In this talk, I will discuss several results obtained in collaboration with Jean-Benoît Bost on the geometry of formal-analytic surfaces, as introduced in Bost's talk, of which this one is a follow-up.

Using numerical invariants of those surfaces, I will discuss finiteness results under suitable positivity conditions, with applications to qualitative finiteness results for certain algebras of power series satisfying "twisted" integrality conditions, motivated by concrete situations in the arithmetic geometry of modular curves.

I will also explain how to use formal-analytic arithmetic surfaces to obtain finiteness results for fundamental groups of certain arithmetic surfaces, including some integral models of modular curves.