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

Mohamed Saïdi

doi:10.1007/s00209-020-02553-1

Abstract

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 Usubset X with the property that {text {period}}(X)=1, and {text {index}}(X) equals 1 or 2 (resp. {text {period}}(X)={text {index}}(X)=1, assuming {text {period}}(X)={text {index}}(X)), if (resp. if and only if) the exact sequence of the geometrically abelian fundamental group of Usplits. 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 {text {Pic}}^0 (resp. {text {Pic}}^{wedge }). As a consequence we observe that the non-geometric (geometrically pro-p) section constructed by Hoshi [3] is orthogonal to {text {Pic}}^0.

Highlights

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 d=ef k(X )
We have an exact sequence of fundamental groups 1 → π1(Uk, η) → π1(U, η) → Gk d=ef Gal(k/k) → 1
For U = X \{P1, . . . , Pn} as above let JU be the generalised jacobian of U which sits in the following exact sequence

Summary

B Mohamed Saïdi

As was observed in [1] 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. Pic0) if given a finite extension /k and the induced section s : G → π1(X , η) of the projection π1(X , η) G , where X d=ef X ×k , s is orthogonal to Pic∧ In case period(X ) = 1, 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. Our main result is the following which characterises sections of arithmetic fundamental groups of curves over p-adic local fields which are orthogonal to Pic0.

Proof of Theorem A

Proof of Theorem B

Proof of Theorem C

Proof of Theorem D