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Description
Complex analytic geometry is both a field of research in its own right and a powerful tool. However, from an arithmetic point of view, the field C is just one valued field among others that are equally important. This has historically led to the need for an analytic theory on non-Archimedean fields, particularly p-adic fields.
Such a theory cannot be a naive copy of complex theory because, as the topology of these fields is completely discontinuous, there are far too many functions that are “locally developable in entire series.” Currently, there is essentially only one non-Archimedean analytic theory, but several different “languages” (Tate, Raynaud, Berkovich, Huber, Fujiwara, etc.), each with its own advantages depending on the intended applications.
We will begin by reviewing the theory of absolute values to see why it is not possible to simply copy the definition of complex varieties to construct an analytic theory on non-Archimedean fields. We will then present the theory of analytic spaces in the sense of Berkovich, which has the advantage of being closer to complex geometric intuition and therefore seems to be the natural framework for the development of the theory “for its own sake” and for the transfer of classical complex geometry techniques. The objective will be to motivate the construction of Berkovich spaces in such a way as to be able to give examples.
