What are Robba rings and why are they important?

Question

If we study $p$-adic Galois representations we come across the so called "Robba rings", defined as the ring of Laurent series which converge in some annulus $\{z| \,\,R<|z|<1\}$ for some positive number $R<1$. I looked at the literature a bit to see why such rings are important, but I could not find any that explains the genesis of the concept itself. Apparently these rings are intimately associated to $(\Phi,\Gamma)$- modules. Any reference or any hint why Robba rings are studied? Perhaps the motivation is from the usual complex analysis?

Answer 1

One sentence of buzzwords to help your literature search: it connects the field of norms (or, more fashionably, tilting) with Fontaine's functors of $p$-adic Hodge theory. The original work in this direction is Laurent Berger's thesis, "Représentations $p$-adiques et équations différentielles, Invent. Math. 148 (2002), no. 2, 219–284.

But there are many papers from the 00s, say, that take this relationship further. Notably, the Robba ring makes possible the notion of trianguline object due to Pierre Colmez, which shows up both in the families of Galois representations over eigenvarieties and in the $p$-adic local Langlands correspondence, at least for $GL_2(Q_p)$.