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

TL;DR

This paper develops a universal formal group law with monoid actions, connecting perfectoid rings, p-adic fields, and Mochizuki's anabelian ideas, providing a unified algebraic framework for these areas.

Contribution

It introduces a universal monoid formal group law that interpolates additive structures of rings with fixed multiplicative monoids, linking perfectoid and anabelian geometries.

Findings

Existence of universal monoid formal group law for p-perfect monoids.

Application to Lubin-Tate formal groups and p-adic fields.

Connection to Fontaine's A_inf ring in p-adic Hodge theory.

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).

Equations

$$G_{K_1}\simeq G_{K_2}$$

$${\mathcal{O}}_{K_1}^{▹}\simeq {\mathcal{O}}_{K_2}^{▹}$$

$$(x,y)⟼x+y$$

$${\iota }_M:M\to {\mathrm{End}}_{\mathrm{R}}(\mathcal{F}).$$

$$[m](x)={\alpha }_1(m)\cdot x+\sum _{i\ge 2}{\alpha }_i(m)x^i,$$

$$\mathcal{F}(x,y)=x+y.$$

$$[m](x)=m\cdot x$$

$$\mathcal{F}(x,y)=g(f(x)+f(y)),$$

$$[m](x):=g(m\cdot f(x)).$$

$$M^{⊛}=M\cup \{0\}$$

$${\iota }^{⊛}:M^{⊛}\to {\mathrm{End}}_{\mathrm{R}}(\mathcal{F})\text{ with }{\iota }^{⊛}(0)=0\in {\mathrm{End}}_{\mathrm{R}}(\mathcal{F}).$$

$$[m_1+m_2](x)$$

$$[m_1\cdot m_2]$$

$${\mathcal{O}}_K\simeq {\mathrm{End}}_{{\mathcal{O}}_{\mathrm{K}}}(\mathcal{F}).$$

$$\mathcal{F}(x,y)=x+y+\sum _{i+j\ge 2}{c}_{i,j}{x}^i{y}^j$$

$$g_m(x)=m\cdot x+\sum _{i\ge 2}{d}_{m,i}{x}^i$$

$$\mathcal{F}(x,y)-\mathcal{F}(y,x)=\sum _{i+j\ge 2}(c_{i,j}-c_{j,i})x^i{y}^j$$

$$c_{i,j}=c_{j,i}$$

$$\mathcal{F}(x,\mathcal{F}(y,z))-\mathcal{F}(\mathcal{F}(x,y),z)=\sum _{i,j,k}{P}_{i,j,k}{x}^i{y}^j{z}^k,$$

$$g_m(\mathcal{F}(x,y))-\mathcal{F}(g_m(x),g_m(y))=\sum _i{Q}_{m,i}{x}^i$$

$$g_{m^{\prime }}(g_m(\mathcal{F}(x,y))-g_{m\cdot m^{\prime }}(\mathcal{F}(x,y)))=\sum _i{Z}_{m,m^{\prime },i}{x}^i,$$

$$g_{m^{\prime }\cdot m}(x)-g_{m\cdot m^{\prime }}(x)=\sum _{i\ge 2}(d_{m\cdot m^{\prime },i}-d_{m^{\prime }\cdot m,i})x^i.$$

$$L_M=A[\{c_{i,j}\},\{d_{m,i}\}]/I$$

$$I=\left(c_{i,j}-c_{j,i},P_{i,j,k},Q_{m,i},Z_{m,m^{\prime },i},d_{m\cdot m^{\prime },i}-d_{m^{\prime }\cdot m,i}\right),$$

$$\tilde{\mathcal{F}}(x,y)=x+y+\sum _{i,j}{\tilde{c}}_{i,j}{x}^i{y}^j$$

$${\tilde{g}}_m(x)={\alpha }_1(m)\cdot x+\sum _{i\ge 2}{\tilde{d}}_{m,i}{x}^i$$

$$\tilde{\mathcal{F}}(x,y)=x+y+\sum _{i,j}f({\tilde{c}}_{i,j})x^i{y}^j$$

$$[\alpha ](x)\in L_{{\mathcal{O}}_K^{▹}}[[x]]$$

$$[\alpha ]\in {\mathrm{End}}_{{\mathrm{L}}_{{\mathcal{O}}_{\mathrm{K}}^{▹}}}(\mathcal{F})$$

$$\alpha +\beta :=\mathcal{F}([\alpha ](x),[\beta ](x))\text{ for any }\alpha ,\beta \in {\mathcal{O}}_K^{▹}.$$

Full text

1. 1 A prelude
2. 2 Formal groups with monoid actions
3. 3 Existence of formal groups with monoid actions
4. 4 Strict $M$-formal groups
5. 5 Strict ${\mathscr{O}}^{\triangleright}_{K}$-formal groups and rings structures on ${\mathscr{O}}^{\triangleright}_{K}\cup\{0\}$
6. 6 Existence of universal $M$-formal groups
7. 7 Existence of a universal ${\mathscr{O}}^{\triangleright}$-formal group and its consequences
8. 8 Universal strict ${\mathscr{O}}^{\triangleright}$-formal group
9. 9 Monoid pairs and Mochizuki’s Indeterminacies Ind1 and Ind2
10. 10 Geometric applications

1 A prelude

Shinichi Mochizuki has shown that a $p$-adic field $K$ (the term $p$-adic field in this paper will mean a finite extension of ${\mathbb{Q}}_{p}$ for some prime number $p$) can be recovered from its absolute Galois group $G_{K}$ (as a topological group) equipped with its filtration by inertia subgroups (“the upper numbering filtration”); later he has refined this result and shown that $K$ may be recovered from the topological group $G_{K}$ and one Lubin-Tate character of $G_{K}$ (see [14] and [20]). On the other hand, given a $p$-adic field $K_{1}$, the Jarden-Ritter Theorem (see [8]) provides a characterization of all $p$-adic fields $K_{2}$ such that one has a topological isomorphism $G_{K_{2}}\simeq G_{K_{1}}$ of their absolute Galois groups and it is well-known that for every prime $p$, pairs of fields with this property always exists.

Mochizuki’s anabelian reconstruction yoga (see [22] and its references) provides, starting with the topological group $G\simeq G_{K}$, the reconstruction amphora of $G$ (see [6, 5, 7, 20, 21, 22] and its references for the contents of the reconstruction amphora; in [9] I introduced this term as a convenient short-form and memory-aid) which contains all quantities related to $K$ which are reconstructed from the topological group $G$ such as the prime $p$, the topological monoids ${\mathscr{O}}^{*}_{K}$ (the group of units of the ring of integers ${\mathscr{O}}_{K}$ of $K$) and ${\mathscr{O}}_{K}^{\triangleright}$ the multiplicative monoid of non-zero elements of ${\mathscr{O}}_{K}$ (this notation is due to Mochizuki). However the ring ${\mathscr{O}}_{K}$ is not contained in the reconstruction amphora of $G$.

Moreover Mochizuki’s Reconstruction yoga also asserts that if one has an isomorphism of topological groups

[TABLE]

then an isomorphism of the topological monoids

[TABLE]

may also be reconstructed from it (see [6, Section 2]).

So one could say that the set ${\mathscr{O}}^{\triangleright}_{K}\cup\{0\}$ can be equipped with many non-isomorphic ring structures in such a way that ${\mathscr{O}}_{K}^{\triangleright}$ is (isomorphic to) the multiplicative monoid of non-zero elements. As was impressed upon me by Mochizuki, the existence of this variation of ring structures was discovered in the nineteen seventies (for it is a consequence of existence of local field counter examples to the global theorems of Neukirch and Uchida (see [25, 26] and its references) which assert that absolute Galois group of a global field determines the field), but as far as I am aware, Mochizuki appears to be the first to recognize its foundational importance and utility in fundamental problems of arithmetic. This anabelian fluidity of the ring structures on the multiplicative monoid ${\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$ forms an integral part of the fabric of the theory developed by Mochizuki in [19, 20, 21, 21, 22] which in turn form the backdrop of [10, 11, 12, 13].

In this paper I consider a different approach to this problem of understanding the fluidity of ring structures and in particular to the problem of quantifying the fluidity of the additive structures on the set ${\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$ for a $p$-adic field $K$. I began thinking of this problem in Kyoto (Spring 2018) and my preoccupation with it became more or less permanent on my return from Kyoto.

The idea, which I elaborate here, occurred to me in a recent lecture by Michael Hopkins at the Arizona Winter School (2019). In one of his lectures, Hopkins narrated an anecdote about Daniel Quillen’s discovery of the role of formal groups in topological cohomology theories: in particular Quillen’s assertion (to Hopkins) that “as addition rule for Chern classes fails to hold, it must therefore fail in worst possible way–namely by means of a formal group” (I am paraphrasing both Hopkins and Quillen here).

