Let me start out by urging you to take seriously that whatever I write about the papers surrounding IUTT really are questions. If you would like to use it as a guide to the mathematics in any way, they should be regarded as 'introductions to introductions'. I've wondered about the wisdom of writing in such a loose and ambiguous style based upon a few days worth of cursory skimming. However, I believe the circumstance justifies the effort somewhat. In any case, caveat lector. This might be especially true when it comes to the étale theta function, which touches on many technical aspects of arithmetic geometry that require careful checking to feel confident about. The main reason I am posting this question hastily is to keep myself on the job, since worry and inertia tend to build up as I let things sit.

Much of anabelian geometry deals with recovering arithmetic geometry from group theory. The situation is somewhat reminiscent of Klein's programme, with the important difference that the geometry there was of a far more homogeneous nature. However, that the analogy is not so far fetched is captured by the notion of *rigidity* that was employed in Grothendieck's letter to Faltings, and which has been ubiquitous also in Mochizuki's papers.
Grothendieck's usage took its inspiration from rigidity theorems of Mostow-Margulis type, whereby an isomorphism between fundamental groups of hyperbolic manifolds of dimension at least 3 is necessarily induced by an isometry. The original anabelian conjectures, proved by Nakamura, Tamagawa, and Mochizuki in the 1990's, stated that hyperbolic curves over number fields are similarly rigid, so that isomorphisms between their arithmetic fundamental groups are necessarily induced by maps of schemes. In slightly fanciful terms, the arithmetic structure endows the curve with a rigidity similar to a hyperbolic metric in higher dimensions. This makes a modicum of sense since (some ring of integers in) the base field is adding dimensions, creating a bundle of which the geometric curve is a fibre.

Even when the geometry can't be entirely recovered from group theory, one might still ask about certain geometric invariants or characteristics. For example, an isomorphism of groups $$\pi_1(X_1)\simeq \pi_1(X_2)$$ induces isomorphisms on group cohomology $$H^n(\pi_1(X_1), \hat{\mathbb{Z}})\simeq H^n(\pi_1(X_2), \hat{\mathbb{Z}}).$$ On $K(\pi,1)$-spaces, one might consider the subgroups $$Ch(X_i)\subset \oplus_{n}H^n(\pi_1(X_i), \hat{\mathbb{Z}})$$ arising as Chern classes of vector bundles and ask if these classes are preserved under the isomorphism. (Some discussion of Tate twists is necessary for this to make sense.) This specific question was an important one in the resolution of the anabelian conjectures.

Here is another example of a more category-theoretic nature.
Suppose $G_1$ and $G_2$ are absolute Galois groups of local fields $F_1$ and $F_2$ that are finite extensions of $\mathbb{Q}_p$. An interesting phenomenon is that $G_1$ and $G_2$ can be isomorphic even when the fields are not. Nevertheless, other refined questions are possible. For example, denote by $Rep_i$ the category of continuous finite-dimensional $\mathbb{Q}_p$-representations of $G_i$.
An isomorphism between the $G_i$ will of course induce an equivalence of categories
$$F: Rep_1\simeq Rep_2.$$
We might then consider various subcategories of interest, for example, the sub-categories
$$HT_i\subset Rep_i$$ of *Hodge-Tate* representations. The definition of this subcategory depends explicitly on the field, so that it's not at all obvious that the functor will preserve it. In fact, one of Mochizuki's easier theorems says that $F$ will carry $HT_1$ to $HT_2$ if and only if the fields are isomorphic.

Mochizuki's recent papers have generally been concerned with refining the dictionary

geometry $\leftrightarrow$ group theory

into something like

geometry $\leftrightarrow$ category theory

Sometimes, this is straightforward, such as replacing the fact that a number field $F$ can be recovered from its absolute Galois group $\mbox{Gal}(\bar{F}/F)$ by the statement that $F$ can be recovered from the Galois category (opposite to that) of finite separable extensions of $F$. However, the category can be made more elaborate, or 'stretched out', so that more refined operations can be performed on it. Roughly speaking, this idea informs the theory of Frobenioids.

The theory of the étale theta function, as expounded in the paper *The etale theta function on its Frobenioid-theoretic manifestations*, carries both group-theoretical and category-theoretical aspects. We will mostly ask about the former, putting off even a superifical description of the category theory until it is needed for IUTT.

