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For anyone who wants to really get going, I recommend as starting point some familiarity with two papers, 'The Hodge-Arakelov theory of elliptic curves (HAT)' and 'The Galois-theoretic Kodaira-Spencer morphism of an elliptic curve (GTKS).' [It has been noted here and there that the 'Survey of Hodge Arakelov Theory I,II' papers might be reasonable alternatives.] These papers depart rather little from familiar language, are essential prerequisites for the current series on IUTT, and will take you a long way towards a grasp at least of the motivation behind Mochizuki's imposing collected works. This was the impression I had from conversations six years ago, and then Mochizuki himself just pointed me to page 10 of IUTT I, where exactly this is explained. The goal of the present answer is to decipher just a little bit those few paragraphs.

I can't emphasize enough times that much of what I write is based on faulty memory and guesswork. At best, it is superficial, while at worst, it is (not even) wrong. [In particular, I am no longer sure that the GTKS map is used in an entirely direct fashion.] I have not yet done anything with the current papers than give them a cursory glance. If I figure out more in the coming weeks, I will make corrections. But in the meanwhile, I do hope what I wrote here is mostly more helpful than misleading.

For anyone who wants to really get going, I recommend as starting point some familiarity with two papers, 'The Hodge-Arakelov theory of elliptic curves (HAT)' and 'The Galois-theoretic Kodaira-Spencer morphism of an elliptic curve (GTKS).' [It has been noted here and there that the 'Survey of Hodge Arakelov Theory I,II' papers might be reasonable alternatives.][I've just examined them again, and they really might be the better way to begin.] These papers depart rather little from familiar language, are essential prerequisites for the current series on IUTT, and will take you a long way towards a grasp at least of the motivation behind Mochizuki's imposing collected works. This was the impression I had from conversations six years ago, and then Mochizuki himself just pointed me to page 10 of IUTT I, where exactly this is explained. The goal of the present answer is to decipher just a little bit those few paragraphs.

I can't emphasize enough times that much of what I write is based on faulty memory and guesswork. At best, it is superficial, while at worst, it is (not even) wrong. [In particular, I am no longer sure that the GTKS map is used in an entirely direct fashion.] I have not yet done anything with the current papers than give them a cursory glance. If I figure out more in the coming weeks, I will make corrections. But in the meanwhile, I do hope what I wrote here is mostly more helpful than misleading.

For anyone who wants to really get going, I recommend as starting point some familiarity with two papers, 'The Hodge-Arakelov theory of elliptic curves (HAT)' and 'The Galois-theoretic Kodaira-Spencer morphism of an elliptic curve (GTKS).' These papers depart rather little from familiar language, are essential prerequisites for the current series on IUTT, and will take you a long way towards a grasp at least of the motivation behind Mochizuki's imposing collected works. This was the impression I had from conversations six years ago, and then Mochizuki himself just pointed me to page 10 of IUTT I, where exactly this is explained. The goal of the present answer is to decipher just a little bit those few paragraphs.

The beginning of the investigation is indeed the function field case (over $\mathbb{C}$, for simplicity), where one is given a family $$f:E \rightarrow B$$ of elliptic curves over a compact base, best assumed to be semi-stable and non-isotrivial. There is an exact sequence $$0\rightarrow \omega_E \rightarrow H^1_{DR}(E) \rightarrow H^1(O_E)\rightarrow0,$$ which is moved by the logarithmic Gauss-Manin connection of the family. (I hope I will be forgiven for using standard and non-optimal notation without explanation in this note.) That is to say, if $S\subset B$ is the finite set of images of the bad fibers, there is a log connection $$H^1_{DR}(E) \rightarrow H^1_{DR}(E) \otimes \Omega_B(S),$$ which * does not preserve* $\omega_E$. This fact is crucial, since it leads to an $O_B$-linear Kodaira-Spencer map $$KS:\omega \rightarrow H^1(O_E)\otimes \Omega_B(S),$$ and thence to a non-trivial map $$\omega_E^2\rightarrow \Omega_B(S).$$ From this, one easily deduces Szpiro's inequality: $$\deg (\omega_E) \leq (1/2)( 2g_B-2+|S|).$$ At the most simple-minded level, one could say that Mochizuki's programme has been concerned with replicating this argument over a number field $F$. Since it has to do with differentiation on $B$, which eventually turns into $O_F$, some philosophical connection to $\mathbb{F}_1$-theory begins to appear. I will carry on using the same notation as above, except now $B=Spec(O_F)$.