It was immediately clear to me that the answer to the problem of fluidity of additive structure of a $p$-adic field must similarly lie more naturally in formal groups. To be sure the idea of using formal groups had occurred to me in Kyoto, but I had rejected it as I did not see how to implement it (at that time) and secondly Mochizuki’s multiplicative theory, it appeared to me, had no room for addition. However I realized, after listening to Hopkin’s anecdote, that the problem of fluidity of additive structures is also a problem of deforming the tautological addition law

[TABLE]

of a field and it was immediately clear that a formal group could provide the required deformation of this tautological addition law. So it seemed to me that one could replace Lubin-Tate characters by formal groups–even though the former are more natural from Mochizuki’s multiplicative point of view. Of course, it took some time to realize this idea into a precise theory, and it was not altogether a straight forward path as my description might lead readers to believe.

This paper is a record of these ruminations. I will assume that readers are familiar with basic aspects of the theory of formal groups as documented in [4] and I will freely use notation of that book. The main idea is that the variation of additive structure of ${\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$ is encoded in the existence of a universal formal group law with the action of a multiplicative monoid ${\mathscr{O}}_{K}^{\triangleright}$ (Theorem 7.1). In particular this proves (see Corollary 7.3) the existence of universal additive expression for any $\alpha\in{\mathscr{O}}_{K}^{\triangleright}$ which is independent of the additive structure of the chosen field $K$. More precisely for any $\alpha\in{\mathscr{O}}_{K}^{\triangleright}$ there is an endomorphism $[\alpha]$ of the universal formal group with ${\mathscr{O}}_{K}^{\triangleright}$-action (provided by Theorem 7.1) in which $\alpha$ acts by the endomorphism of this formal group by $[\alpha]$ (also see Remark 7.2 for additional comments). Mochizuki’s paper [12] develops the remarkable idea of writing elements $\alpha$ in a manner independently of ring structures (called multi-radial representations in loc. cit.) in an entirely different way. Conjecture 8.1 suggests that in fact there is a universal ring structure on ${\mathscr{O}}^{\triangleright}_{K}\cup\{0\}$ which is given using a suitable formal group. This conjecture is natural given Theorem 7.1 but I do not have any other compelling evidence for it at the moment.

In Theorem 7.6 I prove that every commutative ring $R$ of characteristic zero (with unity) such that $R^{\triangleright}\simeq{\mathscr{O}}^{\triangleright}_{K}$ arises from the universal ${\mathscr{O}}_{K}^{\triangleright}$-formal group in a manner prescribed by Proposition 4.1 and more importantly that every discrete valuation ring $R$ of characteristic zero with a finite residue field such that $R^{\triangleright}\simeq{\mathscr{O}}^{\triangleright}_{K}$ arises from a non-trivial (i.e. not isomorphic to the additive group) ${\mathscr{O}}_{K}^{\triangleright}$-formal group and hence in particular from the universal ${\mathscr{O}}_{K}^{\triangleright}$-formal group in the manner prescribed by Proposition 4.1. So one could say that the universal ${\mathscr{O}}_{K}^{\triangleright}$-formal group gives rise to all ring structures of interest on ${\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$.

Let me emphasize: this paper is completely self-contained (except for my use of the basic theory of formal groups as detailed in [4]) and requires no results from Anabelian geometry except for two standard and elementary facts (1) If $G$ is a topological group isomorphic to an absolute Galois group of a $p$-adic field $K$ then one can reconstruct from $G$, a topological multiplicative monoid ${\mathscr{O}}^{\triangleright}$ isomorphic to ${\mathscr{O}}_{K}^{\triangleright}$ (see [6, Section 3] where ${\mathscr{O}}^{\triangleright}$ is denoted by ${\mathscr{O}}^{\triangleright}(G)$) and (2) if $G_{K_{1}}\simeq G_{K_{2}}$ as topological groups then one can construct from it an isomorphism of topological monoids ${\mathscr{O}}^{\triangleright}_{K_{1}}\simeq{\mathscr{O}}^{\triangleright}_{K_{2}}$ (this follows from the previous assertion). Proofs of these basic facts can be found in Hoshi’s papers [6, Section 2], [7] or in Mochizuki’s [20]. In [9] I have called a $p$-adic field $K_{2}$ with the property $G_{K_{2}}\simeq G_{K_{1}}$ an unjilt of $K_{1}$ or that $K_{1},K_{2}$ are a jilt-unjilt pair (see [9] to understand why this terminology is natural) and conjectured in, analogy with [2, 24], that for a fixed $p$-adic field $K_{1}$, there is a natural topological space constructed using $G_{K_{1}}$, which parametrizes all the unjilts $K_{2}$ of $K_{1}$ (see [9]). The results of this paper give me hope that perhaps a suitable adic space parameterizing all unjilts of a fixed $p$-adic field might also exist (strengthening the analogy with [2, 24]).

Now let me turn to geometric applications of the preceding ideas. Let $X/K$ be a geometrically connected, smooth, hyperbolic curve over a $p$-adic field $K$ and let $\Pi_{X}$ be its étale fundamental group with respect to some base point. Then Mochizuki has shown (see [16, 22]) that from the surjection of profinite groups $\Pi_{X}\to G_{K}$, one can reconstruct the monoid $K(X)^{*}$ of non-zero meromorphic functions on $X$ as well as the monoid of non-zero constants $K^{*}$ together with an inclusion $K^{*}\hookrightarrow K(X)^{*}$ (see [16, 22]). The isomorphism class of $\Pi_{X}\to G_{K}$ does not determine the ring $K(X)$, of meromorphic functions on $X$, uniquely (just as isomorphism class of $G_{K}$ does not determine $K$) and so one sees that the ring structure on $K(X)^{*}\cup\{0\}$ varies while the multiplicative structure of $K(X)^{*}$ is fixed (just as the ring structure of $K^{*}\cup\{0\}$ varies while the multiplicative structure of $K^{*}$ remains fixed). In Section 10, I show that the formalism developed here can be applied to this geometric variation of ring structures. Theorem 10.1 shows that there exists a universal $K(X)^{*}$-formal group which binds together all the additive structures on $K(X)^{*}\cup\{0\}$ and this binding is compatible with the additive structures on $K^{*}\cup\{0\}$ provided by the universal formal group constructed in Theorem 7.1. In particular Corollary 10.2 shows that any non-zero function $f$ on $X$ is given by a universal additive expression independent of ring structures on $K(X)^{*}\cup\{0\}$ in the sense explained earlier (for elements of $K^{*}$).

In particular one deduces for example on taking $X=E-\{O\}$ with $E/K$ a semi-stable elliptic curve and $O\in E(K)$ the identity of the group law of $E$, then theta function(s) on $E$ have a representation independent of the ring structure of $K(X)^{*}\cup\{0\}$ (that such representations exist is an important idea in [10, 11, 12, 13]). Note that $X$ is hyperbolic as required in Mochizuki’s Theorem (see [16]) i.e. one cannot use $\Pi_{E}$ in the reconstruction of $K(X)^{*}=K(E)^{*}$.

As was also pointed out to me by Taylor Dupuy, Mochizuki recognized a long time ago (see for instance [15, Section 4]) that arithmetic applications of anabelian geometry lead naturally to the deep and difficult problem of understanding the line bundles and (Arakelov) degrees (or Arakelov Chern classes) in the presence of anabelian variation of ring structures and he resolved this problem by means of his theory of Frobenioids and realified Frobenioids [18] and Arakelov-Hodge theoretic evaluation methods culminating in [10, 11, 12, 13]. I do not know at the moment how to apply my algebraic methods to study variation of (Arakelov) Chern classes (with respect to variation of ring structures) but as I have mentioned earlier this paper began with Hopkin’s anecdote about Quillen’s discovery ([23]) that the simple and familiar formula for Chern classes of line bundles $c_{1}(L_{1}\otimes L_{2})=c_{1}(L_{1})+c_{1}(L_{2})$ fails to hold in some topological cohomology theories and one could remedy this defect by replacing this simple addition rule by a formal group. To put it differently, Quillen applied formal groups to study variation of cohomology ring structures and here I consider formal groups to study anabelian variation of ring structures, so it seems reasonable that my methods can ultimately be applied to understanding Arakelov Chern classes (at least for curves) in the presence of anabelian variation of ring structures.

One could say that Mochizuki’s approach is ultimately group-theoretic, while my approach is algebro-geometric and leads to a surprising discovery, not manifest from Mochizuki’s extant and preponderant work on anabelian variation of ring structures, that there is in fact an algebro-geometric deformation space which captures anabelian variation of ring structures (see Theorem 7.1).