The group theory is that of the tempered etale fundamental group $\Pi^{tp}_X$ of a once punctured curve $X$ of geometric genus 1 over a finite extension $K$ of $\mathbb{Q}_p$ with split stable reduction. (That is, an elliptic curve minus the origin.) The tempered fundamental group is quite similar to the profinite one, except in the use of either rigid analytic or formal geometry to allow infinite covers. In particular, it admits a surjection $$\Pi^{tp}_X\rightarrow \mathbb{Z}$$ to the discrete group of integers corresponding to a formal cover $$Y\rightarrow X,$$ which is essentially the Tate uniformization. I should warn you that it's rather important to move efficiently between various formal schemes and their (algebraized) generic fibers, but I will mostly ignore the distinction to avoid clutter in this exposition. Also important is that Mochizuki is using log curves rather than punctured ones. But these as well I will pretend are the same, since most people will not be familiar with log geometry.

The analytic theta function is a function on $Y$, to which one can associate a cohomology class
$$\ddot{\eta}^{\Theta}\in H^1(\Pi_{\ddot{Y}}^{tp}, \hat{\mathbb{Z}}(1)),$$
on a double cover $\ddot{Y}$ of (some base-change of) $Y$, the *étale theta function*.

It's not unreasonable to say that the main goal of the paper is to

(1) characterize this class group-theoretically; and

(2) give it a Frobenioid theoretic interpretation.

A precise characterization doesn't quite work, and much of the paper is devoted to formalizing the indeterminacy via the notion of the *mono-theta environment*, a sophisticated invariant of the theta function that does have a group-theoretic construction.

As a small exercise, you might try to characterize the cyclotomic character that appears in the coefficient group-theoretically. One way is to use the exact sequence $$1\rightarrow \Delta^{tp}_X\rightarrow \Pi^{tp}_X\rightarrow G_K\rightarrow 1,$$ which can be constructed group-theoretically from $\Pi^{tp}_X$ by a theorem from Mochizuki's paper 'Absolute anabelian geometry of hyperbolic curves.' (We will discuss such matters more carefully later.) One then has the theta quotient $$(\Delta^{tp})^{\Theta}_X:=\Delta^{tp}_X/\overline{ [ \Delta^{tp}_X , [\Delta^{tp}_X ,\Delta^{tp}_X ] ] },$$ which fits into an exact sequence $$1\rightarrow \Delta_{\Theta}\rightarrow (\Delta^{tp}_X)^{\Theta}\rightarrow (\Delta^{tp}_X)^{ell}\rightarrow 1,$$ where the superscript `$ell$' denotes the abelianization. The group $(\Delta^{tp}_X)^{ell}$ is essentially the fundamental group of the elliptic curve compactification of $X$, except for the 'tempered' nature. The group $ \Delta_{\Theta}$, on the other hand, equipped with the conjugation action of $\Pi^{tp}_X$, is canonically isomorphic to $\hat{\mathbb{Z}}(1)$ (the fundamental group of the punctured tangent space to the elliptic curve at the missing point). Note that $(\Delta^{tp}_X)^{\Theta}$ is a group of Heisenberg type.

We outline briefly the construction of the class $\ddot{\eta}^{\Theta}$. There are compatible towers of covers

$$\begin{array}{ccc} Z_N & \rightarrow & Z_M \\ \downarrow & & \downarrow \\ Y_N & \rightarrow & Y_M\end{array}$$

for $M|N$, where $Y_1=Z_1=Y$. All these covers come from the theta quotient $(\Delta^{tp})^{\Theta}_Y$ of $\Delta^{tp}_Y$ which fits into an exact sequence $$1\rightarrow \Delta_{\Theta}\rightarrow (\Delta^{tp}_Y)^{\Theta}\rightarrow (\Delta^{tp}_Y)^{ell}\rightarrow 1$$ with $(\Delta^{tp}_Y)^{\Theta}$ now profinite abelian of rank 2.

It's important not to be too intimidated by the notation for all the covers in the paper. The towers just mentioned are the main ones. The other covers are $$\ddot{Z}_N\rightarrow \ddot{Y}_N\rightarrow Y_N,$$ where $$\ddot{Y}_N=Y_{2N}\otimes \ddot{J}_N,$$ and the field extension $\ddot{J}_N$ is obtained by adjoining suitable $2N$-th roots. Base changes of this sort will also be ignored for the purpose of this question. The cover $\ddot{Z}_N\rightarrow \ddot{Y}_N$ is obtained as a compositum of $\ddot{Y}_N\rightarrow Y_N$ and $Z_N\rightarrow Y_N$. The importance of these variants to $Y_N$ and $Z_N$ has to do with eliminating sign ambiguities of the sort that come up quite frequently even in the classical geometric theory of theta functions (as in Mumford's papers on algebraic theta functions).

