While understanding an analytic construction of $p$-adic numbers is relatively easy (as are understanding Hensel decomposition and basic operations — addition, multiplication…), the field seems to get incredibly hard if we want to go further.
I was looking for what a $p$-adic geometry would look like, hoping for a new playfield, but all I could find are papers from P. Scholze which are way beyond my level (and my motivation).
Is there a nice and gentle $p$-adic geometry, like for instance a $p$-adic triangle geometry ?
Googling these terms lead me to a presentation of B. Le Stum stating that all triangles are isosceles in $\mathbb{Q}_p$, which is true but seems to me as relevant as saying that all triangles verify « the length of the longest side is the sum of the lengths of the other two's » in $\mathbb{R}$. I was rather looking for a triangle geometry in $\mathbb{Q}_p^2$ or maybe $\mathbb{C}_p$.
Does this question even make sense ? Or is there no such thing, at least none that would be relevant* ?
(*For instance, though it is totally possible to define an order over $\mathbb{C}$, there is no interest in doing so, as no order will ever respect the structure of $\mathbb{C}$ — and we should definitively think of $\mathbb{C}$ as a set with no order.
Maybe, likewise it is possible to define a metric over $\mathbb{Q}_p^2$, but none that would have any interest ?)