This paper would not be possible without Shinichi Mochizuki’s bold, audacious, deep and profoundly original ideas which continue to be a source of inspiration for me–in particular his truly remarkable and astounding discovery that there are arithmetic properties of non-zero elements of a fixed $p$-adic field which are independent of the ring structure of this field. I am deeply indebted to him for many conversations on many topics surrounding his ideas and for his continued support and encouragement. I am also indebted to Yuichiro Hoshi for many insightful conversations on anabelian geometry and for explaining to me a number of subtle ideas which lie at the foundations of this subject and which form the backdrop of this paper. I began thinking about the problem addressed in this paper while I was enjoying the support and hospitality of Research Institute of Mathematical Sciences, Kyoto and I thank RIMS for the same. Thanks are also due to Machiel van Frankenhuijsen (who was also visiting RIMS at the time) for many conversations on Mochizuki’s $p$-adic reconstruction theorem and other related topics. Thanks are also due to Taylor Dupuy for many conversations on this paper (and related matters) and for many suggestions, notably that I discuss the problem of Chern classes, which led me to add Section 10, where I discuss a few geometric applications which should viewed as a starting point for understanding anabelian variation of line bundles and Arakelov Chern classes (though this remains distant at the moment). Dupuy also provided a number of references and many corrections which have improved this manuscript. Finally it is a pleasure to thank Michael Hopkins for his insightful lectures at the Arizona Winter School 2019 and also the organizers of the School for an excellently organized meeting.

2 Formal groups with monoid actions

Let $R$ be a ring. (this will be an auxiliary ring for us and eventually a choice will be made). I will work with formal groups over $R$. All formal groups I deal with will be one dimensional commutative formal groups over the chosen ring $R$ (I will not repeat these assumptions again) . Let $\mathscr{F}$ be such a formal group. So $\mathscr{F}(x,y)$ is a formal power series in $x,y$ with coefficients in $R$ with certain properties (see [4] for the definition).

Let $M$ be a commutative monoid with identity (always written multiplicatively in this paper). I say that $\mathscr{F}$ is a formal group with an action of the monoid $M$ (or more simply that * is an $M$-formal group over $R$*) if one has a homomorphism of multiplicative monoids

[TABLE]

For $m\in M$, write $[m]\in\rm{End}_{R}(\mathscr{F})$ for the endomorphism $\iota_{M}(m)$ corresponding to $m\in M$. Note that this means $[m]$ is a formal series over $R$ in some auxiliary variable say $x$ and when one wants to emphasize this, I will write $[m](x)$ for an auxiliary variable $x$. So $[m]$ is of the form

[TABLE]

One checks easily, using the fact that $m\longmapsto[m]$ is a homomorphism of multiplicative monoids, that the mapping given by $m\longmapsto\alpha_{1}(m)$ is a homomorphism of multiplicative monoids denoted again by $\alpha_{1}:M\to R$.

3 Existence of formal groups with monoid actions

First let me prove that given any commutative monoid $M$, there exists a ring $R$ with no ${\mathbb{Z}}$-torsion and an $M$-formal group $\mathscr{F}$ over $R$. Let $R={\mathbb{Q}}[M]$ be the monoid ring over ${\mathbb{Q}}$. Then I claim that there always exists an $M$-formal group over $R$.

Theorem 3.1**.**

Let $M$ be a commutative monoid and let ${\mathbb{Q}}[M]$ be the monoid ring over ${\mathbb{Q}}$. Let $R\supseteq{\mathbb{Q}}[M]$ then there exists an $M$-formal group $\mathscr{F}$ over $R$.

Proof.

This is quite trivial if one uses the fact that over any ${\mathbb{Q}}$-algebra any one-dimensional commutative formal group is isomorphic to the additive formal group with the group law

[TABLE]

Then it is clear that

[TABLE]

defines an endomorphism of our additive group as our ring $R\supset{\mathbb{Q}}[M]\supset M$ and it is clear that $m\longmapsto[m]\in\rm{End}_{R}(\mathscr{F})$ provides a homomorphism of multiplicative monoids. Thus an $M$-formal group exists over any $R\supset{\mathbb{Q}}[M]$.

Here is another constructive proof. The proof closely follows the proof of existence of formal groups (see [4]). Let $f(T)\in R\llbracket T\rrbracket$ be a power series in $T$ with $f(T)=T\bmod{T^{2}}$. Let $g(T)\in R\llbracket T\rrbracket$ be the unique power series such that $f(g(T))=T=g(f(T))$. Such a $g$ always exists as ${\mathbb{Q}}\subseteq{\mathbb{Q}}[M]\subseteq R$.

Now define

[TABLE]

and for any $m\in M$, let $[m]$ be the endomorphism of $\mathscr{F}$ defined by

[TABLE]

Note that $m\cdot f(x)$ makes sense as $m\in R={\mathbb{Q}}[M]$ and $f(x)\in R\llbracket x\rrbracket$. ∎

Imitating the construction of universal formal groups, one can expect to prove that there exists a universal $M$-formal group for some universal ring containing ${\mathbb{Z}}[M]$, which gives rise to all the $M$-formal group laws over any commutative ring. This will be proved in Theorem 6.1 below.

4 Strict $M$-formal groups

Let

[TABLE]

with multiplication $0\cdot m=0=m\cdot 0$ for all $m\in M$. Note that $M^{\circledast}$ is obviously a commutative monoid with this rule. Suppose $\mathscr{F}$ is an $M$-formal group over $R$. Then clearly $\iota:M\to\rm{End}_{R}(\mathscr{F})$ extends to a homomorphism of multiplicative monoids

[TABLE]

Proposition 4.1**.**

Suppose $\mathscr{F}$ is an $M$-formal group over $R$. Suppose that $\iota_{M}$ induces an isomorphism of multiplicative monoids $\iota^{\circledast}_{M}:M^{\circledast}\overset{\sim}{\longrightarrow}\rm{End}_{R}(\mathscr{F})$. Then there exists a unique structure of a commutative ring on $M^{\circledast}$ given by $\mathscr{F}$, such that $M$ is the multiplicative monoid of non-zero elements. The additive structure of $M^{\circledast}$ is given by

[TABLE]

Proof.

The proof is entirely trivial: $\rm{End}_{R}(\mathscr{F})$ is a ring, and if $\iota^{\circledast}:M^{\circledast}\to\rm{End}_{R}(\mathscr{F})$ is a bijection of multiplicative monoids, then $M^{\circledast}$ carries a ring structure induced from the target of $\iota^{\circledast}$ and in this ring structure $M$ is multiplicative monoid of non-zero elements. As $M^{\circledast}$ is a commutative monoid and as $\iota^{\circledast}_{M}:M^{\circledast}\overset{\sim}{\longrightarrow}\rm{End}_{R}(\mathscr{F})$ is an isomorphism of multiplicative monoids, this ring structure on $M^{\circledast}$ is that of a commutative ring. This ring structure is given by the stated formulae which simply state the rules which hold in the target of $\iota^{\circledast}$. So for example, to add $m_{1},m_{2}\in M$, one adds the endomorphisms $[m_{1}],[m_{2}]$ of our formal group $\mathscr{F}$ and the result gives $m_{1}+m_{2}$ (uniquely). ∎

This proposition motivates the following definition. I say that $\mathscr{F}$ is a strict $M$-formal group over $R$ if $\iota_{M}^{\circledast}$ induces a isomorphism of multiplicative monoids $\iota^{\circledast}_{M}:M^{\circledast}\overset{\sim}{\longrightarrow}\rm{End}_{R}(\mathscr{F})$.

5 Strict ${\mathscr{O}}^{\triangleright}_{K}$-formal groups and rings structures on ${\mathscr{O}}^{\triangleright}_{K}\cup\{0\}$

Now let me explain why the notion $M$-formal groups is germane to our story. Let us take for our monoid $M$ to be $M={\mathscr{O}}_{K}^{\triangleright}$ the multiplicative monoid of non-zero elements of the ring of integers of a $p$-adic field $K$. One of Mochizuki’s Reconstruction Machines produces from the topological group $G_{K}$, the multiplicative monoid ${\mathscr{O}}_{K}^{\triangleright}$.

Let us record the following which will be used in the sequel:

Lemma 5.1**.**

Let $K$ be any $p$-adic field. Then there exists a strict ${\mathscr{O}}_{K}^{\triangleright}$-formal group law over ${\mathscr{O}}_{K}$.

Proof.

Indeed by the central result of Lubin-Tate Theory there exists a one-dimensional Lubin-Tate formal group $\mathscr{F}$ over ${\mathscr{O}}_{K}$ such that one has an isomorphism of rings (see [4, Section 20.1.21])

[TABLE]

Then the restriction of this homomorphism of rings to the multiplicative monoid ${\mathscr{O}}_{K}^{\triangleright}$ makes $\mathscr{F}$ into a strict ${\mathscr{O}}_{K}^{\triangleright}$-formal group over ${\mathscr{O}}_{K}$. Hence there exist natural, strict ${\mathscr{O}}_{K}^{\triangleright}$-formal group laws over ${\mathscr{O}}_{K}$. ∎

Proposition 4.1 and Lemma 5.1 can be applied to our context and let me record the preceding discussion in the following:

Theorem 5.2**.**

Let $G$ be the absolute Galois group of some $p$-adic field $K$. Let ${\mathscr{O}}^{\triangleright}\simeq{\mathscr{O}}_{K}^{\triangleright}$ be the multiplicative monoid constructed from $G$.