A large part of HAT is exactly concerned with the set-up necessary to implement this idea, where, roughly speaking, the Galois action has to play the role of the GM connection. Obviously, $G_F$ doesn't act on $H^1_{DR}(E)$. But it does act on $H^1_{et}(\bar{E})$ with various coefficients. The comparison between these two structures is the subject of $p$-adic Hodge theory, which sadly works only over local fields rather than a global one. But Mochizuki noted long ago that something like $p$-adic Hodge theory should be a key ingredient at some level because over $\mathbb{C}$, the comparison isomorphism $$H^1_{DR}(E)\simeq H^1(E(\mathbb{C}), \mathbb{Z})\otimes_{\mathbb{Z}} O_B$$ allows us to completely recover the GM connection, by the condition that the topological cohomology generates the flat sections.

In order to get a global arithmetic analogue, Mochizuki has to formulate a discrete non-linear version of the comparison isomorphism. What is non-linear? This is the replacement of $H^1_{DR}$ by the universal extension $$E^{\dagger}\rightarrow E,$$ (the moduli space of line bundles with flat connection on $E$) whose tangent space is $H^1_{DR}$ (considerations of this nature already come up in usual p-adic Hodge theory). What is discrete is the 'etale cohomology, which will just be $E[\ell]$ with global Galois action, where $\ell$ can eventually be large, on the order of the height of $E$ (that is $\deg (\omega_E)$). The comparison isomorphism in this context takes the following form: $$\Xi: A_{DR}=\Gamma(E^{\dagger}, L)^{<\ell}\simeq L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}.$$ (I apologize for using the notation $A_{DR}$ for the space that Mochizuki denotes by a calligraphic $H$. I can't seem to write calligraphic characters here.) Here, $L$ is a suitably chosen torsion line bundle on $E$, which can then be pulled back to $E^{\dagger}$. The inequality refers to the polynomial degree in the fiber direction of $E^{\dagger} \rightarrow E$. The isomorphism is effected via evaluation of sections at $$E^{\dagger}[\ell]\simeq E[\ell].$$ Finally, $$ L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}$$ comes from Mumford's theory of theta functions. The interpretation of the statement is that it gives an isomorphism between the space of functions of some bounded fiber degree on non-linear De Rham cohomology and the space of functions on discrete 'etale cohomology. This kind of statement is entirely due to Mochizuki. One sometimes speaks of $p$-adic Hodge theory with finite coefficients, but that refers to a theory that is not only local, but deals with linear De Rham cohomology with finite coefficients.

$\Xi^{Lag}$, in contrast to $\Xi$ is free of the Gausssian poles

As another example, one might take a usual global object, such as $$ (E, O_F, E[l], V)$$ (where $V$ denotes a collection of valuations of $F(E[l])$ that restrict bijectively to the valuations $V_0$ of $F$), and associate to it a collection of local categories indexed by $V_0$ (something like Frobenioids corresponding to the $E_v$ for $v\in V_0$). One can then try to glue them together in non-standard ways along sub-categories, after performing a number of non-standard transformations. My rough impression at the moment is that the 'Hodge theatres' arise in this fashion. [This is undoubtedly a gross oversimplification, which I will correct in later amendments.] You might further imagine that some construction of this sort will eventually retain the data necessary to get the height of $E$, but also have data corresponding to the $G^{\mu}$, necessary for the Lagrangian KS map. In any case, I hope you can appreciate that a good deal of 'dismantling' and 'reconstructing' will be necessary.

I can't emphasize enough times that much of what I write is based on faulty memory and guesswork. At best, it is superficial, while at worst, it is (not even) wrong. I have not yet done anything with the current papers than give them a rough glance. If I figure out more in the coming weeks, I will make corrections. But in the meanwhile, I do hope what I wrote here is mostly more helpful than misleading.

For anyone who wants to really get going, I recommend as starting point some familiarity with two papers, 'The Hodge-Arakelov theory of elliptic curves (HAT)' and 'The Galois-theoretic Kodaira-Spencer morphism of an elliptic curve (GTKS).' [It has been noted here and there that the 'Survey of Hodge Arakelov Theory I,II' papers might be reasonable alternatives.] These papers depart rather little from familiar language, are essential prerequisites for the current series on IUTT, and will take you a long way towards a grasp at least of the motivation behind Mochizuki's imposing collected works. This was the impression I had from conversations six years ago, and then Mochizuki himself just pointed me to page 10 of IUTT I, where exactly this is explained. The goal of the present answer is to decipher just a little bit those few paragraphs.

