Arithmetic of $p$-adic curves and sections of geometrically abelian fundamental groups

Let $X$ be a proper, smooth, and geometrically connected curve of genus $g(X)\ge 1$ over a $p$-adic local field. We prove that there exists an effectively computable open affine subscheme $U\subset X$ with the property that $period (X)=1$, and $index (X)$ equals $1$ or $2$ (resp. $period(X)=index (X)=1$, assuming $period (X)=index (X)$), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of $U$ splits. We compute the torsor of splittings of the exact sequence of the geometrically abelian absolute Galois group associated to $X$, and give a new characterisation of sections of arithmetic fundamental groups of curves over $p$-adic local fields which are orthogonal to $Pic^0$ (resp. $Pic^{\wedge}$). As a consequence we observe that the non-geometric (geometrically pro-$p$) section constructed by Hoshi in [Hoshi] is orthogonal to $Pic^0$.

§0. Introduction/Main Results. Let k be a field of characteristic 0 and X a proper, smooth, and geometrically connected curve over k of genus g(X) ≥ 1 with function field K def = k(X). Let η be a geometric point of X with values in its generic point. Thus, η determines an algebraic closure K (resp. k) of K (resp. k). Let U ⊆ X be a non-empty open subscheme and U k def = U × k k. We have an exact sequence of fundamental groups 1 → π 1 (U k , η) → π 1 (U, η) → G k def = Gal(k/k) → 1 (here η is the geometric point of U , U k , naturally induced by η). By pushing this sequence by the maximal abelian quotient π 1 (U k , η) ։ π 1 (U k , η) ab of π 1 (U k , η) we obtain an exact sequence where π 1 (U, η) (ab) def = π 1 (U, η)/ Ker(π 1 (U k , η) ։ π 1 (U k , η) ab ) is the geometrically abelian fundamental group of U . Similarly, by pushing the exact sequence of absolute Galois groups 1 → G k(X) def = Gal(K/kK) → G k(X) def = Gal(K/K) → G k → 1 by the maximal abelian quotient G k(X) ։ G ab k(X) of G k(X) we obtain an exact sequence where G (ab) k(X) def = G k(X) / Ker(G k(X) ։ G ab k(X) ) is the geometrically abelian absolute Galois group of X. For U ⊆ X as above we have exact sequences where I U def = Ker(π 1 (U ) (ab) ։ π 1 (X) (ab) ) = Ker(π 1 (U k ) ab ։ π 1 (X k ) ab ), and (4) 1 → I → G (ab) k(X) → π 1 (X) (ab) → 1, where I def = Ker(G (ab) k(X) ։ π 1 (X) (ab) ) = Ker(G ab k(X) ։ π 1 (X k ) ab ). Note that G subschemes U ⊆ X. Moreover, if P 1 , . . . , P n ∈ X are closed points and U def = X \ {P 1 , . . . , P n } then we have an exact sequence Ind k k(P i )Ẑ (1) → I U → 0, as follows from the well-known structure of π 1 (U k , η) ab , and (by passing to the projective limit we obtain) the exact sequence of G k -modules, where in (6) the product is over all closed points P ∈ X cl . More precisely, for U = X \ {P 1 , . . . , P n } as above let J U be the generalised jacobian of U which sits in the following exact sequence Res k(P i )/k G m is a torus and J def = Jac(X) is the jacobian of X. We have an exact sequence of Tate modules and T J U is identified with π 1 (U k , η) ab (as G k -modules).
As was observed in [Esnault-Wittenberg] Remark 2.3(ii), in the case where k is a p-adic local field, index(X) = 1 (i.e., X possesses a divisor of degree 1) if and only if the exact sequence (2) splits. Our first main result is the following. (See [Lichtenbaum] for the definition of the period of a curve.) Theorem A. Assume that k is a p-adic local field for some prime integer p ≥ 2 (i.e., k/Q p is a finite extension). Then there exists an effectively computable nonempty open affine subscheme U ⊂ X with the following properties.