1. (1)

Any strict ${\mathscr{O}}_{K}^{\triangleright}$-formal group law $\mathscr{F}$ over $R$ equips the multiplicative monoid ${\mathscr{O}}^{\circledast}={\mathscr{O}}^{\triangleright}\cup\{0\}$ with the structure of a commutative ring with ${\mathscr{O}}^{\triangleright}$ as the multiplicative monoid of non-zero elements. 2. (2)

Suppose $K_{1},K_{2}$ are a jilt-unjilt pair (so $K_{1},K_{2}$ are $p$-adic $p$-adic fields with $G_{K_{1}}\simeq G_{K_{2}}$).

1. (a)

Then through the isomorphism ${\mathscr{O}}^{\triangleright}_{K_{1}}\simeq{\mathscr{O}}^{\triangleright}_{K_{2}}$, any ${\mathscr{O}}^{\triangleright}_{K_{1}}$-formal group over any ring $R$ can be viewed as an ${\mathscr{O}}^{\triangleright}_{K_{2}}$-formal group over $R$ and vice versa. 2. (b)

Any strict ${\mathscr{O}}^{\triangleright}_{K_{1}}$-formal group over ${\mathscr{O}}_{K_{2}}$ equips ${\mathscr{O}}^{\circledast}_{K_{1}}$ with a ring structure (in general distinct from its given ring structure) provided by the given formal group law. 3. (c)

More precisely any strict ${\mathscr{O}}_{K_{1}}^{\triangleright}$-formal Lubin-Tate group over ${\mathscr{O}}_{K_{2}}$ equips ${\mathscr{O}}^{\circledast}_{K_{1}}$ with a ring structure isomorphic to ${\mathscr{O}}_{K_{2}}$.

6 Existence of universal $M$-formal groups

The above theorem shows very explicitly that the fluidity of the ring structures on ${\mathscr{O}}_{K}$ arises from the fluidity of strict ${\mathscr{O}}_{K}^{\triangleright}$-formal group laws and in this sense fluidity of additive structures of ${\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$ is encoded in the fluidity of certain formal group laws. This leads one to suspect that fluidity of local additive structures might be a reflection of existence of universal ${\mathscr{O}}^{\triangleright}$-formal group law over some suitably universal ring. This hope is realized in the theorem proved below. This result is weaker than what can probably be proved using a more sophisticated techniques–namely an appropriate version of the functional equation lemma of [4]. But I think this result is more than adequate for my illustrative purpose here.

Theorem 6.1**.**

Let $M$ be any commutative monoid. Then there exists a universal $M$-formal group over a ring $L_{M}$ and homomorphism of multiplicative monoids $M\to L_{M}$ with the following universal property. If $\tilde{\mathscr{F}}$ is an $M$-formal group over some ring $R$ then there is a ring homomorphism $f:L_{M}\to R$ and $\tilde{\mathscr{F}}(x,y)$ is obtained by applying $f$ to the coefficients of $\mathscr{F}(x,y)$. Conversely any homomorphism of rings $L_{M}\to R$ provides an $M$-formal group on $R$.

Proof.

Let $L_{M}$ be the ring which is being sought. The assertion will be proved if one can prove the existence of a power series in two variables

[TABLE]

with coefficients in this ring $L_{M}$. This power series $\mathscr{F}(x,y)$ is required to satisfy the following list of properties which make it into a formal group over $L$:

1. (1)

$\mathscr{F}(x,y)=\mathscr{F}(y,x)$, 2. (2)

$\mathscr{F}(x,\mathscr{F}(y,z))=\mathscr{F}(\mathscr{F}(x,y),z)$,

and the following list of properties which make $\mathscr{F}$ into an $M$-formal group over $L_{M}$:

1. (3)

for every $m\in M$ an endomorphism $g_{m}(x)=m\cdot x+\sum_{i\geq 2}d_{m,i}x^{i}$ such that

1. (a)

$g_{1}(x)=x$ 2. (b)

$g_{m}(\mathscr{F}(x,y))=\mathscr{F}(g_{m}(x),g_{m}(y)),$ 3. (c)

$g_{m_{1}}(g_{m_{2}}(x))=g_{m_{2}}(g_{m_{1}}(x))$

The idea of the proof is to start with an arbitrary formal power series

[TABLE]

and for each $m\in M$, a formal power series

[TABLE]

with $g_{1}(x)=x$ and all these with arbitrary coefficients. Now let us write down all the relations which must hold and then declare all the coefficients to be variables and kill off all the obstructions to the relations. The first property is commutativity which says

[TABLE]

this gives the condition on the coefficients

[TABLE]

which for commutativity to hold. Next associativity gives

[TABLE]

where $P_{i,j,k}$ are polynomials in the coefficients of $\mathscr{F}$ and hence in particular associativity holds if and only the polynomials $P_{i,j,k}=0$. Next let us understand the condition that for every $m\in M$ we have an endomorphism of our formal group. This condition means the following: for every $m\in M$, one wants a power series $g_{m}(x)=m\cdot x+\sum_{i\geq 2}d_{m,i}x^{i}$ which satisfies

[TABLE]

where $Q_{m,i}$ are polynomials in the coefficients of the power series $g_{m},\mathscr{F}$. For instance for $1\in M$ (our monoids are written multiplicatively) one obviously sets $g_{1}(x)=x$. These power series $\{g_{m}\}_{m\in M}$ are required to satisfy the property of composition of endomorphisms:

[TABLE]

where $Z_{m,m^{\prime},i}$ is a polynomial in coefficients of $g_{m},g_{m^{\prime}},\mathscr{F}$ and are required to satisfy commutativity of endomorphisms:

[TABLE]

So now let

[TABLE]

where all the indices run over all the relevant indexing sets and where $I$ is the ideal

[TABLE]

where all the indices run over all the relevant indexing sets.

Then $\mathscr{F}$ is clearly a formal group with a family of endomorphisms $m\longmapsto g_{m}$ for all $m\in M$. Since $g_{m}(x)=g_{m^{\prime}}(x)$ if and only if $d_{m,i}=d_{m^{\prime},i}$ for all $i\geq 2$, so $m\longmapsto g_{m}$ is an injective homomorphism of monoids making $\mathscr{F}$ into an $M$-formal group over $L$ with an $\iota_{M}:M\to\rm{End}_{L}(\mathscr{F})$ given by $m\longmapsto g_{m}$.

Now let me show that this formal group law has the following universal property: given a $M$-formal group $\tilde{\mathscr{F}}$ over a ring $R$, there exists a homomorphism $f:L_{M}\to R$ which provides $\tilde{\mathscr{F}}$ by applying $f$ to the coefficients of $\mathscr{F}$.

To prove this suppose that $\tilde{\mathscr{F}}$ is an $M$-formal group over $R$ given by

[TABLE]

with $\tilde{c}_{i,j}$ in $R$ and suppose that $\tilde{\iota}_{M}:M\to\rm{End}_{R}(\tilde{\mathscr{F}})$ is given by power series

[TABLE]

(as has been remarked earlier, note that $m\longmapsto\alpha_{1}(m)$ is a homomorphism of monoids $M\to R$). Now define $f:{\mathbb{Z}}[M][c_{i,j},d_{m,i}]\to R$ as follows: $m\longmapsto\alpha_{1}(m)\in R$ and extend it linearly to ${\mathbb{Z}}[M]$. Next map the variables $f(c_{i,j})=\tilde{c}_{i,j}\in R$, and the variables $f(d_{m,i})=\tilde{d}_{m,i}\in R$. Then clearly $g_{m}(x)\longmapsto\tilde{g}_{m}(x)\in R\llbracket x\rrbracket$. Since $\tilde{\mathscr{F}}$ is an $M$-formal group, its coefficients satisfy all the relevant relations which make it into an $M$ formal group over $R$ (in other words this homomorphism maps the ideal $I$ constructed above to zero), hence this map $f:{\mathbb{Z}}[M][c_{i,j},d_{m,i}]\to R$ factors through the quotient ring $L_{M}\to R$ (again denoted by $f$). Then

[TABLE]

gives rise to $\tilde{\mathscr{F}}$ through this homomorphism of rings.

Now suppose $f:L_{M}\to R$ is any homomorphism of rings. Then by applying $f$ to coefficients of the power series $\mathscr{F}(x,y)$, and to the coefficients of the endomorphisms $[m]$ (for $m\in M$) of $\mathscr{F}$, which give the universal $M$-formal group law $\mathscr{F}$ over $L_{M}$, one obtains an $M$-formal group law over $R$.

This proves the assertion. ∎

Let me record here the following elementary functorial property which will be used in later sections.

