Bogomolov’s proof of the geometric version of the Szpiro Conjecture from the point of view of interuniversal Teichmüller theory
 Shinichi Mochizuki^{1}Email author
https://doi.org/10.1186/s406870160057x
© Mochizuki. 2016
Received: 23 September 2015
Accepted: 14 January 2016
Published: 5 June 2016
Abstract
The purpose of the present paper is to expose, in substantial detail, certain remarkable similarities between interuniversal Teichmüller theory and the theory surrounding Bogomolov’s proof of the geometric version of the Szpiro Conjecture. These similarities are, in some sense, consequences of the fact that both theories are closely related to the hyperbolic geometry of the classical upper halfplane. We also discuss various differences between the theories, which are closely related to the conspicuous absence in Bogomolov’s proof of Gaussian distributions and theta functions, i.e., which play a central role in interuniversal Teichmüller theory.
Keywords
1 Background
 (IU1)
the geometry of \(\Theta ^{\pm {{{\text {ell}}}}}{} \mathbf{NF}\)Hodge theaters (cf. [6, Definition 6.13]; [6, Remark 6.12.3]),
 (IU2)
the precise relationship between arithmetic degrees—i.e., of q pilot and \(\Theta \) pilot objects—given by the \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\) link (cf. [8, Definition 3.8, (i), (ii)]; [8, Remark 3.10.1, (ii)]), and
 (IU3)
the estimates of logvolumes of certain subsets of logshells that give rise to diophantine inequalities (cf. [9, §1, §2]; [8, Remark 3.10.1, (iii)]) such as the Szpiro Conjecture
After reviewing, in Sects. 2–4, the theory surrounding Bogomolov’s proof from a point of view that is somewhat closer to interuniversal Teichmüller theory than the point of view of [1, 10], we then proceed, in Sects. 5 and 6, to compare, by highlighting various similarities and differences, Bogomolov’s proof with interuniversal Teichmüller theory. In a word, the similarities between the two theories revolve around the relationship of both theories to the classical elementary geometry of the upper halfplane, while the differences between the two theories are closely related to the conspicuous absence in Bogomolov’s proof of Gaussian distributions and theta functions, i.e., which play a central role in interuniversal Teichmüller theory.
2 The geometry surrounding Bogomolov’s proof
First, we begin by reviewing the geometry surrounding Bogomolov’s proof, albeit from a point of view that is somewhat more abstract and conceptual than that of [1, 10].
3 Fundamental groups in Bogomolov’s proof
Next, we discuss the various fundamental groups that appear in Bogomolov’s proof.
4 Estimates of displacements subject to indeterminacies
We conclude our review of Bogomolov’s proof by briefly recalling the key points of the argument applied in this proof. These key points revolve around estimates of displacements that are subject to certain indeterminacies.