The beginning of the investigation is indeed the function field case (over $\mathbb{C}$, for simplicity), where one is given a family $$f:E \rightarrow B$$ of elliptic curves over a compact base, best assumed to be semi-stable and non-isotrivial. There is an exact sequence $$0\rightarrow \omega_E \rightarrow H^1_{DR}(E) \rightarrow H^1(O_E)\rightarrow0,$$ which is moved by the logarithmic Gauss-Manin connection of the family. (I hope I will be forgiven for using standard and non-optimal notation without explanation in this note.) That is to say, if $S\subset B$ is the finite set of images of the bad fibers, there is a log connection $$H^1_{DR}(E) \rightarrow H^1_{DR}(E) \otimes \Omega_B(S),$$ which does not preserve $\omega_E$. This fact is crucial, since it leads to an $O_B$-linear Kodaira-Spencer map $$KS:\omega \rightarrow H^1(O_E)\otimes \Omega_B(S),$$ and thence to a non-trivial map $$\omega_E^2\rightarrow \Omega_B(S).$$ From this, one easily deduces Szpiro's inequality: $$\deg (\omega_E) \leq (1/2)( 2g_B-2+|S|).$$ At the most simple-minded level, one could say that Mochizuki's programme has been concerned with replicating this argument over a number field $F$. Since it has to do with differentiation on $B$, which eventually turns into $O_F$, some philosophical connection to $\mathbb{F}_1$-theory begins to appear. I will carry on using the same notation as above, except now $B=Spec(O_F)$.

A large part of HAT is exactly concerned with the set-up necessary to implement this idea, where, roughly speaking, the Galois action has to play the role of the GM connection. Obviously, $G_F$ doesn't act on $H^1_{DR}(E)$. But it does act on $H^1_{et}(\bar{E})$ with various coefficients. The comparison between these two structures is the subject of $p$-adic Hodge theory, which sadly works only over local fields rather than a global one. But Mochizuki noted long ago that something like $p$-adic Hodge theory should be a key ingredient because over $\mathbb{C}$, the comparison isomorphism $$H^1_{DR}(E)\simeq H^1(E(\mathbb{C}), \mathbb{Z})\otimes_{\mathbb{Z}} O_B$$ allows us to completely recover the GM connection by the condition that the topological cohomology generates the flat sections.

In order to get a global arithmetic analogue, Mochizuki has to formulate a discrete non-linear version of the comparison isomorphism. What is non-linear? This is the replacement of $H^1_{DR}$ by the universal extension $$E^{\dagger}\rightarrow E,$$ (the moduli space of line bundles with flat connection on $E$) whose tangent space is $H^1_{DR}$ (considerations of this nature already come up in usual p-adic Hodge theory). What is discrete is the 'etale cohomology, which will just be $E[\ell]$ with global Galois action, where $\ell$ can eventually be large, on the order of the height of $E$ (that is $\deg (\omega_E)$). The comparison isomorphism in this context takes the following form: $$\Xi: A_{DR}=\Gamma(E^{\dagger}, L)^{<\ell}\simeq L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}.$$ (I apologize for using the notation $A_{DR}$ for the space that Mochizuki denotes by a calligraphic $H$. I can't seem to write calligraphic characters here.) Here, $L$ is a suitably chosen line bundle of degree $\ell$ on $E$, which can then be pulled back to $E^{\dagger}$. The inequality refers to the polynomial degree in the fiber direction of $E^{\dagger} \rightarrow E$. The isomorphism is effected via evaluation of sections at $$E^{\dagger}[\ell]\simeq E[\ell].$$ Finally, $$ L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}$$ comes from Mumford's theory of theta functions. The interpretation of the statement is that it gives an isomorphism between the space of functions of some bounded fiber degree on non-linear De Rham cohomology and the space of functions on discrete 'etale cohomology. This kind of statement is entirely due to Mochizuki. One sometimes speaks of $p$-adic Hodge theory with finite coefficients, but that refers to a theory that is not only local, but deals with linear De Rham cohomology with finite coefficients.

$\Xi^{Lag}$, in contrast to $\Xi$, is free of the Gaussian poles

As another example, one might take a usual global object, such as $$ (E, O_F, E[l], V)$$ (where $V$ denotes a collection of valuations of $F(E[l])$ that restrict bijectively to the valuations $V_0$ of $F$), and associate to it a collection of local categories indexed by $V_0$ (something like Frobenioids corresponding to the $E_v$ for $v\in V_0$). One can then try to glue them together in non-standard ways along sub-categories, after performing a number of non-standard transformations. My rough impression at the moment is that the 'Hodge theatres' arise in this fashion. [This is undoubtedly a gross oversimplification, which I will correct in later amendments.] You might further imagine that some construction of this sort will eventually retain the data necessary to get the height of $E$, but also have data corresponding to the $G^{\mu}$, necessary for the Lagrangian KS map. In any case, I hope you can appreciate that a good deal of 'dismantling' and 'reconstructing,' what Mochizuki calls surgery, will be necessary.

