Algebraization of Mochizuki's anabelian variation of ring structures, perfectoid geometry and formal groups

Abstract

Let $M$ be a multiplicative monoid with identity. Then I show that there is a universal one dimensional formal group law equipped with an action of $M$. If $M$ is $p$-perfect (i.e. $m\mapsto m^p$ is an isomorphism for some prime number $p$) then the universal $M$-formal group law comes equipped with a natural Frobenius endomorphism. There are a number of concrete applications of this result. If $K$ is a $p$-adic field and $\mathcal{O}=\mathcal{O}_K$ is the multiplicative monoid of the ring of integers of $K$, then there is a universal formal group (over a suitable (non-zero) ring) which is equipped with an action of the multiplicative monoid $\mathcal{O}$. Lubin-Tate formal groups arise from this universal monoid formal group law. This has applications to Mochizuki's anabelian ideas: if two p-adic fields have isomorphic absolute Galois groups then they have isomorphic multiplicative monoids $\mathcal{O}$ (but possibly non-isomorphic ring structures). The existence of the universal monoid formal group law for the monoid $\mathcal{O}$ implies that the additive structures of a ring can be interpolated into a universal algebraic family (while keeping the multiplicative structure of the ring fixed). Here is another important example covered by my result: let $R$ be a perfectoid ring and let $R^\flat$ be its tilt and the multiplicative monoid $R^\flat$ of $R^\flat$. Then there exists a universal monoid formal group law for this monoid which interpolates the additive structures of untilts with tilt $R^\flat$. Thus in some sense one has a unified approach to various phenomenon which are well-known in anabelian geometry and in perfectoid geometry. These results also provide a natural number field version of Fontaine's fundamental ring $A_{inf}$ of $p$-adic Hodge Theory (Section 4.3).