Let $K$ be a finite extension of $\mathbb{Q}_p$ and let $V$ be a p-adic representation of $\mathrm{Gal}(\overline{\mathbb{Q}}_p/K)$. Write $K_\infty=K(\mu_{p^{\infty}})$ for the cyclotomic extension of $K$. A construction of Sen attaches to $V$ a $K_\infty$-vector space $\mathrm{D}_\mathrm{Sen}(V)$ of dimension $\mathrm{dim}_{\mathbb{Q}_p}(V)$; it is defined as the union of all finite dimensional $\mathrm{Gal}(K_\infty/K)$-stable subspaces of $(\mathbb{C}_p\otimes V)^{\mathrm{Gal}(\overline{K}/K_\infty)}$.

Later, Fontaine similarly defined a $K_{\infty}((t))$-vector space $\mathrm{D}_\mathrm{dif}(V)$, by letting $\mathrm{D}^+_\mathrm{dif}(V)$ be the union of all finite rank $\mathrm{Gal}(K_\infty/K)$-stable $K_\infty[[t]]$-submodules of $(\mathrm{B}_{\mathrm{dR}}^+\otimes V)^{\mathrm{Gal}(\overline{K}/K_\infty)}$ and $\mathrm{D}^+_\mathrm{dif}(V)=\mathrm{D}^+_\mathrm{dif}(V)\otimes_{K_\infty[[t]]}K_\infty((t))$.

In his paper "Sur un resultat de S. Sen", Colmez constructs a period ring $\mathrm{B}_{\mathrm{Sen}}$ such that $(\mathrm{B}_{\mathrm{Sen}}\otimes_{\mathbb{Q}_p} V)^{\mathrm{Gal}(\overline{K}/K_\infty)}$ is isomorphic to $\mathrm{D}_\mathrm{Sen}(V)$. In fact $\mathrm{B}_{\mathrm{Sen}}$ is the ring $\mathbb{C}_p\{\{{\log{t}\}\}}$ of power series with nonzero radius of convergence in the variable $\log{t}$, where $t$ is Fontaine's period.

Does there exist a similar ring $\mathrm{B}_{\mathrm{dif}}$ which would be a period ring for $\mathrm{D}_\mathrm{dif}$? If so, what is it?




Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.