 (B1)Every unipotent element \(\tau \in \textit{SL}(E)\) lifts uniquely to an elementthat stabilizes and restricts to the identity on some \(({\widetilde{\tau }}^{\angle })^ {\mathbb Z} \) orbit of \((E^\angle )^\sim \). Such a \({\widetilde{\tau }}\) is minimal and satisfies$$\begin{aligned} {\widetilde{\tau }}\ \in \ {\widetilde{\textit{SL}}}(E)\end{aligned}$$$$\begin{aligned} \delta ^{\sup }\left( {\widetilde{\tau }}\right) < [\pi ]. \end{aligned}$$
 (B2)Every commutator \([{\widetilde{\alpha }},{\widetilde{\beta }}]\in {\widetilde{\textit{SL}}}(E)\) of elements \({\widetilde{\alpha }},{\widetilde{\beta }}\in {\widetilde{\textit{SL}}}(E)\) satisfies$$\begin{aligned} \delta ^{\sup }\left( \left[ {\widetilde{\alpha }},{\widetilde{\beta }}\right] \right) < [2\pi ]. \end{aligned}$$
 (B3)Let \({\widetilde{\tau }}_\infty ,{\widetilde{\tau }}_0\in {\widetilde{\textit{SL}}}(E_ {\mathbb Z} )\) be liftings of \(\tau _\infty ,\tau _0\in \textit{SL}(E_ {\mathbb Z} )\) as in (B1). ThenIn particular,$$\begin{aligned} {\widetilde{\tau }}_\infty \cdot {\widetilde{\tau }}_0={\widetilde{\tau }}_\theta ,\quad \hbox {and} \quad \theta =\tfrac{1}{3}\pi >0. \end{aligned}$$$$\begin{aligned} \left( {\widetilde{\tau }}_\infty \cdot {\widetilde{\tau }}_0\right) ^3={\widetilde{\tau }}^{\measuredangle },\quad \chi \left( {\widetilde{\tau }}_\infty \right) =\chi \left( {\widetilde{\tau }}_0\right) =1, \chi \left( {\widetilde{\tau }}^\measuredangle \right) =2\cdot \chi ({\widetilde{\tau }}^{\measuredangle })=12. \end{aligned}$$
Next, observe that since \(\pi < 2\pi \tfrac{1}{3}\pi \), it follows immediately that \(\{[0],[(0,\pi )]\}\ \cap \ \delta ({\widetilde{\tau }}_\theta \cdot ({\widetilde{\tau }}^\measuredangle )^n)\ =\ \emptyset \) for \(n\not =0\). On the other hand, (B1) implies that \([0]\in \delta ({\widetilde{\tau }}_0)\) and \(\delta ^{\sup }({\widetilde{\tau }}_\infty )<[\pi ]\), and hence that \(\{[0],[(0,\pi )]\} \cap \delta ({\widetilde{\tau }}_\infty \cdot {\widetilde{\tau }}_0) \not = \emptyset \). Thus, the relation \({\widetilde{\tau }}_\infty \cdot {\widetilde{\tau }}_0={\widetilde{\tau }}_\theta \) of observation (B3) follows immediately; the positivity of \(\theta \) follows immediately from the clockwise nature (cf. the definition “\({\widetilde{\tau }}^\measuredangle \)” in the final portion of Sect. 2) of the assignments \({(\begin{array}{c}1\\ 0\end{array})}\mapsto {(\begin{array}{c}0\\ 1\end{array})}\), \({(\begin{array}{c}0\\ 1\end{array})}\mapsto {(\begin{array}{c}1\\ 1\end{array})}\) determined by \(\tau _\infty \cdot \tau _0\).
 (B4)The “orders of qparameters” \(v_1,\ldots ,v_r\) satisfy the equality—where \({n^\measuredangle }\in {\mathbb Z} \) is the quantity defined in the above discussion.$$\begin{aligned} \sum \limits _{i=1}^r\ v_i\ =\ 12{n^\measuredangle }\end{aligned}$$
 (B5)The “orders of qparameters” \(v_1,\ldots ,v_r\) satisfy the estimate—where (g, r) is the type of the hyperbolic Riemann surface S.$$\begin{aligned} \frac{1}{6}\cdot \sum \limits _{i=1}^r\ v_i < 2g+r \end{aligned}$$
5 Similarities between the two theories
We are now in a position to reap the benefits of the formulation of Bogomolov’s proof given above, which is much closer “culturally” to interuniversal Teichmüller theory than the formulation of [1, 10].