Proposition 6.2**.**

Let $M^{\prime}\to M$ be a morphism of monoids. Let $\mathscr{F}_{M}$ (resp. $\mathscr{F}_{M^{\prime}}$) be the universal $M$-formal group (resp. $M^{\prime}$-formal group) over $L_{M}$ (resp. $L_{M^{\prime}}$). Then one has a homomorphism of rings $L_{M^{\prime}}\to L_{M}$ such that the $M^{\prime}$-formal group $\mathscr{F}_{M}$ (given by $M^{\prime}\to M$) arises from the universal $M^{\prime}$-formal group $\mathscr{F}_{M^{\prime}}$ by applying the morphism $L_{M^{\prime}}\to L_{M}$ to the coefficients of $\mathscr{F}_{M^{\prime}}$. To put it differently the $M^{\prime}$-formal group $\mathscr{F}_{M}$ is the pull-back of the universal $M^{\prime}$-formal group $\mathscr{F}_{M^{\prime}}$ over ${\rm Spec}(L_{M^{\prime}})$ by the morphism ${\rm Spec}(L_{M})\to{\rm Spec}(L_{M^{\prime}})$.

Proof.

The assertion is immediate from the universal property of $(L_{M^{\prime}},\mathscr{F}_{M^{\prime}})$. The composite $M^{\prime}\to M\to\rm{End}(\mathscr{F}_{M})$ provides an $M^{\prime}$-formal group over $L_{M}$. By the universal property of $(\mathscr{F}_{M^{\prime}},L_{M^{\prime}})$ one has a ring homomorphism $L_{M^{\prime}}\to L_{M}$ corresponding to the $M^{\prime}$-formal group $\mathscr{F}_{M}$ over $L_{M}$. This proves the assertion. ∎

7 Existence of a universal ${\mathscr{O}}^{\triangleright}$-formal group and its consequences

Let us note that the rules of Mochizuki’s anabelian game permit the construction of the monoid ring ${\mathbb{Z}}[{\mathscr{O}}_{K}^{\triangleright}]$ of ${\mathscr{O}}_{K}^{\triangleright}$ as it makes no reference to the additive structure of $K$ (or ${\mathscr{O}}_{K}$) and so this ring is available to us and so our constructions will be independent of the additive structure of ${\mathscr{O}}_{K}$. The following theorem is a consequence of Theorem 6.1 and Theorem 5.2 as applied to the monoid ${\mathscr{O}}_{K}^{\triangleright}$ for a $p$-adic field $K$.

Theorem 7.1**.**

Let $K$ be a $p$-adic field and let $M={\mathscr{O}}_{K}^{\triangleright}$ be the monoid of non-zero elements of the ring of integers ${\mathscr{O}}_{K}$.

1. (1)

Then there is a universal ${\mathscr{O}}_{K}^{\triangleright}$-formal group law over the ring $L_{{\mathscr{O}}_{K}^{\triangleright}}$. 2. (2)

Let $\mathscr{F}_{1}$ be any ${\mathscr{O}}_{K}^{\triangleright}$-formal group over a ring $R$ (for example a Lubin-Tate group). Then there is a homomorphism of rings $L_{{\mathscr{O}}_{K}^{\triangleright}}\to R$ such that $\mathscr{F}_{1}$ arises from the universal ${\mathscr{O}}_{K}^{\triangleright}$-formal group $\mathscr{F}$ over $L_{{\mathscr{O}}_{K}^{\triangleright}}$.

Let me stress that the construction of the ring $L_{{\mathscr{O}}_{K}^{\triangleright}}$ and the universal formal group law over $L_{{\mathscr{O}}_{K}^{\triangleright}}$ makes no explicit or implicit use of the ring structure of $K$–the ring which is used to start the construction is ${\mathbb{Z}}[{\mathscr{O}}_{K}^{\triangleright}]$. So its construction is independent of the ring structures on ${\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$.

Thus the universal ${\mathscr{O}}_{K}^{\triangleright}$-formal group law over $L_{{\mathscr{O}}_{K}^{\triangleright}}$ should be viewed as encoding the variation of additive structures on the set ${\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$ keeping its multiplicative structure fixed.

Remark 7.2**.**

The constructions carried out here should be viewed as being global (and also integral) in the sense that the universal formal group constructed here binds together all the unjilts of (any) $p$-adic field $K$ simultaneously (the only fixed datum is the monoid ${\mathscr{O}}^{\triangleright}\simeq{\mathscr{O}}_{K}^{\triangleright}$). The universal formal group law provides a (global) access to the additive structure. In contrast Mochizuki uses the $p$-adic logarithm to access the additive structure but this necessitates working over ${\mathbb{Q}}$ (as the $p$-adic logarithm is defined over ${\mathbb{Q}}$), and also forces one to work with individual unjilts of $K$. On the other hand I ignore the topology of the monoid ${\mathscr{O}}_{K}^{\triangleright}$ completely but presumably more refined versions of my constructions could over come this defect. Nevertheless let me point out that there is a close resemblance between the two approaches: in Mochizuki’s reconstruction theorem for $p$-adic fields the ring structure of a $p$-adic field emerges from the endomorphism ring of the Lubin-Tate character; and here the ring structure emerges from the endomorphism ring of the formal group.

Let us note one further consequence which is relevant to us:

Corollary 7.3**.**

Let $K$ be a $p$-adic field. For any $\alpha\in{\mathscr{O}}_{K}^{\triangleright}$, there exists a power series

[TABLE]

representing the endomorphism

[TABLE]

of the universal ${\mathscr{O}}_{K}^{\triangleright}$-formal group. This power series expresses $\alpha$ independently of the ring structures on ${\mathscr{O}}_{K}^{\circledast}={\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$.

Also important is the following corollary:

Corollary 7.4**.**

Let $G$ be topological group isomorphic to $G_{K}$ for some $p$-adic field $G_{K}$. Then any ${\mathscr{O}}_{K}$-scheme is a scheme over the scheme ${\rm Spec}(L_{{\mathscr{O}}^{\triangleright}})$. In particular schemes $X_{1},X_{2}$ defined over any jilt-unjilt pair $K_{1},K_{2}$ (respectively) are schemes over the scheme ${\rm Spec}(L_{{\mathscr{O}}^{\triangleright}})$.

Proof.

Mochizuki’s reconstruction theory provides from $G$ the multiplicative monoid ${\mathscr{O}}^{\triangleright}$ and an isomorphism of monoids ${\mathscr{O}}^{\triangleright}\simeq{\mathscr{O}}_{K}^{\triangleright}$. This isomorphism together with a choice of a strict ${\mathscr{O}}_{K}^{\triangleright}$-formal group over ${\mathscr{O}}_{K}$ provides a morphism of rings $L_{{\mathscr{O}}^{\triangleright}}\to{\mathscr{O}}_{K}$ equivalently the arrow ${\rm Spec}({\mathscr{O}}_{K})\to{\rm Spec}(L_{{\mathscr{O}}^{\triangleright}})$. So any ${\mathscr{O}}_{K}$ scheme can be viewed as a scheme over ${\rm Spec}(L_{{\mathscr{O}}^{\triangleright}})$. The remaining part of the assertion follows from this. ∎

Proposition 7.5**.**

Let $M$ be a monoid and suppose that for some ring $R$, there exists an $M$-formal group $\mathscr{F}$ over $R$ such that $\mathscr{F}$ is not $R$-isomorphic to the additive formal group $\hat{\mathbb{G}}_{a/{R}}$. Then the universal $M$-formal group over $\mathscr{F}_{M}$ is not $L_{M}$-isomorphic to $\hat{\mathbb{G}}_{a/{L_{M}}}$. In particular if $M={\mathscr{O}}^{\triangleright}$ then $\mathscr{F}_{{\mathscr{O}}^{\triangleright}}$ is not $L_{{\mathscr{O}}^{\triangleright}}$-isomorphic to $\hat{\mathbb{G}}_{a/{L_{{\mathscr{O}}^{\triangleright}}}}$.

Proof.

Suppose $\mathscr{F}_{M}\simeq\hat{\mathbb{G}}_{a/{L_{M}}}$ over $L_{M}$. By the universal property of $\mathscr{F}_{M}$, there exists a homomorphism of rings $L_{M}\to R$ such that the $M$-formal group $\mathscr{F}$ is obtained by applying this homomorphism to the coefficients of $\mathscr{F}_{M}$. As $\mathscr{F}_{M}\simeq\hat{\mathbb{G}}_{a/{L_{M}}}$ one sees that $\mathscr{F}\simeq\hat{\mathbb{G}}_{a/{R}}$. But by hypothesis the $M$-formal group $\mathscr{F}$ is not $R$-isomorphic to $\hat{\mathbb{G}}_{a/{R}}$. This is a contradiction.

