In this talk, I will present various results, established in collaboration with François Charles, concerning formal-analytic arithmetic surfaces. These play the role, in Arakelov geometry, of germs of analytic or formal surfaces in complex algebraic geometry. Firstly I will discuss various motivations for the study of formal-analytic arithmetic surfaces, notably algebraization results in Diophantine geometry and their recent developments by Calegari-Dimitrov-Tang. I will also recall classical theorems in complex geometry, due to Andreotti and Nori, of which our results are arithmetic analogues. I will present in some detail the arithmetic counterpart, in Arakelov geometry, of Nori’s degree bound. This arithmetic bound notably involves a new archimedean invariant, the overflow, attached to an analytic map from a pointed compact Riemann surface to another Riemann surface. Explicit expressions for this invariant play a key role in various applications of our results in Diophantine geometry. Further applications of our arithmetic Nori’s bound will be presented in the talk of François Charles