the additive \({\mathbb F}_l^{\rtimes \pm }\)symmetry portion of a \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)Hodge theater as corresponding to the unipotent transformations \(\tau _\infty \), \(\tau _0\), \(\gamma _i\)

the multiplicative \({\mathbb F}_l^{\divideontimes }\) symmetry portion of a \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)Hodge theater as corresponding to the toral/“typically nonunipotent” transformations \(\tau _\infty \cdot \tau _0\), \(\alpha _i\), \(\beta _i\)
 (L1)
The object \((\omega ^\measuredangle _{\mathcal M})^\sim \) that appears in Bogomolov’s proof may be thought of as corresponding to the holomorphic logshells of interuniversal Teichmüller theory, i.e., in the sense that it may be thought of as a sort of “logarithm” of the “holomorphic family of copies of the group of units \({\mathbb S}^1\)” constituted by \(\omega ^\measuredangle _{\widetilde{\mathcal M}}\)—cf. the discussion of variation of complex structure in Sect. 2.
 (L2)
Each fiber over \({\widetilde{\mathcal M}}\) of the “holomorphic logshell” \((\omega ^\measuredangle _{\mathcal M})^\sim \) maps isomorphically (cf. Fig. 1) to \((E^\angle )^\sim \), an essentially real analytic object that is independent of the varying complex structures discussed in (L1), hence may be thought of as corresponding to the monoanalytic logshells of interuniversal Teichmüller theory.
 (L3)
Just as in the case of the monoanalytic logshells of interuniversal Teichmüller theory (cf., especially, the proof of [9, Theorem 1.10]), \((E^\angle )^\sim \) serves as a container for estimating the various objects of interest in Bogomolov’s proof, as discussed in (B1), (B2), objects which are subject to the indeterminacies constituted by the action of \({{\text {Aut}}}_{\pi }( {\mathbb R} )\), \({{\text {Aut}}}_{\pi }( {\mathbb R} _{\ge 0})\) [cf. the indeterminacies (Ind\({1}\)), (Ind\({2}\)), (Ind\({3}\)) in interuniversal Teichmüller theory].
 (L4)
In the context of the estimates of (L3), the estimates of unipotent transformations given in (B1) may be thought of as corresponding to the estimates involving theta values in interuniversal Teichmüller theory, while the estimates of “typically nonunipotent” transformations given in (B2) may be thought of as corresponding to the estimates involving global number field portions of \(\Theta \)pilot objects in interuniversal Teichmüller theory.
 (L5)As discussed in the [6, §I1], the Kummer theory surrounding the theta values is closely related to the additive symmetry portion of a \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)Hodge theater, i.e., in which global synchronization of ±indeterminacies (cf. [6, Remark 6.12.4, (iii)]) plays a fundamental role. Moreover, as discussed in [8, Remark 2.3.3, (vi), (vii), (viii)], the essentially local nature of the cyclotomic rigidity isomorphisms that appear in the Kummer theory surrounding the theta values renders them free of any ±indeterminacies. These phenomena of rigidity with respect to ±indeterminacies in interuniversal Teichmüller theory are highly reminiscent of the crucial estimate of (B1) involvingfor the action of \(\{\pm 1\}\) on \(E^\angle \) (i.e., as opposed to the volume \(2\pi \) of the \(\{\pm 1\)orbit \(\pm {\overline{D}}^\angle \) of \({\overline{D}}^\angle \)!), as well as of the uniqueness of the minimal liftings of (B1). In this context, we also recall that the additive symmetry portion of a \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)Hodge theater, which depends, in an essential way, on the global synchronization of ±indeterminacies (cf. [6, Remark 6.12.4, (iii)]), is used in interuniversal Teichmüller theory to establish conjugate synchronization, which plays an indispensable role in the construction of bicoric monoanalytic logshells (cf. [8, Remark 1.5.1]). This state of affairs is highly reminiscent of the important role played by \(E^\angle \), as opposed to \(E^{\angle }=E^\angle {/}\{\pm 1\}\), in Bogomolov’s proof.$$\begin{aligned} \hbox {the}~\mathbf{volume}~\pi ~\hbox {of a}~\mathbf{fundamental~domain}~ {\overline{D}}^\angle \end{aligned}$$
 (L6)As discussed in the [6, §I1], the Kummer theory surrounding the number fields under consideration is closely related to the multiplicative symmetry portion of a \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)Hodge theater, i.e., in which one always works with quotients via the action of ±1. Moreover, as discussed in [8, Remark 2.3.3, (vi), (vii), (viii)] (cf. also [7, Remark 4.7.3, (i)]), the essentially global nature—which necessarily involves at least two localizations, corresponding to a valuation [say, “0”] and the corresponding inverse valuation [i.e., “\(\infty \)”] of a function field—of the cyclotomic rigidity isomorphisms that appear in the Kummer theory surrounding number fields causes them to be subject to ±indeterminacies. These ±indeterminacy phenomena in interuniversal Teichmüller theory are highly reminiscent of the crucial estimate of (B2)—which arises from considering products of two noncommuting unipotent transformations, i.e., corresponding to “two distinct localizations”—involvingfor the action of \(\{\pm 1\}\) on \(E^\angle \) (i.e., as opposed to the volume \(\pi \) of \({\overline{D}}^\angle \)!).$$\begin{aligned} \hbox {the}~\mathbf{volume}~2\pi ~\hbox {of the}~\{\pm 1\}\hbox {}{} \mathbf{orbit} \pm {\overline{D}}^\angle ~\hbox {of a}~\mathbf{fundamental~domain}~{\overline{D}}^\angle \end{aligned}$$
 (L7)
The analytic continuation aspect (say, from “\(\infty \)” to “0”) of interuniversal Teichmüller theory–i.e., via the technique of Belyi cuspidalization as discussed in [6, Remarks 4.3.2, 5.1.4]—may be thought of as corresponding to the “analytic continuation” inherent in the holomorphic structure of the “holomorphic logshell \((\omega ^\measuredangle _{\mathcal M})^\sim \),” which relates, in particular, the localizations at the cusps “\(\infty \)” and “0.”