Now suppose $M={\mathscr{O}}^{\triangleright}$ and let $K$ be a $p$-adic field such that ${\mathscr{O}}_{K}^{\triangleright}\simeq{\mathscr{O}}^{\triangleright}$. Then by Proposition 5.1 there exists an ${\mathscr{O}}^{\triangleright}$-formal group over ${\mathscr{O}}_{K}$ which is not ${\mathscr{O}}_{K}$-isomorphic to the additive formal group $\hat{\mathbb{G}}_{a/{{\mathscr{O}}_{K}}}$. Thus one sees that $\mathscr{F}_{{\mathscr{O}}^{\triangleright}}$ is not $L_{{\mathscr{O}}^{\triangleright}}$-isomorphic to $\hat{\mathbb{G}}_{a/{L_{{\mathscr{O}}^{\triangleright}}}}$. ∎

Let $R$ be a ring and let $\mathscr{F}$ be a formal group over $R$. I say that $\mathscr{F}$ is the trivial formal group over $R$ if $\mathscr{F}\simeq\hat{\mathbb{G}}_{a/{R}}$ is an isomorphism of formal groups over $R$. If no such isomorphism exists then I say that $\mathscr{F}$ is a non-trivial formal group over $R$. Preceding result says that the universal ${\mathscr{O}}^{\triangleright}$-formal group $\mathscr{F}_{{\mathscr{O}}^{\triangleright}}$ is a non-trivial formal group over $L_{{\mathscr{O}}^{\triangleright}}$.

Now let me prove the following converse to Theorem 5.2(1) which shows that ring structures of interest on ${\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$ always arise from my constructions:

Theorem 7.6**.**

Let ${\mathscr{O}}^{\triangleright}\simeq{\mathscr{O}}_{K}^{\triangleright}$ be a multiplicative monoid of non-zero elements of the ring of integers some $p$-adic field. Let ${\mathscr{O}}^{\circledast}={\mathscr{O}}^{\triangleright}\cup\{0\}$ be the multiplicative monoid defined earlier.

1. (1)

Let $R$ be a ring which satisfies the following hypothesis:

1. (i)

Suppose $R$ is a domain of characteristic zero, 2. (ii)

one has an isomorphism ${\mathscr{O}}^{\triangleright}\simeq R^{\triangleright}$.

Then the choice of the trivial formal group over $R$ equips ${\mathscr{O}}^{\circledast}$ with a ring structure which is isomorphic to the ring $R$. 2. (2)

Now suppose that $R$ is a ring which satisfies the following hypothesis:

1. (i)

$R$* is a discrete valuation ring of characteristic zero,* 2. (ii)

the residue field of $R$ is finite, 3. (iii)

one has an isomorphism of multiplicative monoids ${\mathscr{O}}^{\triangleright}\to R^{\triangleright}$.

Then there is a non-trivial, strict ${\mathscr{O}}^{\triangleright}$-formal group $\mathscr{F}$ over $R$ such that ${\mathscr{O}}^{\circledast}\to\rm{End}_{R}(\mathscr{F})\to R$ is an isomorphism of rings. 3. (3)

At any rate up to isomorphism,

1. (i)

Every commutative domain $R$ of characteristic zero such that ${\mathscr{O}}^{\triangleright}\simeq R^{\triangleright}$ arises from some ${\mathscr{O}}^{\triangleright}$-formal group (and hence from universal pair $(L_{{\mathscr{O}}^{\triangleright}},\mathscr{F}_{L_{{\mathscr{O}}^{\triangleright}}})$) in the manner described by Proposition 4.1. 2. (ii)

Every discrete valuation ring $R$ with a finite residue field and such that ${\mathscr{O}}^{\triangleright}\simeq R^{\triangleright}$ arises from some non-trivial ${\mathscr{O}}^{\triangleright}$-formal group (and hence from universal pair $(L_{{\mathscr{O}}^{\triangleright}},\mathscr{F}_{L_{{\mathscr{O}}^{\triangleright}}})$) in the manner described by Proposition 4.1.

Proof.

Suppose ${\mathscr{O}}^{\circledast}={\mathscr{O}}^{\triangleright}\cup\{0\}$ is equipped with the structure of a discrete valuation ring of characteristic zero with a finite residue field such that the monoid ${\mathscr{O}}^{\triangleright}\subset{\mathscr{O}}^{\circledast}={\mathscr{O}}^{\triangleright}\cup\{0\}$ is the multiplicative monoid of non-zero elements of this ring. Let us write $R$ for this ring. Let us suppose for the moment that one can find a strict ${\mathscr{O}}^{\triangleright}$-formal group $\mathscr{F}$ over $R$ such that $\rm{End}_{R}(\mathscr{F})\simeq R$. Then by the universal property of the pair $(L_{{\mathscr{O}}^{\triangleright}},\mathscr{F}_{L_{{\mathscr{O}}^{\triangleright}}})$, there is a homomorphism of rings $L_{{\mathscr{O}}^{\triangleright}}\to R$ such that $\mathscr{F}$ is the ${\mathscr{O}}^{\triangleright}$-formal group over $R$ obtained by applying $L_{{\mathscr{O}}^{\triangleright}}\to R$ to the coefficients of $\mathscr{F}_{L_{{\mathscr{O}}^{\triangleright}}}$. The ring structure on ${\mathscr{O}}^{\circledast}$ given by Proposition 4.1 (by virtue of existence of a strict ${\mathscr{O}}^{\triangleright}$-formal group $\mathscr{F}$ over $R$) is by definition the ring structure of $\rm{End}_{R}(\mathscr{F})$. As $\rm{End}_{R}(\mathscr{F})\simeq R$ is an isomorphism of rings, one has an isomorphism of rings ${\mathscr{O}}^{\circledast}\to\rm{End}_{R}(\mathscr{F})\to R$.

Thus to complete the proof it suffices to prove that there exists a strict ${\mathscr{O}}^{\triangleright}$-formal group $\mathscr{F}$ over $R$ with $\rm{End}_{R}(\mathscr{F})\simeq R$. This is a consequence of our hypothesis that $R$ is a discrete valuation ring of characteristic zero with a finite residue field and of Lubin-Tate theory (see [4, Section 20.1.19–20.1.21]). ∎

8 Universal strict ${\mathscr{O}}^{\triangleright}$-formal group

The preceding results lead me to make the following very tempting conjecture (but this may be difficult to prove–if true):

Conjecture 8.1**.**

Let $K$ be a $p$-adic field. Then there exists a universal strict ${\mathscr{O}}_{K}^{\triangleright}$-formal group over some ring $S$.

It is of course possible that this is too strong and perhaps some additional hypotheses are need for some assertion of this sort to be true. But let us understand the fundamental consequence of this conjecture: the set ${\mathscr{O}}_{K}^{\triangleright}\cup\{0\}$ carries a universal ring structure given by the existence of this universal formal group over $S$. Multiplication structure of this ring is given by the multiplication rule in ${\mathscr{O}}_{K}^{\triangleright}$ and the addition rule is defined by a universal power series $\mathscr{F}(x,y)\in S\llbracket x,y\rrbracket$ by the rule:

[TABLE]

In particular one should think of the expression $[\alpha](x)\in S\llbracket x\rrbracket$ as a way of expressing a $p$-adic number $\alpha\in K$ in a manner independent of the additive structure of the field $K$!

9 Monoid pairs and Mochizuki’s Indeterminacies Ind1 and Ind2

Let $G$ be a topological group isomorphic to the absolute Galois group of some $p$-adic field $K$. Let $\bar{K}$ be an algebraic closure of $K$ and let $\mu(K)\subset{\mathscr{O}}^{*}_{K}$ be the group of roots of unity in $K$. Then Mochizuki’s anabelian reconstruction machine [20, 21, 22] (and also see [6]) provides the explicit construction of the following four fundamental monoids: these are denoted in loc. cit. by ${\mathscr{O}}^{\triangleright}(G),{\mathscr{O}}^{*}(G),{\mathscr{O}}^{\mu}(G)$ and $K(G)^{*}$ and which will be denoted here by ${\mathscr{O}}^{\triangleright},{\mathscr{O}}^{*},{\mathscr{O}}^{\mu},\mathscr{K}^{*}$ (respectively). The reason for introduction of new notation $\mathscr{K}^{*}$ for $K^{*}$ is simply to remind the reader that $\mathscr{K}^{*}$ is independent of the ring $K$ which it arises from (just as ${\mathscr{O}}^{\triangleright}$ is a copy of the monoid ${\mathscr{O}}_{K}^{\triangleright}$ but separated from the ring structure of ${\mathscr{O}}_{K}$). These four monoids have the property that if $G\simeq G_{K}$ as topological groups then one has isomorphisms

[TABLE]

As a consequence of Theorem 5.2 one gets universal $M$-formal groups $\mathscr{F}_{{\mathscr{O}}_{K}^{\triangleright}},\mathscr{F}_{{\mathscr{O}}^{*}},\mathscr{F}_{{\mathscr{O}}^{\mu}}$ and $\mathscr{F}_{\mathscr{K}^{*}}$, for each of these four monoid ${\mathscr{O}}^{\triangleright},{\mathscr{O}}^{*},{\mathscr{O}}^{\mu}$, and $\mathscr{K}^{*}$, over the rings $L_{{\mathscr{O}}^{\triangleright}},L_{{\mathscr{O}}^{*}},L_{{\mathscr{O}}^{\mu}}$ and $L_{\mathscr{K}^{*}}$. The inclusions ${\mathscr{O}}^{\triangleright}\hookrightarrow\mathscr{K}^{*}$ and ${\mathscr{O}}^{*}\hookrightarrow{\mathscr{O}}^{\triangleright}$ provided, by Propositon 6.2, homomorphisms of rings $L_{{\mathscr{O}}^{\triangleright}}\to L_{\mathscr{K}^{*}}$ and $L_{{\mathscr{O}}^{*}}\to L_{{\mathscr{O}}^{\triangleright}}$ and the morphism $L_{{\mathscr{O}}^{*}}\to L_{\mathscr{K}^{*}}$ which factors as the composite

[TABLE]

respectively. These homomorphisms should be understood as expressing compatibility morphisms: the universal formal group laws on $L_{{\mathscr{O}}^{*}},L_{{\mathscr{O}}^{\triangleright}}$ are compatible with the universal formal group law on $L_{\mathscr{K}^{*}}$.

