On Mochizuki’s Idea of Anabelomorphy and its applications

The term anabelomorphy (pronounciation guide anabel-o-morphy; the root of this term is in Alexander Grothendieck’s Anabelian Program) is coined and introduced here as a concise way of expressing Shinichi Mochizuki’s notion of an anabelian way of changing base fields or base rings. Roughly speaking, one may understand anabelomorphy as the branch of arithmetic in which one studies arithmetic by fixing the absolute Galois group of a field rather than the field itself, and is firmly grounded in the well-known theorem of Mochizuki which asserts that a $p$-adic field is determined by its absolute Galois group equipped with the upper numbering ramification filtration [Mochizuki, 1997].

The case of $p$-adic fields is already quite non-trivial and hence this Introduction will focus on this case. Two $p$-adic fields $K,L$ are anabelomorphic if and only if their absolute Galois groups are topologically isomorphic (see Definition 2.1.1). Anabelomorphy is a new equivalence relation on $p$-adic fields. Isomorphic $p$-adic fields are anabelomorphic, but there exist many non-isomorphic but anabelomorphic $p$-adic fields (Lemma 4.4).

A quantity (resp. a property, an algebraic structure) associated with a $p$-adic field is said to be amphoric if it depends only on the anabelomorphism class of $K$ i.e. if two $p$-adic fields $K,L$ in the same anabelomorphism class have the same quantity (resp. same property, isomorphic algebraic structures). The amphora of $G_{K}$ is the collection of all quantities, properties, algebraic structures associated with $K$ which depend only on the anabelomorphism class of $K$. For example if $K$ is a $p$-adic local field, then the following are examples of an amphoric quantity, property, algebraic structures respectively: the residue characteristic $p$ of $K$, the property of unramifiedness of $K/{\mathbb{Q}}_{p}$, and the topological group $K^{*}$ (Theorem 2.4.3).

The existence of anabelomorphic $p$-adic fields which are not isomorphic means that within the anabelomorphism class of $K$, the additive structure of a $p$-adic field $K$ deforms or wiggles around while the multiplicative structure of $K$ (i.e. the topological group $K^{*}$) remains fixed. This provides us with a new degree of freedom which can be exploited in number theory. For a picturesque way of thinking about anabelomorphy see § 1.8.

The theory of perfectoid fields and perfectoid varieties [Scholze, 2012] also provides highly non-trivial examples of anabelomorphy (see § 18). While this paper deals mostly with the group theoretic aspects of anabelomorphy, in [Joshi, 2019], I demonstrate that Mochizuki’s idea of keeping multiplicative structures of fields (and rings) fixed while allowing the ring structure to vary, can in fact be algebraized.

If two $p$-adic fields $K,L$ are anabelomorphic, one can view representations of $G_{K}$ as representations of $G_{L}$. Thus Anabelomorphy of $p$-adic fields has immediate applications to the theory of representations of the absolute Galois group of a $p$-adic field. Thanks to the main theorem of [Mochizuki, 1997], an important realization on which this paper is founded is that the upper numbering ramification inertia filtration of the absolute Galois group of a $p$-adic field is a Galois theoretic stand-in for the additive structure of a $p$-adic field and through this stand-in the additive structure (of the field) makes its presence felt in the theory of Galois representations.

This theme is explored here in many different ways in this paper starting with establishing the amphoricity of ordinary representations (Theorem 6.1). One knows, by [Mochizuki, 1997], that the property of a representation being Hodge-Tate is unamphoric, so amphoricity of ordinary representations (Theorem 6.1, Theorem 6.2) assumes arithmetic/geometric significance. Theorem 7.1 and Theorem 7.2 deal with properties of $\Phi_{\rm Sen}$ under anabelomorphy. In Theorem 8.1, I show that the $\mathfrak{L}$-invariant (which given, for a Tate elliptic curve with Tate parameter $q$, by $\mathfrak{L}=\frac{\log_{K}(q)}{\mathop{\rm ord}\nolimits_{K}({q})}$) is unamphoric. Another important observation is that the Fontaine subspace of ordinary crystalline two dimensional representations of $G_{K}$ is also amphoric (Theorem 14.1.4).