The term effectively computable in Theorem A means that one can effectively compute U if one can effectively compute a set of topological generators of the group of k-rational points J(k) of the jacobian J def = Jac(X) (cf. proof of Theorem A and Lemma 1.1). For a p-adic local field ℓ, write (ℓ × ) ∧ for the profinite completion of its multiplicative group ℓ × def = ℓ \ {0}. Our second main result is the following, in which we compute the torsor of splittings of the exact sequence (2). 2 Theorem B. With the assumptions in Theorem A, assume that index(X) = 1.
Then there exists an exact sequence where the projective limit is over all open subschemes U ⊆ X, and J U (k) ∧ def = J U (k) ⊗ ZẐ . Moreover, if U = X \ {P 1 , . . . , P n } is affine, then we have an exact sequence and (by passing to the projective limit we obtain) an exact sequence where the product is over all closed points P ∈ X cl .
Next, let s : G k → π 1 (X, η) be a section of the projection π 1 (X, η) ։ G k . Recall that the section s is called orthogonal to Pic ∧ (resp. Pic 0 ) if the homomorphism (1)) is naturally identified with H 2 (π 1 (X, η),Ẑ(1)) (cf. [Mochizuki], Proposition 1.1)] annihilates the Picard part Pic(X) ∧ def = Pic(X) ⊗ ZẐ (resp. the (image in Pic(X) ∧ of the) degree 0 part Pic 0 (X)) of H 2 et (X,Ẑ(1)) (cf. [Saïdi], Definition 1.4.1). We say that the section s is strongly orthogonal to Pic ∧ (resp. Pic 0 ) if for every neighbourhood X i def = X i [s] → X of the section s and the induced section s i : G k → π 1 (X i , η) of the projection π 1 (X i , η) ։ G k (cf. loc. cit. 1.3) the section s i is orthogonal to Pic ∧ (resp. Pic 0 ), i ≥ 1. (Note that the above definition differs slightly from the definition in loc. cit. where the notion of having a cycle class orthogonal to Pic ∧ was defined as being strongly orthogonal to Pic ∧ in the above sense.) We say that the section s is uniformly orthogonal to Pic ∧ (resp. Pic 0 ) if given a finite extension ℓ/k and the induced section s ℓ : G ℓ → π 1 (X ℓ , η) of the projection π 1 (X ℓ , η) ։ G ℓ , where X ℓ def = X × k ℓ, then s ℓ is orthogonal to Pic ∧ (resp. Pic 0 ). The above definitions carry out in a similar way in the case of sections of geometrically pro-Σ arithmetic fundamental groups, where Σ is a non-empty set of prime integers (cf. loc. cit.).
To a section s : G k → π 1 (X, η) as above one associates naturally, by considering the composite morphism of s and the natural projection π 1 (X, η) ։ π 1 (X, η) (ab) , a section s ab : G k → π 1 (X, η) (ab) of the projection π 1 (X, η) (ab) ։ G k . Let J 1 def = Pic 1 X which is a torsor under J. There is a natural morphism X → J 1 . In case period(X) = 1, hence J 1 (k) = ∅, we identify J 1 and J via the isomorphism J 1 ∼ → J which maps a point z ∈ J 1 (k) to the zero section 0 ∈ J(k) and consider the composite morphism X → J 1 ∼ → J. We then obtain a commutative diagram where the vertical maps are isomorphisms. We fix compatible base points of the torsors of splittings of the horizontal sequences in the above diagram. For example, the splitting s z : G k → π 1 (X, η) (ab) of the upper sequence arising from the above point z ∈ J 1 (k) (once we identify π 1 (X, η) (ab) and π 1 (J 1 , η)), and the induced splitting s 0 : G k → π 1 (J, η) of the lower sequence which arises from the zero section 0 ∈ J(k). The section s ab : G k → π 1 (X, η) (ab) gives rise to a section s ab : G k → π 1 (J, η) of the lower sequence in the above diagram, we will denote by the cohomology class (i.e., the cohomology class of the 1-cocycle s ab − s 0 : G k → π 1 (J k , η)) associated to s ab , where T J is the Tate module of J which we identify with π 1 (J k , η). Recall the Kummer exact sequence , for a detailed discussion). If k is a p-adic local field then the natural map J(k) → J(k) ∧ is an isomorphism as follows from the well-known structure of J(k) in this case. In this paper, if k/Q p is a finite extension, we will identify J(k) and J(k) ∧ via this isomorphism. Our next main result is the following which characterises sections of arithmetic fundamental groups of curves over p-adic local fields which are orthogonal to Pic 0 .