Let me point out that the formalism developed here extends naturally to the formalism of monoid pairs considered by Mochizuki. Let $G$ be a group and $M$ a commutative monoid with an action of $G$. Such a pair will be written as $G\curvearrowright M$ and will be called a monoid pair. Typical example of these pairs which arise in anabelian geometry are $G\curvearrowright\bar{{\mathscr{O}}}^{\triangleright},G\curvearrowright\bar{{\mathscr{O}}}^{*},G\curvearrowright\bar{{\mathscr{O}}}^{\mu}$ and $G\curvearrowright\bar{\mathscr{K}^{*}}$ where $\bar{{\mathscr{O}}}^{\triangleright}$ is a monoid isomorphic to $\bar{{\mathscr{O}}}^{\triangleright}_{\bar{K}}$ etc. i.e. four fundamental monoids constructed for the field $\bar{K}$.

Let $G\curvearrowright M$ be a monoid pair. Let $\sigma\in G$, then $\sigma:M\to M$ is a homomorphism of monoids such that the composite M{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 10.00006pt\raise 4.50694pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-1.50694pt\hbox{\scriptstyle{\sigma}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces}M{\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern 3.0pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern 0.0pt\offinterlineskip\halign{\entry@#!@&&\entry@@#!@\cr&\crcr}}}\ignorespaces{\hbox{\kern-3.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{{}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern 8.33339pt\raise 5.15138pt\hbox{{}\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern 0.0pt\raise-2.15138pt\hbox{\scriptstyle{\sigma^{-1}}}}}\kern 3.0pt}}}}}}\ignorespaces{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern 27.0pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise 0.0pt\hbox{\textstyle{}}}}}}}}\ignorespaces}}}}\ignorespaces}M is the identity. By the universality of the pair $(\mathscr{F},L_{M})$, consisting of the universal formal group law over the ring $L_{M}$, one gets for each $\sigma\in G$ and $m\in M$ an endomorphism $\sigma(m)\in\rm{End}_{L_{M}}(\mathscr{F})$ and hence an action of $G$ on $M$ considered as a monoid of endomorphisms of the universal formal group law over $L_{M}$. By functoriality of $L_{M}$ this provides an action of $G$ on $L_{M}$.

These considerations can be applied to the four fundamental monoid pairs in anabelian geometry. In particular one has a natural action of $G$ on the rings $L_{\bar{{\mathscr{O}}}^{\triangleright}},L_{\bar{{\mathscr{O}}}^{*}},L_{\bar{{\mathscr{O}}}^{\mu}}$ and $L_{\bar{\mathscr{K}^{*}}}$

Thus one can consider $G\curvearrowright M$ as a pair consisting of a group $G$ with action of $G$ on the monoid $M$ considered now as the monoid of endomorphisms of the universal $M$-formal group law over $L_{M}$. This allows us to transfer $G$-action on these monoids in a manner compatible with the additive structure provided by the universal formal group law over these rings.

Remark 9.5**.**

Finally let me summarize the significance of these constructions of $L_{\bar{{\mathscr{O}}}^{\triangleright}},L_{\bar{{\mathscr{O}}}^{*}},L_{\bar{{\mathscr{O}}}^{\mu}}$ and $L_{\bar{\mathscr{K}^{*}}}$ and their relationships. The rings $L_{\bar{{\mathscr{O}}}^{\triangleright}},L_{\bar{{\mathscr{O}}}^{*}}$ should be understood as providing an algebraic ‘locale’ for what Mochizuki has called indeterminacies of type Ind1 and Ind2 (respectively). In Mochizuki’s theory Ind1 is the variation of ring structures on ${\mathscr{O}}^{\triangleright}\cup\{0\}$ arising from outer automorphisms of the topological group $G_{K}$ and Ind2 is the variation corresponding to the outer automorphisms of the monoid pair $(G_{K},\bar{{\mathscr{O}}}^{*})$. In other words $L_{\bar{{\mathscr{O}}}^{\triangleright}}$ (resp. $L_{\bar{{\mathscr{O}}}^{*}}$) is the (universal) scheme parameterizing indeterminacies of type Ind1 (resp. Ind2) in [10, 11, 12, 13].

Remark 9.6**.**

Let me now explain the role of the morphism ${\rm Spec}(L_{{\mathscr{O}}^{\triangleright}})\to{\rm Spec}(L_{{\mathscr{O}}^{*}})$ vis-a-vis Mochizuki’s Theory. Observe that a choice of a valuation (or a uniformizer) provides a splitting of the inclusion ${\mathscr{O}}^{*}\to{\mathscr{O}}^{\triangleright}$. This splitting provides an isomorphism ${\mathbb{Z}}[{\mathscr{O}}^{\triangleright}]\simeq{\mathbb{Z}}[{\mathscr{O}}^{*}][\mathbb{N}]$ of monoid rings. Choosing an identification ${\mathbb{Z}}[\mathbb{N}]\simeq{\mathbb{Z}}[x]$ for a variable $x$, one sees that ${\mathbb{Z}}[{\mathscr{O}}^{\triangleright}]\simeq{\mathbb{Z}}[{\mathscr{O}}^{*}][x]$ i.e. ${\mathbb{Z}}[{\mathscr{O}}^{\triangleright}]$ is a polynomial ring in one variable over ${\mathbb{Z}}[{\mathscr{O}}^{*}]$. Hence in particular $L_{{\mathscr{O}}^{\triangleright}}\simeq L_{{\mathscr{O}}^{*}}[x]$ (this is easily proved by verifying that $L_{{\mathscr{O}}^{*}}[x]$ satisfies the universal property of $L_{{\mathscr{O}}^{\triangleright}}$ with the action of $x$ identified with the action of the uniformizer of ${\mathscr{O}}^{\triangleright}$). So $L_{{\mathscr{O}}^{\triangleright}}$ is a polynomial ring in one variable over $L_{{\mathscr{O}}^{*}}$. In particular splittings of ${\mathscr{O}}^{*}\to{\mathscr{O}}^{\triangleright}$ provides an isomorphism ${\rm Spec}(L_{{\mathscr{O}}^{\triangleright}})\simeq{\rm Spec}(L_{{\mathscr{O}}^{*}})\times_{{\mathbb{Z}}}\mathbb{A}^{1}$. In Mochizuki’s theory splittings of the inclusion ${\mathscr{O}}^{*}\to{\mathscr{O}}^{\triangleright}$ corresponds to a splitting into “Frobenius-like” (i.e ${\mathscr{O}}^{\triangleright}/{\mathscr{O}}^{*}\simeq\mathbb{N}$) and “étale” (i.e. ${\mathscr{O}}^{*}$) portions and the morphism ${\rm Spec}(L_{{\mathscr{O}}^{\triangleright}})\to{\rm Spec}(L_{{\mathscr{O}}^{*}})$ is the universal “Frobenius-like” portion i.e. parameterizing splittings of ${\mathscr{O}}^{*}\to{\mathscr{O}}^{\triangleright}$ (without reference to ring structures).

10 Geometric applications

In this section I want to consider geometric applications of the theory developed in the preceding sections. Let me emphasize that readers do not need familiarity with Mochizuki’s Reconstruction Theory Trilogy ([20, 21, 22]) for reading this section as the theory of preceding sections depends only on monoids and will now be applied to geometric monoids such as the field of non-zero meromorphic functions on a geometrically irreducible, smooth curve over a field; but it should be noted that the main examples which I have in my mind are the ones arising from anabelian geometry which is viewed here as being a source of interesting (anabelian) variation of ring structures.