[Theorem 20.6, should be thought of as the Ordinary Synchronization Theorem at archimedean primes, I provide the archimedean analog of Mochizuki’s theory of étale theta functions [Mochizuki, 2009] (which deals with non-archimdean primes of semi-stable reduction). This approach is quite different from Mochizuki’s treatment of archimedean primes and I believe that my approach provides a certain aesthetic symmetry by bringing the theory at archimedean primes on par with the theory at semi-stable primes. ]

Now let me say a few words about the relationship between Anabelomorphy and the local Langlands Correspondence. The local Langlands Correspondence deals with representations of the absolute Galois group of a $p$-adic field (or a local field of characteristic $p>0$). So the natural question which arises is this: If two $p$-adic fields $K,L$ are anabelomorphic, then how are the corresponding automorphic representations related? The amphoricity of ordinary Galois representations (Theorems 6.1 and 6.2) suggested to me that there might be a portion of the Local Langlands Correspondence which is amphoric. This leads me to the following results. In Theorem 15.2.2, I show that for anabelomorphic $p$-adic fields $K\leftrightsquigarrow L$, one can also synchronize or match local automorphic principle series representations of ${\rm GL}_{n}(K)$ and ${\rm GL}_{n}(L)$ i.e. the principal series representations are amphoric (here $K,L$ are only assumed to be anabelomorphic and there may be not exist any (abstract) field isomorphism between them at all). If $p\neq 2$ and $n=2$, then one can also synchronize supercuspidal representations of ${\rm GL_{2}}(K)$ and ${\rm GL_{2}}(L)$ in a manner compatible with the Local Langlands Correspondence (§ 15, Theorem 15.2.2, Theorem 15.3.2). The situation for ${\rm GL}_{n}$ ($n>2$) needs substantial clarification.

Another important observation of this paper is that several arithmetic invariants of Galois representations such as the different and the discriminant Theorem 4.1, the Swan conductor Theorem 9.1, are unamphoric. This has not appeared in the existing literature on anabelian geometry.

The idea of anabelomorphy can be extended to higher dimensions from the zero dimensional case of fields, by means of fundamental groups of various types. The two principal ones discussed here and [Joshi, 2021, 2022, 2023b, 2023a, 2024b] are the tempered fundamental group of a rigid analytic space arising from a reasonable quasi-projective variety over a $p$-adic field and the case of étale fundamental groups of varieties over number fields or $p$-adic fields. Because of this, anabelomorphy also enjoys a close relationship with the absolute Grothendieck Conjecture and one obtains non-trivial geometric examples of anabelomorphy when the said conjecture fails. A non-trivial example of this is [Joshi, 2020]–which is important from the point of view of [Mochizuki, 2021a, b, c, d] and [Joshi, 2021, 2022, 2023b, 2023a, 2024b].

The validity of Grothendieck’s Anabelian Conjecture for number fields (Theorem 2.4.1) means that a number field $M$ is anabelomorphically rigid. So the question of globalizing local changes of arithmetic into global arithmetic a geometry is quite a subtle one. Local Anabelomorphy, may be thought of as Galois-theoretic surgery on number fields. This leads to the notion of anabelomorphically connected number fields (see 13.1.1, basic example is in Example 13.1.2). The notion of anabelomorphically connected number fields is an important stepping stone in incorporating local anabelomorphic changes into global geometry (Theorem 13.2.2, Theorem 13.2.4).

To begin the discuss, recall that in many results related to automorphic forms and Galois representations (for example [Taylor, 2002]), a theorem of [Moret-Bailly, 1989] plays a central role in incorporating local changes into global arithmetic. An important insight of this paper is that Moret-Bailly’s Theorem can be viewed as arising from a trivial case of Anabelomorphy.

This observation, together with Grothendieck’s Section Conjecture, suggests an anabelomorphic version of Moret-Bailly’s Theorem about density of global points in $p$-adic topologies for anabelomorphically connected number fields. Simplest version of this anabelomorpphic version of Moret-Bailly’s Theorem is Theorem 16.1.1 (for ${\mathbb{P}}^{1}-\{0,1,\infty\}$). Since Grothendieck’s Section Conjecture remains open, the general anabelomorphic version of Moret-Bailly’s Theorem for anabelomorphically connected number fields is largely conjectural Theorem 16.3.3 and Conjecture 16.3.5. The theorem of [Moret-Bailly, 1989] emerges as a very special case of these results. However, Corollary 16.3.4 shows that this general anabelomorphic version of Moret-Bailly’s Theorem is true unconditionally for projective and affine spaces.

