Department of MathematicsAlthough Berkovich spaces usually appear in a non-archimedean setting, their general definition actually allows arbitrary Banach rings as base rings, e.g. Z endowed with the usual absolute value. Over the latter, Berkovich spaces look like fibrations that contain complex analytic spaces as well as p-adic analytic spaces for every prime number p. We will try to show what those spaces look like and explain their main properties: topological (local path-connectedness, etc.) or algebraic (noetherianity of the local rings, etc.).
We will also spend some time talking about global functions on those spaces: they are typically convergent arithmetic power series, i.e. power series with coefficients in Z with a positive radius of convergence. D. Harbater introduced these series (in one variable) in the late 1980’s in his study of the inverse Galois problem and investigated the properties of the corresponding rings. We will explain how the nice properties of Berkovich spaces over Z translate into nice properties of rings of convergent arithmetic power series and prove a noetherianity result in several variables.