# Literature on non-Archimedean analogues of basic complex analysis results

It looks like there is some literature out there on what might be called 'non-Archimedean complex analysis' e.g. Benedetto - An Ahlfors Islands Theorem for non-archimedean meromorphic functions and Cherry - Lectures on Non-Archimedean Function Theory. I am mainly working on non-Archimedean functional analysis right now and need to become better acquainted with the non-Archimedean analogues of basic results that might be encountered in a first course on complex analysis, up to and including Liouville's Theorem e.g. Cauchy integral formulas, holomorphic functions etc. for a few spectral theory proofs.

Of course I am well aware that with many results there will be no such analogue. What I would like to know is if there exists a good introduction to this area that I could look at, that starts with the fundamentals. For example, is there an analogue of 'holomorphic iff analytic'? Any advice much appreciated.

This relates to my earlier question: Non-emptiness of spectrum σ(a) in non-Archimedean Banach algebras

• By the way, there is a talk on YouTube named after New foundations for functional analysis (youtube.com/watch?v=rec9uHzrDM8).
– Z. M
Jan 2 at 15:35
• There are a lot of book on p-adic analysis, but on the "complex analytic" analogues I recall the following. The Appendix of Koblitz's "P-adic Analysis: A Short Course on Recent Work" (and the references). The chapter "Analytic Functions and Elements" in Robert's "A Course in p-adic Analysis". Also, I recall that Alain Escassut has some books on the topic, but I haven't look at them.
– efs
Jan 4 at 0:54

Benedetto has a textbook that discusses basic $$p$$-adic analysis, although his aim is to study $$p$$-adic dynamics. And it's for a single variable. But might be a good place to get some information.

Dynamics in One Non-Archimedean Variable, Robert L. Benedetto, Graduate Studies in Mathematics, Volume 198, 2019, American Mathematical Society

I'll also mention that these days a lot of $$p$$-adic analysis is done on Berkovich space, rather than on $$\mathbb Q_p$$ or $$\mathbb C_p$$. An introduction to analysis on the Berkovich line can be found in the book of Baker and Rumely.

Potential Theory and Dynamics on the Berkovich Projective Line, Matthew Baker, Robert Rumely, Mathematical Surveys and Monographs Volume 159, 2010, American Mathematical Society.

You might like Nonarchimedean Functional Analysis by Peter Schneider; while it’s not directly concerned with $$\mathbb{C}$$, it does discuss properties of nonarchimedean Banach spaces and the like.

• On the same line, there is the book by Van Rooij and the one by Monna.
– efs
Jan 4 at 13:56