Kasprowitz, Ralf Algebraic monodromy groups of vector bundles on \(p\)-adic curves. (English) Zbl 1279.14043 J. Reine Angew. Math. 681, 83-99 (2013). Let \({\mathbb Q}_p\) be the field of \(p\)-adic numbers, \(\bar{\mathbb Q}_p\) an algebraic closure of it and \({\mathbb C}_p\) the completion of \(\bar{\mathbb Q}_p\). If \(X\) is a smooth, connected, projective curve over \(\bar{\mathbb Q}_p\), let \(X_{{\mathbb C}_p}:=X\otimes _{\bar{\mathbb Q}_p}\bar{\mathbb Q}_p\). Let \(\mathfrak o\) be the ring of integers of \({\mathbb C}_p\). One says that a vector bundle \(E\) on \(X_{{\mathbb C}_p}\) is trivial modulo \(p^q\), for \(q \in {\mathbb Q}^+\) if “there is a model \(\mathfrak X\) of \(X\) and a model \(\mathcal E\) of \(E\) on \({\mathfrak X}_0:={\mathfrak X}\otimes _{\bar{{\mathbb Z}}_p }{\mathfrak o}\) such that the reduction modulo \(p^q\) of …\(\mathcal E\) is trivial”. If \(x \in X({\mathbb C}_p)\) is a geometric point, denote by \(G_{\rho _E}\) the image of the representation \(\rho _E : \pi _1(X,x) \to \text{GL}(E_x)\) and by \(\bar{G}_{\rho E}\) its Zariski closure. “If \(E\) is a semistable bundle of degree 0, let \(G_E\) be the Tannaka dual group of the Tannaka subcategory generated by \(E\).”The main result of the paper is Theorem 3.12.: “Let \(E\) be a vector vector bundle of rank \(r\) on the smooth, connected and projective curve \(X_{{\mathbb C}_p}\). If \(E\) is a trivial modulo \(p^q\) with \(q > \frac{1}{p-1}\), then \(G_E\) and \({\bar{G}}_{\rho _E}\) are connected.”The main tool is the use of the “partial \(p\)-adic analogue of the classical Narasimhan-Seshadri correspondence between vector bundles and representations of the fundamental group” realized by C. Deninger and A. Werner in [Ann. Sci. Éc. Norm. Supér. (4) 38, No. 4, 553–597 (2005; Zbl 1087.14026); Lond. Math. Soc. Lect. Note Ser. 344, 94–111 (2007; Zbl 1141.14016)].In the last section of the paper one applies “this result to the restriction of certain stable vector bundles on the projective space” and one computes “the Tannaka dual groups in this case”. Reviewer: Nicolae Manolache (Bucureşti) MSC: 14H60 Vector bundles on curves and their moduli 11G20 Curves over finite and local fields 14D05 Structure of families (Picard-Lefschetz, monodromy, etc.) Keywords:\(p\)-adic fields; vector bundle; fundamental group; monodromy; (semi)stable vector bundle; Tannaka dual group; (linearly) reductive group Citations:Zbl 1087.14026; Zbl 1141.14016 PDFBibTeX XMLCite \textit{R. Kasprowitz}, J. Reine Angew. Math. 681, 83--99 (2013; Zbl 1279.14043) Full Text: DOI arXiv