Theorem C. With the assumptions in Theorem A, let s : G k → π 1 (X, η) be a section of the projection π 1 (X, η) ։ G k . Then the followings hold.
(i) The section s is orthogonal to Pic 0 if (resp. assuming index(X) = 1, if and only if) the section s ab : G k → π 1 (X, η) (ab) lifts to a sections ab : G k → G (ab) k(X) of the exact sequence (2).
(ii) Assume that X(k) = ∅. Then s is orthogonal to Pic 0 if and only if [s ab ] ∈ J(k).
The assumption that X(k) = ∅ in Theorem C(ii) is rather mild. Indeed, in order to verify that s is orthogonal to Pic ∧ (resp. Pic 0 ) one can pass to a finite extension ℓ/k, and the corresponding section s ℓ : G ℓ → π 1 (X ℓ , η) of the projection π 1 (X ℓ , η) ։ G ℓ (cf. proof of Theorem C(i)). Thus, Theorem C (especially Theorem C(ii)) can be in principle used to detect if a section s as above is (strongly) orthogonal to Pic 0 . As an illustration of this fact we observe that the non-geometric (geometrically pro-p) section constructed by Hoshi over p-adic local fields in [Hoshi] is orthogonal to Pic 0 (cf. Proposition 3.3). Finally, we observe the following characterisation of sections s as above which are strongly orthogonal to Pic ∧ .
Theorem D. With the assumptions in Theorem A, let s : G k → π 1 (X, η) be a section of the projection π 1 (X, η) ։ G k . Then the following two conditions are equivalent.
(i) The section s is strongly orthogonal to Pic ∧ .
Acknowledgment. I would like to thank the referee for his/her careful reading of the paper and useful comments. I thank Akio Tamagawa for the several interesting discussions we had on the topic of this paper, especially around Theorem A. §1. Proof of Theorem A. In this section we prove Theorem A. First, note that if index(X) = 1 (i.e., X possesses a divisor of degree 1) then the exact sequence (1) (as well as the exact sequence (2)) splits for every open subscheme U ⊆ X, as follows from a restriction and corestriction argument in Galois cohomology. We start with the following Lemmas.
arises from (the Galois cohomology of ) the exact sequence (*).
Proof of Lemma 1.2. Let U = X \ {P 1 , . . . , P n } be an open affine subscheme (P 1 , . . . , P n ∈ X are closed points). We have an exact sequence (where Br denotes Brauer groups) → 0 arising from the long Galois cohomology exact sequence associated to the exact se- and we identify the Brauer group of a p-adic local field with Q/Z. The Pontryagin The dual of H 1 (G k , J) is (by Tate duality) J(k), and the dual of the map H 1 (G k , J) → H 2 (G k , H U ) is the homomorphism Div 0 (X \U ) ∧ → J(k) ∧ which is induced by the map Div 0 (X \ U ) → J(k) which maps a divisor of degree 0 on X supported on X \ U to its class in J(k). Further, J(k) is topologically finitely generated as is well-known (cf. [Mattuck]). Let {x 1 , . . . , x t } be topological generators of J(k). There exists an integer r ≥ 1 depending only on g (for example 2 if g = 1, or . Then Im(Div 0 (X \ U ) ∧ → J(k)) has finite index in J(k), and by duality Ker(H 1 (G k , J) → H 2 (G k , H U )) is finite.