As an arithmetic application of Theorem 16.1.1, I prove an Anbelomorphic Connectivity Theorem for Elliptic Curves (see Theorem 16.2.1) which shows that if $E/K$ is an elliptic curve such that $E$ has semi-stable reduction at ${v}_{1},\ldots,{v}_{n}$ and if $\left({K},\{{v}_{1},\ldots,{v}_{n}\}\right)\leftrightsquigarrow\left({K^{\prime}},\{{w}_{1},\ldots,{w}_{n}\}\right)$ is any anabelomorphically connected number field then there exists an elliptic curve $E^{\prime}/K^{\prime}$ with $\mathop{\rm ord}\nolimits_{v_{i}}(j_{E})=\mathop{\rm ord}\nolimits_{w_{i}}(j_{E^{\prime}})$ and with potentially good reduction at all other non-archimedean primes of $K^{\prime}$.

Since I have suggested that anabelomorphy should be roughly understood as providing an anabelian way of base-change, so it is interesting to study the behavior of (say) a variety over ${\mathbb{Q}}_{p}$ when viewed over two anabelomorphic extensions of ${\mathbb{Q}}_{p}$. This leads to the notion of weak or basal anabelomorphy studied in § 17. In Theorem 17.2.1, I show that for an elliptic curve $E$ over a $p$-adic field, all the four quantities: the exponent of the discriminant, the exponent of the conductor, the Kodaira Symbol and the Tamagawa Number are weakly unamphoric. In particular, the bad reduction type of an elliptic curve appears to be sensitive to the differences between the arithmetic of strictly anabelomorphic $p$-adic fields. In Theorem 17.4.2, I show that the Artin and Swan conductors of a higher genus curve are also weakly unamphoric. In particular, the phenomena observed for elliptic curves also occur in higher genus situation. One way to think about these results is that many familiar and frequently used operations, such as choosing a minimal equation for an elliptic curve over two anabelomorphic fields, are strongly tied to the subtle differences between the intertwining of addition and multiplication in the two fields.

In § 19, I show that $p$-adic differential equations (in the sense of [André, 2003]) on a geometrically connected, smooth, quasi-projective and anabelomorphic varieties can also be synchronized under anabelomorphy. This should be thought of as “gluing $p$-adic differential equations by their monodromy.” In particular, the Riemann-Hilbert Correspondence of [André, 2003] can be synchronized with respect to this gluing.

In § 18, I show that anabelomorphy also appears non-trivially in the theory of perfectoid fields and perfectoid spaces considered in [Scholze, 2012].

[This subsection was written by Shinichi Mochizuki and explains how the idea of anabelomorphy discussed in this paper relates to the idea of “Indeterminacy Ind1” [Mochizuki, 2021c, Page 416] (also see [Mochizuki, 2020, Page 104]) which plays a central role in [Mochizuki, 2021a, b, c, d].]

In the parlance of [Mochizuki, 2021a, b, c, d], anabelomorphy, in the case of absolute Galois groups of p-adic local fields, is closely related to Mochizuki’s indeterminacy (Ind1), i.e., to the ${\rm Aut}(G)$-indeterminacy, where $G$ denotes the absolute Galois group of a p-adic local field, which, in [Mochizuki, 2021a, b, c, d], occurs at all nonarchimedean primes. In particular, the following results of the present paper: Theorem 4.1 (and the table following it), Theorem 9.1, and Theorem 17.2.1; (and the data tables after Theorem 17.2.1) provide explicit numerical insight concerning how automorphisms of $G$ that do not arise from field automorphisms, i.e., concerning automorphisms of the sort that arise in the (Ind1) indeterminacy of [Mochizuki, 2021a, b, c, d], can act in a fashion that fails to preserve differents, discriminants, and the Swan and Artin conductors, as well as several other quantities associated to elliptic curves and Galois representations that depend, in an essential way, on the additive structure of the p-adic field.

One could think of anabelomorphy in the following picturesque way:

It should be clear to the readers, after reading this paper, that assimilation of this idea (and the idea of anabelomorphic connectivity) into the theory of Galois representations should have interesting consequences for number theory. Here I have considered anabelomorphy for number fields but interpolating between the number field case and my observation that perfectoid algebraic geometry is a form of anabelomorphy, it seems reasonable to imagine that anabelomorphy of higher dimensional fields will have applications to higher dimensional algebraic geometry as well.

