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the entry p-adic number had (and has) its Definition-section filled with a lengthy recollection of the p-adic integers. I have split into two subsections, such as to make it more clear where the actual definition begins.
added to the Idea-section a pointer to Lubicz: “An introduction to the algorithmic of p-adic numbers”
I added some material to p-adic number on duality, with a view towards Tate’s thesis and Euler factors. It’s unfinished; I have to take a break away from this for the time being.
I realized that nowhere on the $n$Lab so far was the actual statement of how the totally disconnectedness of the p-adics motivates the G-topology and so on.
I have added a paragraph on this now at p-adic number – Topological disconnectedness and G-topology.
I have added essentially the same paragraph also to the beginning of rigid analytic geometry.
Could you say a little more on what about total disconnectedness entails a failure of formulating $p$-adic geometry by analogy with complex analytic geometry? What fails exactly?
Right, good point, one should say more.
One thing that fails is analytic continuation. Apparently the original motivation for Tate to introduce what is now commonly called the G-topology on formal duals to $p$-adic Banach algebras is that he could prove analytic continuation using that.
An excellent account of all this traditional story (excluding Berkovich spectra, but most everything else) in in Bosch-Güntzer-Remmert Non-Archimedean Analysis.
Check out the introduction, which hightlights the motivation from analytic continuation
I had a text somewhere that discussed the naive definition “p-adic manifolds” and the various problems and shortcoming. There is Schneider’s p-Adic analysis and Lie groups (pdf) but I seem to remember I had another text. Hm.
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