A nontrivial unipotent element \(\xi \in \textit{PGL}_2(k)\) may be regarded as expressing a local geometry, i.e., the geometry in the neighborhood of a single point [namely the unique fixed point of \(\xi \)]. Such a “local geometry”—that is to say, more precisely, the set \(P^\xi \) of cardinality one—does not admit a reflection, or ±, symmetry.

By contrast, a nontrivial nonunipotent element \(\xi \in PGL_2(k)\) may be regarded as expressing a global geometry, i.e., the “toral” geometry corresponding to a pair of points “0” and “\(\infty \)” [namely the two fixed points of \(\xi \)]. Such a “global toral geometry”—that is to say, more precisely, the set \(P^\xi \) of cardinality two—typically does admit a “reflection, or ±, symmetry” (i.e., which permutes the two points of \(P^\xi \)).

complex holomorphic objects such as the holomorphic line bundle \(\omega _{\mathcal M}\) and the natural surjections \(\omega ^\times _{\mathcal M}\ \twoheadrightarrow \ \omega ^{\times \otimes 12}_{\mathcal M}\ \twoheadrightarrow \ {\mathbb C} ^\times \) arising from the discriminant modular form

the local system \( {\mathcal E} _{\mathcal M}\) and the various fundamental groups [and morphisms between such fundamental groups such as \(\chi \)] that appear in Fig. 3
The analogies discussed above are summarized in Table 1.
Similarities between the two theories
Interuniversal Teichmüller Theory  Bogomolov’s proof 

