Algebraic monodromy groups of vector bundles on \(p\)-adic curves (Q2848377)

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 \dots \(\mathcal E\) 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 \(E\).''NEWLINENEWLINEThe 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.''NEWLINENEWLINEThe 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 \textit{C. Deninger} and \textit{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)].NEWLINENEWLINEIn 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''.