# Rational points of the group of components¶of a Néron model

@article{Bosch1998RationalPO, title={Rational points of the group of components¶of a N{\'e}ron model}, author={Siegfried Bosch and Qing Liu}, journal={manuscripta mathematica}, year={1998}, volume={98}, pages={275-293} }

Abstract:The structure of component groups of Néron models has been investigated on several occasions. Here we admit non-separably closed residue fields and are interested in the subgroup of rational points or, in other terms, in the subgroup of geometrically connected components of a Néron model. We consider Néron models of abelian varieties and of algebraic tori and give detailed computations in the case of Jacobians of curves.

## 43 Citations

### Groups of components of Néron models of Jacobians and Brauer groups

- Mathematics
- 2015

Let X be a proper, smooth, and geometrically connected curve over a non-archimedean local field K. In this paper, we relate the component group of the Neron model of the Jacobian of X to the Brauer…

### Grothendieck’s pairing on component groups of Jacobians

- Mathematics
- 2002

Let R be a discrete valuation ring with field of fractions K . Let AK be an abelian variety over K with dual AK . Denote by A and A ′ the corresponding Neron models and by ΦA and ΦA′ their component…

### ON NÉRON CLASS GROUPS OF SEMIABELIAN VARIETIES

- Mathematics
- 2009

Let F be a global field, write S∞ for the set of archimedean primes of F and let S be a nonempty finite set of primes of F containing S∞. In this paper we study the Néron S-class group CH,F,S of a…

### On the indices of curves over local fields

- Mathematics
- 2006

Fix a non-negative integer g and a positive integer I dividing 2g − 2. For any Henselian, discretely valued field K whose residue field is perfect and admits a degree I cyclic extension, we construct…

### Corrigendum to Néron models, Lie algebras, and reduction of curves of genus one and The Brauer group of a surface

- MathematicsInventiones mathematicae
- 2018

Let X be a proper smooth and connected surface over a finite field. We proved in [LLR2] that the order of the Brauer group Br(X) of X is a perfect square if it is finite. Our proof is based in part…

### Variation of Tamagawa numbers of Jacobians of hyperelliptic curves with semistable reduction

- Mathematics
- 2018

### Self-duality of Selmer groups

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2009

Abstract The first part of the paper gives a new proof of self-duality for Selmer groups: if A is an abelian variety over a number field K, and F/K is a Galois extension with Galois group G, then the…

### Néron models, Lie algebras, and reduction of curves of genus one

- Mathematics
- 2004

Let K be a discrete valuation field. Let OK denote the ring of integers of K , and let k be the residue field of OK , of characteristic p ≥ 0. Let S := SpecOK . Let X K be a smooth geometrically…

### Implementing 2-descent for Jacobians of hyperelliptic curves

- Mathematics, Computer Science
- 2001

This paper gives a fairly detailed description of an algorithm that computes (the size of) the 2-Selmer group of the Jacobian of a hyperellitptic curve over Q. The curve is assumed to have even genus…

### A descent map for curves with totally degenerate semi-stable reduction

- Mathematics
- 2012

Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we…

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