Let $K$ be a $p$-adic field and let $G_{K}$ be its absolute Galois group. Let $X/K$ be a geometrically connected, smooth, hyperbolic curve over $K$ ($X$ need not be proper). Let $\Pi_{X}$ be its étale fundamental group (readers familiar with tempered fundamental groups can choose to work with $\Pi_{X}$ or $\Pi^{{\rm temp}}_{X}$) equipped with its profinite topology (pro-discrete topology in the tempered case). [Note that I will assume a choice of base point has been made (so one can talk about the fundamental group of $X$)]. This comes equipped with a surjection of topological groups

[TABLE]

Let me remark that to Grothendieck (see [3]) one owes the foundations of absolute anabelian geometry (which the study of the relationship $X\longmapsto\Pi_{X}$); on the other hand, Mochizuki was the first to recognize the significance of relative anabelian geometry which includes as a special case the study of the relationship $X/K\longmapsto\mathscr{M}_{X/K}$, and in Mochizuki’s work on anabelian geometry one finds an overwhelming abundance of results which illustrate the importance of this idea.

A $p$-adic Mochizukioid (I introduced this terminology in [9] for the reasons explained above)

[TABLE]

is a surjection of topological groups isomorphic to $\mathscr{M}_{X/K}$ arising from some geometrically connected, smooth, hyperbolic curve $X$ over some $p$-adic field $K$. Morphisms of $p$-adic Mochizukioids are defined in the obvious way. One may similarly define tempered $p$-adic Mochizukioids by replacing $\Pi_{X}$ by the tempered fundamental group of $X$.

Let $\mathscr{K}^{*}_{X}:=K(X)^{*}$ be the multiplicative monoid of non-zero meromorphic functions on $X$ and let $\mathscr{K}^{*}:=K^{*}$ be the multiplicative monoid of non-zero elements of $K$. Let me note that by convention followed in algebraic geometry I should strictly speaking write $K(X)^{*}$ for the former monoid and $K^{*}$ for the latter, but the reasons for my notational convention are twofold: first is notational symmetry with preceding sections and secondly I want to notational simplicity for greater transparency. Note that algebraic geometry provides a natural inclusion of monoids $\mathscr{K}^{*}\to\mathscr{K}^{*}_{X}$. Let $\mathscr{K}^{\circledast}_{X}=\mathscr{K}^{*}_{X}\cup\{0\}$ and let $\mathscr{K}^{\circledast}=\mathscr{K}^{*}\cup\{0\}$.

As mentioned earlier the anabelian variation of ring structures on $\mathscr{K}^{\circledast}$ was discovered in the nineteen seventies, but Mochizuki was first to discover its geometric counterpart: the ring structure on $\mathscr{K}^{\circledast}_{X}$ varies (in general) if one keeps the isomorphism class of $\mathscr{M}=\mathscr{M}_{X}$ fixed.

Mochizuki’s reconstruction theory (see [17, 22]) provides, starting with a $p$-adic Mochizukioid $\mathscr{M}$, a construction of monoids $\mathscr{K}^{*}$ and $\mathscr{K}^{*}_{\bf g}$ and an inclusion of monoids $\mathscr{K}^{*}\to\mathscr{K}^{*}_{\bf g}$ such that if one chooses an isomorphism $\mathscr{M}\simeq\mathscr{M}_{X/K}$ then one has an isomorphism

[TABLE]

here the subscript ${\bf g}$ in $\mathscr{K}^{*}_{\bf g}$ is for geometric and so $\mathscr{K}^{*}_{\bf g}$ is a copy of the monoid $\mathscr{K}^{*}_{X}$ separated from the ring structure of $K(X)$ (just as ${\mathscr{O}}^{\triangleright}$ is a copy of ${\mathscr{O}}_{K}^{\triangleright}$ separated from the ring structure of ${\mathscr{O}}_{K}$). Let $\mathscr{K}^{\circledast}_{\bf g}=\mathscr{K}^{*}_{\bf g}\cup\{0\}$ and let $\mathscr{K}^{\circledast}=\mathscr{K}^{*}\cup\{0\}$.

Mochizuki’s observation (see [15, 20, 21, 22]) is that within the isomorphism class of $\mathscr{M}$, $\mathscr{K}^{\circledast}_{\bf g}$ carries many non-isomorphic ring structures such that its multiplicative structure is given by the monoid $\mathscr{K}^{*}_{\bf g}$. To put it differently $\mathscr{K}^{*}_{\bf g}\cup\{0\}$ has many different additive structures and so provides a geometric example of an anabelian variation of rings structures.

The formalism of $M$-formal groups developed in preceding sections can also be applied to the monoid $M=\mathscr{K}^{*}_{\bf g}$. To see that this provides the variation of ring structures on $\mathscr{K}^{\circledast}_{\bf g}$, I have to show that strict $\mathscr{K}^{*}_{\bf g}$-formal groups exist. This is done as follows. First let $\hat{\mathbb{G}}_{a/{F}}$ denote the formal additive group over a field $F$ of characteristic zero. Then $\hat{\mathbb{G}}_{a/{F}}$ is an $F^{*}$-formal group over $F$ which is also strict $F^{*}$-formal group over $F$ (as $F$ is of characteristic zero).

Now let $X/K$ be a curve as above and let $F=K(X)$ be the field of meromorphic functions on $X$. Then $F$ is a field of characteristic zero so one sees that $\hat{\mathbb{G}}_{a/{K(X)}}$ is a strict $K(X)^{*}=\mathscr{K}^{*}_{X}$-formal group over $F$. Hence strict $\mathscr{K}^{*}_{\bf g}$-formal groups exist. Another (equivalent way) of constructing $F^{*}$-formal group is of the following. Fix a line bundle on $X$ and consider its total space as a $\mathbb{G}_{a}$-bundle over $X$. Let $\mathscr{F}$ be the formal completion of this $\mathbb{G}_{a}$-bundle along its zero section. Then $\mathscr{F}$ is a formal group over $X$. Consider the pull-back of this formal group over $X$ to the generic point of $X$. Then one gets a formal group over the field $F=K(X)$. Since $F\supset{\mathbb{Q}}$ (as $F$ is of characteristic zero), one can trivialize this formal group to a formal group of the above form (i.e. this formal group is $F$-isomorphic to the formal group $\hat{\mathbb{G}}_{a/{F}}$). So at any rate the theory of the previous sections can be applied to the monoid $\mathscr{K}^{*}_{\bf g}$ and one gets the following:

Theorem 10.1**.**

Let $\mathscr{M}$ be a $p$-adic Mochizukioid. Let $\mathscr{K}^{*}\to\mathscr{K}^{*}_{\bf g}$ be the inclusion of monoids constructed from $\mathscr{M}$.

1. (1)

Then strict $\mathscr{K}^{*}_{\bf g}$-formal groups exist. 2. (2)

Any strict $\mathscr{K}^{*}_{\bf g}$-formal group over a ring $R$ equips $\mathscr{K}^{\circledast}_{\bf g}$ with a ring structure (given by Proposition 4.1). 3. (3)

There exists a universal $\mathscr{K}^{*}_{\bf g}$-formal group $\mathscr{F}_{\mathscr{K}^{*}_{\bf g}}$ over a ring $L_{\mathscr{K}^{*}_{\bf g}}$. 4. (4)

The inclusion of monoids $\mathscr{K}^{*}\to\mathscr{K}^{*}_{\bf g}$ provides a ring homomorphism $L_{\mathscr{K}^{*}}\to L_{\mathscr{K}^{*}_{\bf g}}$ such that the $\mathscr{K}^{*}$-formal group $\mathscr{F}_{\mathscr{K}^{*}_{\bf g}}$ arises from the universal formal group $\mathscr{F}_{\mathscr{K}^{*}}$ over $L_{\mathscr{K}^{*}}$ from the morphism $L_{\mathscr{K}^{*}}\to L_{\mathscr{K}^{*}_{\bf g}}$.

The meaning of (3) is that there exists a universal additive law which binds together all the possible additive laws on $\mathscr{K}^{\circledast}_{\bf g}$ and (4) establishes the compatibility of additive laws on $\mathscr{K}^{\circledast}_{\bf g}$ with the additive laws on $\mathscr{K}^{\circledast}$.

The geometric variant of Corollary 7.3 is the following:

Corollary 10.2**.**

Let $\mathscr{M}$ be a (tempered) $p$-adic Mochizukioid (for example $\mathscr{M}=\mathscr{M}_{X/K}$ with $X=E-\{O\}$ with $E/K$ as semi-stable elliptic curve over a $p$-adic field $K$). Let $\mathscr{K}^{*}_{\bf g}$ be the monoid constructed from $\mathscr{M}$. Then for any $f\in\mathscr{K}^{*}_{\bf g}$ (for example a $p$-adic theta function on $E$), there exists a power series

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representing the endomorphism

[TABLE]

of the universal $\mathscr{K}^{*}_{\bf g}$-formal group. This power series expresses $f$ independently of the ring structures on $\mathscr{K}^{*}_{\bf g}$. In other words if $X/K$ is any $p$-adic curve giving rise to $\mathscr{M}$ (i.e. $\mathscr{M}\simeq\mathscr{M}_{X})$, then any non-zero meromorphic function $f\in\mathscr{K}^{*}_{X}$ can be represented by a universal additive expression which is independent of ring structure of $\mathscr{K}^{\circledast}_{X}$.