Equazioni differenziali p-adiche e interpolazione p-adica di formule classiche

We shortly introduce non-archimedean valued fields and discuss the difficulties in the corresponding theory of analytic functions. We motivate the need of p-adic cohomology with the Well Conjectures. We review the two most popular approaches to p-adic analytic varieties, namely rigid and Berkovich, analytic geometries. We discuss the action of Frobenius in rigid cohomology as similar to the classical action of covering transformations. When rigid cohomology is parametrized by twisting characters, Frobenius is a source of interesting p-adic analytic functions of those characters, like Morita's p-adic gamma function Gamma (p). We conclude with some more examples of this phenomenon in connection with Gauss generalized hypergeometric equation and with an integral formula of Selberg.