2019/06/17 by Kirti Joshi, Joshi, Kirti
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1906.06840
openalex publication_date 2019/06/17 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28
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↦ mp 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 O=OK 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 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 O (but possibly non-isomorphic ring structures). The existence of the universal monoid formal group law for the monoid 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 Ainf of p-adic Hodge Theory (Section 4.3).