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On overconvergence in the Lubin-Tate setting and the multivariable Robba ring

Hold Date
2021-05-21 16:00〜2021-05-21 17:00
Object person
Megumi Takata, (Kyushu Sangyo University)

Let p be a prime number, K and F p-adic fields such that F is contained in K. We have a property called F-overconvergence for Galois representations of K over F. If F = Qp, all the Galois representations of K are Qp-overconvergent as proved by Cherbonnier-Colmez. This fact is fundamental and important in the p-adic local Langlands program, which is established in the GL2(Qp) case by Colmez et al. On the other hand, if F ⊇ Qp, Fourquaux-Xie’s work suggests that large part of representations are not F-overconvergent. In this talk, first, we show two results on overconvergence proved by the speaker. Unfortunately, these emphasize that, in the case of F ⊇ Qp, F-overconvergence is a quite strong property. To overcome the obstruction, Berger suggests a method to use the multivariable Robba ring. In the latter part, we discuss recent attempts of the speaker on the multivariable Robba ring.