I can't emphasize enough times that much of what I write is based on faulty memory and guesswork. At best, it is superficial, while at worst, it is (not even) wrong. [In particular, I am no longer sure that the GTKS map is used in an entirely direct fashion.] I have not yet done anything with the current papers than give them a cursory glance. If I figure out more in the coming weeks, I will make corrections. But in the meanwhile, I do hope what I wrote here is mostly more helpful than misleading.

In order to get a global arithmetic analogue, Mochizuki has to formulate a discrete non-linear version of the comparison isomorphism. What is non-linear? This is the replacement of $H^1_{DR}$ by the universal extension $$E^{\dag}\rightarrow E,$$ (the moduli space of line bundles with flat connection on $E$) whose tangent space is $H^1_{DR}$ (considerations of this nature already come up in usual p-adic Hodge theory). What is discrete is the 'etale cohomology, which will just be $E[\ell]$ with global Galois action, where $\ell$ can eventually be large, on the order of the height of $E$ (that is $\deg (\omega_E)$). The comparison isomorphism in this context takes the following form: $$\Xi: A_{DR}=\Gamma(E^{\dag}, L)^{<\ell}\simeq L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}.$$ (I apologize for using the notation $A_{DR}$ for the space that Mochizuki denotes by a calligraphic $H$. I can't seem to write calligraphic characters here.) Here, $L$ is a suitably chosen torsion line bundle on $E$, which can then be pulled back to $E^{\dag}$. The inequality refers to the polynomial degree in the fiber direction of $E^{\dag} \rightarrow E$. The isomorphism is effected via evaluation of sections at $$E^{\dag}[\ell]\simeq E[\ell].$$ Finally, $$ L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}$$ comes from Mumford's theory of theta functions. The interpretation of the statement is that it gives an isomorphism between the space of functions of some bounded fiber degree on non-linear De Rham cohomology and the space of functions on discrete 'etale cohomology. This kind of statement is entirely due to Mochizuki. One sometimes speaks of $p$-adic Hodge theory with finite coefficients, but that refers to a theory that is not only local, but deals with linear De Rham cohomology with finite coefficients.

$F^rA_{DR}= \Gamma(E^{\dag}, L)^{ < r} $

In order to get a global arithmetic analogue, Mochizuki has to formulate a discrete non-linear version of the comparison isomorphism. What is non-linear? This is the replacement of $H^1_{DR}$ by the universal extension $$E^{\dagger}\rightarrow E,$$ (the moduli space of line bundles with flat connection on $E$) whose tangent space is $H^1_{DR}$ (considerations of this nature already come up in usual p-adic Hodge theory). What is discrete is the 'etale cohomology, which will just be $E[\ell]$ with global Galois action, where $\ell$ can eventually be large, on the order of the height of $E$ (that is $\deg (\omega_E)$). The comparison isomorphism in this context takes the following form: $$\Xi: A_{DR}=\Gamma(E^{\dagger}, L)^{<\ell}\simeq L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}.$$ (I apologize for using the notation $A_{DR}$ for the space that Mochizuki denotes by a calligraphic $H$. I can't seem to write calligraphic characters here.) Here, $L$ is a suitably chosen torsion line bundle on $E$, which can then be pulled back to $E^{\dagger}$. The inequality refers to the polynomial degree in the fiber direction of $E^{\dagger} \rightarrow E$. The isomorphism is effected via evaluation of sections at $$E^{\dagger}[\ell]\simeq E[\ell].$$ Finally, $$ L|E[\ell]\simeq (L|e_{E})\otimes O_{E[\ell]}$$ comes from Mumford's theory of theta functions. The interpretation of the statement is that it gives an isomorphism between the space of functions of some bounded fiber degree on non-linear De Rham cohomology and the space of functions on discrete 'etale cohomology. This kind of statement is entirely due to Mochizuki. One sometimes speaks of $p$-adic Hodge theory with finite coefficients, but that refers to a theory that is not only local, but deals with linear De Rham cohomology with finite coefficients.

$F^rA_{DR}= \Gamma(E^{\dagger}, L)^{ < r} $