Proof of Lemma 1.1. Let U ⊂ X be as in Lemma 1.2. We have an exact sequence U ) (arising from the long Galois cohomology exact sequence associated to (*), cf. diagram below). Further, H 1 (G k , H U ) is finite (cf. [Serre], II.5.8 Theorem 6), Ker H 1 (G k , J) → H 2 (G k , H U ) is finite (cf. Lemma 1.2), hence H 1 (G k , J U ) is finite as follows from the exactness of the above sequence. This finishes the proof of Lemma 1.1.
Next, we resume the proof of Theorem A. Let U ⊂ X be an open affine subscheme. We have a commutative diagram 5 where the vertical sequences are exact and arise from the exact sequences (*) and (**), and the horizontal maps are Kummer homomorphisms arising from the Kummer exact sequences in Galois cohomology associated to the algebraic groups J, H U , and J U , respectively. The middle (resp. fourth from the top) horizontal map maps the class [J 1 U ] of the universal torsor J 1 U (of degree 1) (resp. the class [J 1 ] of J 1 = Pic 1 X/k ) to the class [π 1 (U, η) (ab) ] of the group extension π 1 (U, η) (ab) (resp. the class [π 1 (X, η) (ab) ] of the group extension π 1 (X, η) (ab) ) (this is a well-known fact, see for example [Harari-Szamuely] Proposition 2.2 and Remark 2.4). Further, [J 1 U ] (resp. [π 1 (U, η) (ab) ]) maps to [J 1 ] (resp. [π 1 (X, η) (ab) ]) under the left third vertical map from the top (resp. right third vertical map from the top).
Next, we let U be as in Lemma 1.1. We prove that assertions (i) and (ii) in Theorem A are satisfied in this case.
We prove assertion (i). Assume that the class [π 1 (U, η) (ab) ] is trivial in H 2 (G k , T J U ) which implies that the class [J 1 U ] is divisible in H 1 (G k , J U ). (The map H 1 (G k , J U ) → H 2 (G k , T J U ) factors through lim ← − n H 1 (G k , J U )/nH 1 (G k , J U ) and the latter group injects into H 2 (G k , T J U )). As the group H 1 (G k , J U ) is finite the class of [J 1 U ] is then trivial. Thus, [J 1 ] = 0 in H 1 (G k , J) (cf. above discussion) which implies that X possesses a k-rational divisor class of degree 1, i.e., period(X) = 1. The rest of the assertion follows from the fact that either index(X) = period(X) or index(X) = 2 period(X) (cf. [Lichtenbaum], Theorem 7). Assertion (ii) follows from (i) for the if part, and the only if part follows from the observation at the start of the proof of Theorem A. This finishes the proof of Theorem A. §2. Proof of Theorem B. In this section we prove Theorem B. We use the same assumptions as in Theorem A and further suppose that X possesses a degree 1 divisor. We start with the following lemma.
, by Shapiro's Lemma) and by passing to the projective limit over all U ⊂ X open we obtain an exact sequence 0 → lim ← − U H 1 (G k , H U ) → Br(k) → P ∈X cl Br(k(P )). Now Ker(Br(k) → P ∈X cl Br(k(P ))) is finite of cardinality index(X) (cf. [Lichtenbaum], Theorem 3), which equals 1 under our assumption that X possesses a degree one divisor.
