Tilting Correspondence: A Perfect Correspondence (coming very soon)

The tilting correspondence should be my first step towards $p$-adic Hodge theory. The results from tilting and un-tilting process give us truly beautiful and surprising results about the connection between char. 0 and char. $p$. We already saw how to construct  the tilt of a perfectoid field. Via this process, we obtain a complete, perfect, immediate field between $\widehat{L_\infty}^\flat$ and $\mathbb{C}_p^\flat$. Later, we will see how to un-tilt a complete, perfect, immediate field between $\widehat{L_\infty}^\flat$ and $\mathbb{C}_p$. We will again obtain a perfectoid, immediate field between $\widehat{L}_\infty$ and $\mathbb{C}_p$.

Let me explain how it works. The reduction from char. 0 to char. $p$ is often easy, because one just need to take $\mod p$. To do the converse, we need Witt vectors to lift from char. $p$ to char. 0 (For example: $W(\mathbb{F}_p)=\mathbb{Z}_p$). By using Witt vectors, the un-tilting process can be done. This gives us a bijective map between perfectoid fields (of char. 0) and perfect, complete field (of char. $p$). This would be the first step. This step also leads to some relations between the field of norm $E_L$ (it is basically the field $k_L((X))$) and $\mathbb{C}_p^\flat$, and $\widehat{L_\infty}^\flat$.

For the second step, we even go further, to generalize the result of Fontaine as mentioned in Part I. Because we are dealing with metric spaces basically, we need to equip Witt vectors a suitable topology, and it turns out that the weak topology on Witt vectors are natural. By using this, and the field of norm $E_L$, we deduce the topological isomorphism $Gal(\overline{\mathbb{Q}_p}/L_\infty)\cong Gal(k_L((X))^\text{sep}/k_L((X)))$.

The last step is even more interesting, we deduce the correspondence between finite (Galois) extensions of perfectoid fields and finite (Galois) extensions of their tilts. For me, it is a very amazing result, because one actually needs tools from field extensions of char. $p$ to deduce facts of field extension of char. 0. Both, in my mind, before I read these results, have no relation, but it turns out that they have very close relations, and as elementary arithmetic, char. $p$ is very useful, and somewhat easier than char. 0.

The notes will come very soon...