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**Vector bundles on \(p\)-adic curves and parallel transport. II.**
*(English)*
Zbl 1255.14023

Nakamura, Iku (ed.) et al., Algebraic and arithmetic structures of moduli spaces. Proceedings of the conference, Sapporo, Japan, September 2007. Tokyo: Mathematical Society of Japan (MSJ) (ISBN 978-4-931469-59-4/hbk). Advanced Studies in Pure Mathematics 58, 1-26 (2010).

From the abstract: “We extend our previous theory of étale parallel transport to a larger class of slope zero vector bundles on \(p\)-adic curves. The new class is stable under pullback by ramified coverings. We also construct \(p\)-adic representations of a central extension of the fundamental group for certain bundles of non-zero slope.”

This paper significantly extends the interesting results presented by the same authors in a previous paper [Ann. Sci. École Norm. Sup. (4), No. 38, 553–597 (2005; Zbl 1087.14026)], in which they developed a theory of étale parallel transport for vector bundles with “potentially strongly semistable reduction of degree zero” (where “potentially” means “after pullback by a finite étale covering”) and gave a partial \(p\)-adic analogue of the Narasimhan-Seshadri correspondence (M. S. Narasimhan and C. S. Seshadri [Ann. of Math. (2), No. 82, 540–567 (1965; Zbl 0171.04803)]) between stable vector bundles and representations of the fundamental group. In the paper under review, the whole theory is extended to vector bundles with “potentially strongly semistable reduction”. Not only is the degree zero condition dropped, but “potentially” is also defined in a weaker way, meaning “after pullback by an arbitrary ramified covering”.

This work also relates to G. Faltings’ work on a \(p\)-adic analogue of the Simpson correspondence [Adv. Math. 198, No. 2, 847–862 (2005; Zbl 1102.14022)].

The paper under review is very well written, clear and self-contained.

For the entire collection see [Zbl 1193.14002].

This paper significantly extends the interesting results presented by the same authors in a previous paper [Ann. Sci. École Norm. Sup. (4), No. 38, 553–597 (2005; Zbl 1087.14026)], in which they developed a theory of étale parallel transport for vector bundles with “potentially strongly semistable reduction of degree zero” (where “potentially” means “after pullback by a finite étale covering”) and gave a partial \(p\)-adic analogue of the Narasimhan-Seshadri correspondence (M. S. Narasimhan and C. S. Seshadri [Ann. of Math. (2), No. 82, 540–567 (1965; Zbl 0171.04803)]) between stable vector bundles and representations of the fundamental group. In the paper under review, the whole theory is extended to vector bundles with “potentially strongly semistable reduction”. Not only is the degree zero condition dropped, but “potentially” is also defined in a weaker way, meaning “after pullback by an arbitrary ramified covering”.

This work also relates to G. Faltings’ work on a \(p\)-adic analogue of the Simpson correspondence [Adv. Math. 198, No. 2, 847–862 (2005; Zbl 1102.14022)].

The paper under review is very well written, clear and self-contained.

For the entire collection see [Zbl 1193.14002].

Reviewer: Helena Fischbacher-Weitz (Gattikon)

### MSC:

14H60 | Vector bundles on curves and their moduli |

14H30 | Coverings of curves, fundamental group |

11G20 | Curves over finite and local fields |