case of a well-known statement in étale cohomology over a henselian ring (cf.[Mil80, Remark 3.11(a) on p. 116]).

Lemma 2.2. For an étale group scheme
).This map is injective by Hensel's lemma.On the other hand, the inclusion i : F v → O v satisfies q • i = id.It follows that q * must also be surjective. □ Thus, the lemma identifies H 1 (F v , G) with the set of unramified elements in H 1 (K v , G).

Connection to anabelian geometry
Fix a base point x : Spec F → D := D × Spec(F ) Spec(F ).Composing with the canonical maps D → D and D → Spec(F ), this serves as well to fix base points of D and Spec(F ).The base point of Spec(F ) agrees with that determined by the algebraic closure F ⊂ F fixed above.This leads to the fundamental exact sequence where π 1 (−) denotes the étale fundamental group with base point as chosen above.A choice of base point Spec F → C determines a similar sequence for C. The choice of separable closure of K identifies π 1 (D) with the Galois group of the maximal extension K unr of K which is everywhere unramified.For each closed point v ∈ D 1 , the embedding to an open subgroup of a decomposition group of π 1 (C).Let Hom wb (π 1 (D), π 1 (C)) denote the set of well-behaved homomorphisms of profinite groups, and for a subgroup H < π 1 (C), let Hom wb H (π 1 (D), π 1 (C)) denote the quotient of Hom wb (π 1 (D), π 1 (C)) by the action given by composition with an inner automorphism of π 1 (C) coming from an element of H. Remark 3.2.Here is an example of a poorly behaved homomorphism.Suppose the genus of C is at least 2. By [Sti13,Theorem 226], there are uncountably many sections Gal(F ) → π 1 (C) that are not conjugate to any section coming from a point in C(F ).Composing such a section with the canonical surjection π 1 (D) → Gal(F ) gives a continuous morphism π 1 (D) → π 1 (C) that is not well behaved.
Then there exists a well-behaved homomorphism φ : π 1 (D) → π 1 (C) inducing a morphism of exact sequences Here we view G as a constant group scheme over F .By Lemma 2.2, we may view H 1 (F v , G) as a subgroup of H 1 (K v , G).In terms of descent, ρ G corresponds to a torsor in , and ρ G • s v is the evaluation of this torsor at x v ∈ C(F v ).So the fact that (x v ) survives étale descent implies that there is a global class s ∈ H 1 (K, G) such that for all G).Note that such an s must lie in (the image under inflation of) the group H 1 (π 1 (D), G) = Hom G (π 1 (D), G) since the s v are all unramified.
For each v ∈ D 1 , the condition By the Borel-Serre theorem (see [Poo17, Theorem 5.12.29]), the fibers of the map is finite, and it is nonempty by the discussion above.As in the proof of [HS12, Proposition 1.2], it follows that the inverse limit over G of these sets is nonempty.An element of lim with the property that for all v ∈ D 1 , the maps φ • t v and s v are conjugate by an element of π 1 (C F v ) = lim ← − − G v .We claim that φ • t v and s v are in fact π 1 (C)-conjugate.To see this, let p : π 1 (C) → Gal(F ) be the canonical surjection.Suppose To see that γ ′ v conjugates s v to φ • t v , we compute, for arbitrary σ ∈ Gal(F v ), where the final equality uses that Gal(F v ) is abelian.Finally, let us show that φ induces a morphism of exact sequences as in the statement.Write p D : π 1 (D) → Gal(F ) for the canonical map, and use p C similarly.Since p C • s v is the identity on the abelian group Gal(F v ), for any σ ∈ Gal(F v ), we have Remark 3.6.Note that π 1 (C) acts on the set of pro-points w above a given w ∈ C 1 and that any two pro-points above w ∈ C 1 in the same π 1 (C)-orbit have the same image on C F w .It follows that the adelic point (x v ) from Construction 3.5 depends on φ only up to π 1 (C)-conjugacy.Similarly, the image of (x v ) in Map(D 1 , C 1 ) under the map in Lemma 2.1 depends on φ only up to π 1 (C)-conjugacy.
Lemma 3.7.Suppose φ ∈ Hom wb π 1 (C) (π 1 (D), π 1 (C)), and let (x v ) ∈ C(A K,F ) be the locally constant adelic point given by Construction 3.5.Then ( be the section map as defined at the beginning of this section.Define By construction, the image of s v is a decomposition group of π 1 (C) above x v ∈ C(F v ).Let α : C ′ → C be a torsor under a finite group scheme G/F .Then α represents a class in H 1 (C, G) = H 1 (π 1 (C), G(F )), where the action of π 1 (C) on G(F ) is induced by the projection π 1 (C) → Gal(F ).The evaluation of α at x v is the class of α , we see that α • φ lies in the images of the horizontal maps in the following commutative diagram whose vertical maps come from inflation:

Proof of Theorem 1.5
Let x = (x v ) ∈ C(A K ) ét \ C(F ).Since H(C) → C is an étale cover, x lifts to a twist of H(C) by an element ξ ∈ H 1 (K, J C (F )) = Hom(G K , J C (F )).Let L/K be the fixed field of ker(ξ).Then L/K is unramified since, locally, it is given as the extension generated by the roots of (I − Φ)(y) = x v , and L/K is abelian since Gal(L/K) is a subgroup of J C (F ).Thus L is a subfield of the function field K ′ of H(D) (for a suitable embedding D → J D ).Viewing x as an adelic point on C over K ′ , we have x ∈ C(A K ′ ) ét by [Sto07,Proposition 5.15].By the above, this adelic point lifts to H(C)(A K ′ ) ét .
From Theorem 1.3, we get that H(C) and H(D) have the same L-function.Now we are in the same situation as before with H(C), H(D) in place of C, D. Iterating this process, we obtain towers such that H n (D) and H n (C) have the same L-functions.Assuming Conjecture 1.4, this implies C(K) C(F ).
Remark 4.1.The paper [BV20] proves a theorem very close in spirit to Conjecture 1.4 using L-functions with characters.It would be very desirable to have a proof of Conjecture 1.1 in the equigenus case from the main theorem of [BV20] along the lines of the above proof, but we have not succeeded in producing it.
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 ≥ 1. Conjecture 1.4 (cf.[SV19, Conjecture 2.2]).Let C, D be smooth projective curves of equal genus at least 2 over a finite field F .If, for each n, there are choices of twists such that the L-function of H n (C) is equal to the L-function of H n (D) for all n ≥ 0, then C is isomorphic to a conjugate of D. Theorem 1.5.Suppose g(C) = g(D) ≥ 2 and assume Conjecture 1.4.Then C(A K,F ) ét C(F ) if and only if there is a nonconstant morphism D → C.
So for any x ∈ π 1 (D) whose image under p D lies in Gal(F v ), we have p D (x) = p C (φ(x)).As this holds for all v ∈ D 1 , we must have p D = p C • φ.So φ induces a morphism of exact sequences as stated.□ Remark 3.4.The construction of the morphism φ in the preceding proof is similar to the proof of [HS12, Proposition 1.1].However, the verification that it interpolates the s v up to conjugation in π 1 (C) rather than just in π 1 (C) is necessarily different from the approach in the proof of [HS12, Proposition 1.2].Construction 3.5.Let φ : π 1 (D) → π 1 (C) be a well-behaved homomorphism.From this we construct a locally constant adelic point (x v ) ∈ C(A K,F ) as follows.Let D and C denote the universal covers of D and C. The decomposition groups of π 1 (D) and π 1 (C) correspond to closed points on D and C. As we have assumed C to be hyperbolic, the intersection of any two distinct decomposition groups of π 1 (C) is open in neither (see for example [ST11, Proposition 1.5]).So the well-behaved map φ determines a map φ : D1 → C1 by declaring φ( ṽ) to be the point of C whose corresponding decomposition group contains φ(D ṽ ).Given a closed point v ∈ D 1 , the embedding θ v : Gal(K v ) → Gal(K) determines a decomposition group T v above v and consequently a pro-point ṽ ∈ D. Define x v ∈ C(F v ) = C F v (F v ) to be the image of φ( ṽ) on C F v .Ranging over the closed points of D, this determines a locally constant adelic point (x v ) ∈ v∈D 1 C(F v ) = C(A K,F ).