In the paper, the entire discussion and all objects are generalized after a pull-back to $$\underline{\underline{X}}\rightarrow \underline{X} \rightarrow X,$$ a sequence of geometrically $l$-cyclic covers that appear to be important in the eventual applications, especially the question of labeling the components of the special fiber on the regular minimal model of a semi-stable elliptic curves. All the objects under consideration acquire a double underline, like $\underline{\underline{\ddot{Y}}}$. We will stick to the case of $l=1$ for now. (I'm uncertain, however, if some of the results require $l$ to be strictly larger. There are also portions where we need to assume that $X$ is not arithmetic.)

There is a sequence of line bundles $L_N$ on $Y_N$ corresponding to the divisor of special fibers (on the formal schemes) together with natural isomorphisms $$L_M^{\otimes M/N}\simeq L_N|Y_M$$ for $N|M$. Over the tower $\ddot{Z}_N$, there are two compatible sequences $s_N$ and $\tau_N$ of sections, one associated to the special fiber and the other to the cusps. These two trivializations give rise to actions of $\Pi^{tp}_{\ddot{Y}}$ on the line bundles, and the difference between them determines a class $$\ddot{\eta}^{\theta}\in H^1(\Pi^{tp}_{\ddot{Y}}, \Delta_{\Theta}),$$ which is the étale theta function. The 'functional' nature can be thought of in terms of evaluation on points $y\in \ddot{Y}(L)$ over field extensions $L$. We get a class $$\ddot{\eta}^{\theta}|_y\in H^1(G_L, \hat{\mathbb{Z}}(1))\simeq \widehat{L^*},$$ and a comparison with the usual analytic theory can be used to show that the value actually lies in $L^*\subset \widehat{L^*}$. Various choices were made in the construction of the class, and there are two natural group actions on the cohomology accounting for the indeterminacy: One is the conjugation action of $\Pi^{tp}_X/\Pi^{tp}_Y\simeq \underline{\mathbb{Z}}$ (here, I've followed Mochizuki and underlined the $\mathbb{Z}$, to indicate that it's not canonically isomorphic to $\mathbb{Z}$, a point that is eventually important), and the other coming from the translation by the constant integral classes $$\mathcal{O}_{\ddot{K}}^*\hookrightarrow H^1(G_{\ddot{K}}, \hat{\mathbb{Z}}(1))\hookrightarrow H^1(\Pi^{tp}_{\ddot{Y}}, \Delta_{\Theta}).$$ (Here, I believe $\ddot{K}$ is the field obtained by adjoining a square root of $q_X$, the Tate period of $X$, but I'm not entirely sure.) If we denote the orbit of $\ddot{\eta}^{\Theta}$ under these two actions by $$\mathcal{O}_{\ddot{K}}^*[\ddot{\eta}^{\theta}]^{\underline{\mathbb{Z}}}\subset H^1(\Pi^{tp}_{\ddot{Y}}, \Delta_{\Theta}),$$ this set of classes turns out to be a group-theoretic invariant of $\Pi^{tp}_X$. That is to say, if $X_{\alpha}$ and $X_{\beta}$ are two curves defined over fields $K_{\alpha}$ and $K_{\beta}$ with corresponding covers $\ddot{Y}_{\alpha}$ and $\ddot{Y}_{\beta}$, then any isomorphism of topological groups $$\Pi^{tp}_{X_{\alpha}}\simeq\Pi^{tp}_{X_{\beta}}$$ will induce isomorphisms $$\Pi^{tp}_{\ddot{Y}_{\alpha}}\simeq\Pi^{tp}_{\ddot{Y}_{\beta}}$$ and $$(\Delta_{\Theta})_{\alpha}\simeq (\Delta_{\Theta})_{\beta}$$ in such a way that the isomorphism $$H^1(\Pi^{tp}_{\ddot{Y}_{\alpha}}, (\Delta_{\Theta})_{\alpha})\simeq H^1(\Pi^{tp}_{\ddot{Y}_{\beta}}, (\Delta_{\Theta})_{\beta})$$ takes $\mathcal{O}_{\ddot{K}}^*[\ddot{\eta}^{\theta}]^{\underline{\mathbb{Z}}}$ on one side to the other. By using the fundamental group of the orbifold quotient $C=X/\pm 1$, it seems to be possible to designate a class of $\theta$ classes of 'standard type', which eliminate the $\mathcal{O}_{\ddot{K}}^*$ ambiguity up to a sign.