Next, we resume the proof of Theorem B. Consider the morphism X → J as in the introduction, and identify the G k -modules T J and π 1 (X k ) ab . The assertions regarding the structure of H 1 (G k , I U ) and H 1 (G k , I) follow easily from Kummer theory (consider the long cohomology exact sequences associated to the exact sequences (5) and (6) of G k -modules). We establish the exact sequence (7) in the statement of the theorem as well as the isomorphisms lim We have a commutative diagram of group homomorphisms where the vertical sequences are Kummer exact sequences, and the middle and lower horizontal sequences arise from the exact sequences (*) and (**). Note that since H 1 (G k , H U ) is finite (cf. [Serre], II.5.8 Theorem 6), T H 1 (G k , H U ) = 0, and the natural map H 1 (G k , H U ) → H 1 (G k , H U ) ∧ is an isomorphism. The middle horizontal sequence is exact and arises from the long cohomology exact sequence associated to the exact sequence (**). (Note that H 0 (G k , T J) = 0 as follows from the well-known fact that J(k) tor is finite.) The map T H 1 (G k , J U ) → T H 1 (G k , J) is injective as follows easily from the exact sequence H 1 (G k , H U ) → H 1 (G k , J U ) → H 1 (G k , J), the left exactness of the inverse limit functor, and the fact that H 1 (G k , H U ) is finite (cf. [Serre], II.5.8 Theorem 6). We claim that the lower horizontal sequence is exact. Indeed, the map H U (k) ∧ → J U (k) ∧ is injective as follows from the commutativity of the far left lower square, and the injectivity of the maps H U (k) ∧ → H 1 (G k , T H U ) → H 1 (G k , T J U ). Exactness at J U (k) ∧ follows from the commutativity of the lower middle square, the exactness at H 1 (G k , T J U ) of the middle horizontal exact sequence, and the fact that the map β ∈ H 1 (G k , T J U ) by the commutativity of the right lower square and the exactness of the middle horizontal sequence. As α maps to 0 in T H 1 (G k , J), the image of β in T H 1 (G k , J U ) is 0 by the commutativity of the middle upper square and the injectivity of the map T H 1 (G k , J U ) → T H 1 (G k , J). Thus β ∈ J U (k) ∧ maps to α in J(k) ∧ and the lower sequence is exact at J(k) ∧ . By passing to the projective limit over all open subschemes U ⊆ X we obtain a commutative diagram where the middle horizontal sequence is exact and arises from the long exact cohomology sequence associated to the exact sequence (4). The left vertical map is , the second left vertical sequence is exact as follows from the left exactness of the inverse limit functor, the second right vertical sequence is the Kummer exact sequence associated to J, and the right vertical sequence is exact since the H 1 (G k , H U ) are finite (cf. loc. cit.), thus the Mittag-Leffler condition is satisfied. The map lim injective, and the lower horizontal sequences is exact as follows easily from the left exactness of the inverse limit functor and a similar argument as the one used for the previous diagram for the exactness at J(k).
) is an isomorphism and Im(H 1 (G k , G ab k(X) ) → H 1 (G k , T J)) is contained in J(k) (we identify the latter with its image via the injective Kummer map J(k) ֒→ H 1 (G k , T J)). Further, lim ← − U H 1 (G k , H U ) = 0 by Lemma 2.1. Hence Im(H 1 (G k , G ab k(X) ) → H 1 (G k , T J)) = J(k) and we obtain the exact sequence (7) as claimed in Theorem B. This finishes the proof of Theorem B.
Remark 2.2. Let Σ be a non-empty set of prime integers. The same proof as above yields a pro-Σ analog of Theorem B. More precisely, let G ab,Σ k(X) (resp. π 1 (X k , η) ab,Σ ) be the maximal pro-Σ quotient of G ab k(X) (resp. π 1 (X k , η) ab ) which sits in the exact sequence 0 → I Σ → G ab,Σ k(X) → π 1 (X k , η) ab,Σ → 0, where I Σ def = Ker(G ab,Σ k(X) ։ π 1 (X k , η) ab,Σ ). Then, with the same assumptions as in Theorem B, we have an exact sequence 0 → H 1 (G k , I Σ ) → H 1 (G k , G ab,Σ k(X) ) → J(k) Σ → 0, where J(k) Σ is the maximal pro-Σ quotient of J(k). §3. Proof of Theorem C. In this section we prove Theorem C, we use the same assumptions as in Theorem A. The following Lemma will be useful.
Lemma 3.1. Let s : G k → π 1 (X, η) be a section of the projection π 1 (X, η) ։ G k . If s is orthogonal to Pic 0 then s is uniformly orthogonal to Pic 0 .
Proof. Similar to the proof of Proposition 1.6.7 in [Saïdi].