\({\mathbb F}_l^{\rtimes \pm }\) , \({\mathbb F}_l^{\divideontimes }\) symmetries of \(\Theta ^{\pm {{{\text {ell}}}}}{} \mathbf{NF}\)Hodge theaters  Unipotent, toral/nonunipotent symmetries of upper halfplane 
Simulation of global multiplicative subspace  \({\widetilde{\textit{SL}}}(E)\curvearrowright \) \(\textit{SL}(E)\curvearrowright \) \(( {\mathbb R} \ {\overset{\sim }{\rightarrow }{}}\ ) (E^\angle )^\sim \twoheadrightarrow E^\angle \ (\ {\overset{\sim }{\rightarrow }{}}\ {\mathbb S}^1)\) 
Holomorphic logshells, analytic continuation “\(\infty \rightsquigarrow 0\)”  “Holomorphic family” of fibers of \((\omega ^\measuredangle _{\mathcal M})^\sim \rightarrow {\widetilde{\mathcal M}}\), e.g., at “\(\infty \),” “0” 
Multiradial monoanalytic containers via logshells subject to indeterminacies (Ind\({1}\)), (Ind\({2}\)), (Ind\({3}\))  Real analytic \({\widetilde{\textit{SL}}}(E)\ \curvearrowright \ (E^\angle )^\sim \ (\ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} )\) subject to indeterminacies via actions of \({{\text {Aut}}}_{\pi }( {\mathbb R} )\), \({{\text {Aut}}}_{\pi }( {\mathbb R} _{\ge 0})\) 
±Rigidity of “local” Kummer theory, cyclotomic rigidity surrounding theta values, conjugate synchronization  Estimate (B1) via \(\pi \) of unique minimal liftings of unipotent transformations, \(E^\angle \) (as opposed to \(E^{\angle }\)!) 
±Indeterminacy of “global” Kummer theory, cyclotomic rigidity surrounding number fields  Estimate (B2) via \(2\pi \) of commutators of products of two noncommuting unipotent transformations 
Arithmetic degree computations via precise \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\) link versus logshell estimates  Degree computations via precise homomorphism \(\chi \) (B4) versus \(\delta ^{\sup }\) estimates (B1), (B2) 
Frobeniuslike versus étalelike objects  Complex holomorphic objects such as line bundles versus local systems, fundamental groups 
6 Differences between the two theories

Whereas the field \( {\mathbb R} \) over which the symplectic form \(\langle \ \text {}\ ,\ \text {}\ \rangle _E\) is defined may be regarded as a subfield—i.e.,—of the field of definition \( {\mathbb C} \) of the algebraic schemes (or stacks) under consideration, the field \( {{\mathbb F}_l} \) over which the Weil pairing on ltorsion points is defined cannot be regarded as a subfield—i.e.,$$\begin{aligned} \exists \ {\mathbb R} \ \hookrightarrow \ {\mathbb C} \end{aligned}$$—of the number field over which the (algebraic) elliptic curve under consideration is defined.$$\begin{aligned} \not \exists \ {{\mathbb F}_l} \ \hookrightarrow \ {\mathbb Q} \end{aligned}$$

Certain geometric aspects—i.e., aspects that, in effect, correspond to the geometry of the classical upper halfplane (cf. [6, Remark 6.12.3])—of the a priori incompatibility of fields of definition in the case of elliptic curves over number fields are, in some sense, overcome in interuniversal Teichmüller theory by applying various absolute anabelian algorithms to pass from étalelike to Frobeniuslike objects, as well as various cyclotomic rigidity algorithms to pass, via Kummer theory, from Frobeniuslike to étalelike objects.

Certain functiontheoretic aspects of the a priori incompatibility of fields of definition in the case of elliptic curves over number fields are, in some sense, overcome in interuniversal Teichmüller theory by working with Gaussian distributions and theta functions, i.e., which may be regarded, in effect, as exponentiations of the symplectic form \(\langle \ \text {}\ ,\ \text {}\ \rangle _E\) that appears in Bogomolov’s proof.
 (E1)
one central feature of Bogomolov’s proof is the following fundamental difference between the crucial estimate (B1), which arises from the (nonholomorphic) symplectic geometry portion of Bogomolov’s proof, on the one hand, and the homomorphism \(\chi \), on the other: whereas, for integers \(N\ge 1\), the homomorphism \(\chi \) maps Nth powers of elements \({\widetilde{\tau }}\) as in (B1) to multiples by N of elements \(\in {\mathbb Z} \), the estimate \(\delta ^{\sup }()\ <\ [\pi ]\) of (B1) is unaffected when one replaces an element \({\widetilde{\tau }}\) by such an Nth power of \({\widetilde{\tau }}\).
 (E2)
although the multiradial representation of \(\Theta \) pilot objects via monoanalytic logshells in the domain of the \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\)link is related, via the \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\)link, to qpilot objects in the codomain of the \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\)link, the same multiradial representation of the same \(\Theta \)pilot objects may related, in precisely the same fashion, to arbitrary \({\varvec{N}}\) th powers of qpilot objects, for integers \(N\ge 2\)