I met Jean-Marc Fontaine in 1994–1995 at the Tata Institute (Mumbai) where he taught a course on $p$-adic Hodge theory. I was fortunate enough to learn $p$-adic Hodge theory directly from him. In the coming years, Fontaine arranged my stays in Paris (1996, 1997, and 2003) which provided me an opportunity to further my understanding of $p$-adic Hodge Theory from him while he (and a few others) were engaged in creating it. Influence of Fontaine’s ideas on this paper and my work on Arithmetic Teichmuller Spaces detailed in [Joshi, 2021, 2023b, 2023a, 2024b, 2024a] should be obvious. I dedicate this paper to the memory of Jean-Marc Fontaine.

The reflections recorded herein began during my stay at RIMS (Kyoto, Spring 2018). Support and hospitality from RIMS (Kyoto) is gratefully acknowledged. I thank Shinichi Mochizuki for many conversations and correspondence on his results documented in [Mochizuki, 2021a, b, c, d]. After the first version of this paper was posted online in March 2020, some readers strongly asserted that there is no relationship between this paper and [Mochizuki, 2021a, b, c, d], so I invited Mochizuki to explain the relationship between anabelomorphy and his ‘Indeterminacy Ind1’ and he obliged by contributing § 1.7.

I thank Yuichiro Hoshi for answering many questions on anabelian geometry. I also thank Yu Yang for promptly answering my questions about [Mochizuki, 2006]. Thanks are also due to Machiel van Frankenhuijsen for many conversations on the $abc$-conjecture and Mochizuki’s Anabelian Reconstruction Theory [Mochizuki, 2012, 2013, 2015]. I thank Taylor Dupuy for conversations around many topics treated here and for providing versions of his manuscripts [Dupuy and Hilado, 2020a], [Dupuy and Hilado, 2020b]. Taylor carefully read several early versions of this manuscript and provided number of suggestions and improvements for which I am extremely grateful. I also thank Tim Holzschuh for a careful reading of an early version of this manuscript and pointing out many typos. I thank Shinichi Mochizuki and Peter Scholze, for alerting me to some errors in the first version (March 2020) of this manuscript.

In § 2.4, the reader will find examples of illustrating the non-triviality of these definitions.

The definition of anabelomorphy of fields readily extends to smooth varieties of higher dimensions. If $X/K$ is a geometrically connected, smooth quasi-projective variety over $K$ then write $\Pi_{X/K}$ (resp. $\Pi^{temp}_{X/K}$) for its étale (resp. tempered) fundamental group of $X/K$. If $X={\rm Spec}(K)$ then both these groups coincide with $G_{K}$.

Anabelomorphism (resp. tempered anabelomorphism) also defines an equivalence relation on smooth varieties over $p$-adic fields.

Evidently isomorphic varieties over a $p$-adic field are anabelomorphic (over that field).

The following is fundamental in understanding anabelomorphy of varieties:

For examples of amphoric quantities which have been know prior to this paper see section 2.4.

Since I am thinking of applications of anabelomorphy to Galois representations, it would be useful to allow some additional generality. Consider an auxiliary topological field $E$ which will serve as a coefficient field for representations of $G_{K}$ (for example $E={\mathbb{Q}}_{\ell}$ for any prime $\ell$ including $\ell=p$ and $\ell=\infty$ will be more than adequate for my discussion). Let $V/E$ be a finite dimensional $E$-vector space (as a topological vector space). Let $\rho:G_{K}\to{\rm GL}(V)$ be a continuous representation of $G_{K}$. I will say that a quantity or an algebraic structure or a property of the triple $(G_{K},\rho,V)$ is amphoric if it is determined by the anabelomorphism class of $K$.

For the reader’s convenience I provide here five fundamental theorems of anabelian geometry upon which anabelomorphy rests. I have organized the results in a logical manner (as opposed to a chronological order).

The following theorem is a combination of many different results proved by (Neukirch, Uchida, Jarden-Ritter, Mochizuki) in different time periods.

The next assertion is the Jarden-Ritter Theorem [Jarden and Ritter, 1979]. This provides a way of deciding if two fields are anabelomorphic or not in most important cases.

These are five fundamental theorems of classical Anabelomorphy. To this list I would like to add the following elementary but useful result.