It is with this class that one constructs the theta environment for each $\Pi_{X}$.
The basic 'ambient group' is the *cyclotomic envelope*
$$\Pi^{tp}_{Y}[\mu_N]:= \mu_N\rtimes \Pi^{tp}_{Y}$$
of $\Pi^{tp}_{Y}$, where $\Pi^{tp}_Y$ acts on $\mu_N$ through the projection
$\Pi^{tp}_Y\rightarrow G_K$.
There is an action of $H^1(\Pi^{tp}_Y, \mu_N)$ on $\Pi^{tp}_Y[\mu_N]$ whereby a cocycle $c$ sends $(\zeta, g)$ to $(\zeta c(g), g)$. This is independent of the cocycle up to inner automorphism, giving us a homomorphism
$$H^1(\Pi^{tp}_Y, \mu_N)\rightarrow \mbox{Out} (\Pi^{tp}_Y\mu_N]).$$
Hence, $K^*$ also acts on $\Pi^{tp}_Y[\mu_N]$ via the map
$$K^*\rightarrow K^*/(K^*)^N\simeq H^1(G_K, \mu_N)\rightarrow H^1(\Pi^{tp}_Y, \mu_N).$$

There is also a canonical action of the group $\mbox{Gal}(Y/X)$. Denote by
$$\mathcal{D}_{Y}\subset \mbox{Out} (\Pi^{tp}_{Y}[\mu_N])$$ the subgroup generated by the image of $K^*$ and $\mbox{Gal}(Y/X)$.
The theta class $\ddot{\eta}^{\Theta}$ gives rise to a homomorphism
$$s^{\Theta}:\Pi^{tp}_{\ddot{Y}}\rightarrow \Pi^{tp}_{Y}[\mu_N]$$
whose image is independent of the various choices up the $\mu_N$ conjugacy. (This is one of the many points I'm not so clear about.) Denote by $s^{\Theta}_{\Pi}$ the conjugacy class of subgroups of $\Pi^{tp}_{Y}[\mu_N]$ obtained as the image.
The triple, $$(\Pi^{tp}_{Y}[\mu_N], \mathcal{D}_{Y}, s^{\Theta}_{\Pi})$$
is then the $\mod N$ *mono-theta environment* of $\Pi^{tp}_{X}$.
Just to be clear, we note that it consists of:

(1) A group $\Pi$;

(2) A subgroup $\mathcal{D}\subset \mbox{Out}(\Pi)$; and

(3) A set of subgroups $s^{\Pi}$ of $\Pi$.

For the triple given above, there is a compatibility over $N$, and the main group-theoretical theorem is that this system is determined group theoretically by $\Pi^{tp}_{X}$ (or perhaps $\Pi^{tp}_C$).

The Frobenioid-theoretic construction of the mono-theta environment is more or less built into the objects already discussed, although the details are quite elaborate. The 'tempered' Frobenioid is made out of mildly technical functors that assign divisors and meromorphic functions to covers of the formal scheme corresponding to $X$ in the manner of the geometric Frobenioids desrcibed in the previous question. The trivializations discussed above can be viewed as morphisms inside the Frobenioid between $\ddot{Z}_N$ with the 0 divisor and the line bundle $L_N|\ddot{Y}_N$. From this, using the automorphism groups of $L_N|{\ddot{Z}_N}$, there is a complicated but natural way to rebuild the mono-theta environment mod $N$. The main categorical theorem makes a precise statement to the effect that this construction agrees with the group-theoretic one, and depends only on the category theory of the tempered Frobenioid. As far as I know, the category theoretical formulation is to be used in the process of somehow 'globalising' the local theory of theta functions, a goal that came up already in the Hodge-Arakelov papers.

It should be obvious that this question is based on an understanding even more superficial than the previous one. Nevertheless, perhaps it will be useful at least in eliciting answers that provide more background and motivation. Actually, my hope is that even the little bit I explained here will help people read the elaborate explanatory paragraphs in Mochizuki's original papers, which are often considered difficult. One more word of technical advice for now (probably well-known to everyone but me): I find it useful to go through the papers with both a paper copy and an electronic version in hand. This way, you can combine the comfort of the hardcopy with the ease of searching for definitions.

What kind of questions can I ask here?of the help page. $\endgroup$9more comments