Zhiqiang He

Comparison theorems in $p$-adic Hodge theory motivate the study of $p$-adic Galois representations. Fontaine and Cherbonnier–Colmez describe these representations through overconvergent $(\varphi,\Gamma)$-modules, while Berger associates differential equations over the Robba ring to de Rham representations. We introduce the dotted Robba ring and extend this connection to arbitrary $p$-adic Galois representations. I will present initial results on the resulting differential equations, including the $p$-periodicity of their radii of convergence, and describe these radii for a family of examples: representations factoring through the cyclotomic quotient $\Gamma_K=\operatorname{Gal}(K(\mu_{p^\infty})/K)$.