Proof of Theorem C(i). First, assume that s ab : G k → π 1 (X, η) (ab) lifts to a sectioñ s ab : G k → G (ab) k(X) of the exact sequence (2). We will show that s is orthogonal to Pic 0 . Let L ∈ Pic 0 (X) corresponding to the class of a degree zero divisor D = t i=1 n i P i . Given a finite extension ℓ/k, X ℓ def = X × k ℓ, we have a commutative diagram where the left lower and upper horizontal maps arise from Kummer theory (they are injective), the vertical maps are restriction maps, and the map s ⋆ ℓ is induced by the section s ℓ : G k → π 1 (X ℓ , η) of the projection π 1 (X ℓ , η) ։ G k which is induced by s. Identifying both H 2 (G k ,Ẑ(1)) and H 2 (G ℓ ,Ẑ(1)) withẐ, the far right vertical map is multiplication by the degree [ℓ : k] of ℓ/k. In particular, this map is injective. To show that the image of L in H 2 (G k ,Ẑ(1)) is trivial it thus suffices to show that its image in H 2 (G ℓ ,Ẑ(1)) is trivial. We can then, without loss of generality, and after possibly pulling back the line bundle L to X ℓ for a suitable finite extension ℓ/k, assume that the points P 1 , . . . , P t ∈ X are k-rational and deg( Consider the following commutative diagram of horizontal exact sequences. Here the group extension π 1 (U, η) is the pull back of the lower horizontal exact sequence by the map π 1 (X, η) ։ π 1 (X, η) (ab) (i.e., the lower right square is cartesian), π 1 (U, η) c − cn is the geometrically cuspidally central quotient of π 1 (U, η) (cf. [Saïdi] 2.1.1), the surjective map π 1 (U, η) c − cn ։ π 1 (U, η) is the natural one (π 1 (X k , η) acts trivially on I U ), I cn U is the G k -module (1) → I U → 0. By pulling back the group extension π 1 (U, η) c − cn by the section s : G k → π 1 (X, η) we obtain a group extension 1 → I cn U → F U → G k → 1. Further, by pulling back the group extension 1 → I U → π 1 (U, η) (ab) → π 1 (X, η) (ab) → 1 by the section s ab we obtain a group extension 1 → I U → E U → G k → 1, which splits since by assumption s ab lifts to a section s ab U : G k → π 1 (U, η) (ab) of the exact sequence (1). (More precisely, the section s ab U is induced bys ab .) Consider the Galois cohomology exact sequence H 2 (G k ,Ẑ(1)) diag −−→ H 2 (G k , I cn U ) = t i=1 H 2 (G k ,Ẑ(1)) → H 2 (G k , I U ) → 0. The class of the extension F U in H 2 (G k , I cn U ) coincides with (s ⋆ (O(P i )) t i=1 (cf. [Saïdi] proof of Lemma 2.3.1), and the class of the group extension E U in H 2 (G k , I U ) is the image of the class of F U via the above map H 2 (G k , I cn U ) → H 2 (G k , I U ). In particular, since the class of E U vanishes in H 2 (G k , I U ), the class of F U lies in the diagonal image of H 2 (G k ,Ẑ(1)). Thus, we deduce that s ⋆ (O(P i )) is independent of 1 ≤ i ≤ t (i.e., equals the same element of H 2 (G k ,Ẑ(1))), and s ⋆ (L) = 0.
Next, we show that the converse holds assuming index(X) = 1. We assume that s is orthogonal to Pic 0 , index(X) = 1, and show that the section s ab : G k → π 1 (X, η) (ab) lifts to a sections ab : G k → G (ab) k(X) of the exact sequence (2). Recall the exact sequence 1 → I → G (ab) k(X) → π 1 (X, η) (ab) → 1 (resp. 1 → I U → π 1 (U, η) (ab) → π 1 (X, η) (ab) → 1, for U ⊆ X open). By pulling back this group extension by the section s ab we obtain a group extension 1 → I → E → G k → 1 (resp. 1 → I U → E U → G k → 1, for U ⊆ X open), we will show that the group extension E is a split extension which would imply the above assertion. Note that