substitutes Gaussian distributions/theta functions, i.e., in essence, exponentiations of the natural symplectic form \(\langle \ \text {}\ ,\ \text {}\ \rangle _E\), for \(\langle \ \text {}\ ,\ \text {}\ \rangle _E\), and, moreover,

allows for arbitrary iterates of the \({\mathfrak {log}}\) link, which, in effect, allow one to “disguise” the effects of such exponentiation operations,
 (A1)
between the geometry surrounding \(E^\angle \) in Bogomolov’s proof and the combinatorics involving \({\varvec{l}}\) torsion points that underlie the structure of \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)Hodge theaters in interuniversal Teichmüller theory, on the one hand,
 (A2)
between the geometry surrounding \(E^\angle \) in Bogomolov’s proof and the holomorphic/monoanalytic logshells—i.e., in essence, the local unit groups associated to various completions of a number field—that occur in interuniversal Teichmüller theory, on the other
 (GE)
\({\varvec{l}}\) torsion points [cf. (A1)] are, as discussed above, closely related to exponents of functions, such as theta functions or algebraic rational functions (cf. the discussion of [8, Remark 2.3.3, (vi), (vii), (viii)]; [8, Figs. 2.5, 2.6, 2.7]); such functions give rise, via the operation of Galois evaluation (cf. [8, Remark 2.3.3, (i), (ii), (iii)]), to theta values and elements of number fields, which one regards as acting on logshells [cf. (A2)] that are constructed in a situation in which one considers arbitrary iterates of the \({\mathfrak {log}}\) link (cf. [8, Fig. I.6]).
 (P1)
in interuniversal Teichmüller theory, the prime l is regarded as being sufficiently large that the finite field \( {{\mathbb F}_l} \) serves as a “good approximation” for \( {\mathbb Z} \) (cf. [6, Remark 6.12.3, (i)]);
 (P2)
at each nonarchimedean prime at which the elliptic curve over a number field under consideration has stable bad reduction, the copy of “\( {\mathbb Z} \)” that is approximated by \( {{\mathbb F}_l} \) may be naturally identified with the value group associated to the nonarchimedean prime (cf. [7, Remark 4.7.3, (i)]);
 (P3)
at each archimedean prime of the number field over which the elliptic curve under consideration is defined, a monoanalytic logshell essentially corresponds to a closed ball of radius \(\pi \), centered at the origin in a Euclidean space of dimension two and subject to ±indeterminacies (cf. [8, Proposition 1.2, (vii)]; [8, Remark 1.2.2, (ii)]).
Contrasts between corresponding aspects of the two theories
Interuniversal Teichmüller Theory  Bogomolov’s proof 

Gaussians/theta functions play a central, motivating role  Gaussians/theta functions entirely absent 
Weil pairing on l torsion points defined over \( {{\mathbb F}_l} \), \(\not \exists \ {{\mathbb F}_l} \ \hookrightarrow \ {\mathbb Q} \)  Natural symplectic form \(\langle \ \text {}\ ,\ \text {}\ \rangle _E\) defined over \( {\mathbb R} \), \(\exists \ {\mathbb R} \ \hookrightarrow \ {\mathbb C} \) 
Subtle passage between étalelike, Frobeniuslike objects via absolute anabelian algorithms, Kummer theory/ cyclotomic rigidity algorithms  Confusion between étalelike, Frobeniuslike objects via \( {\mathbb R} \ \hookrightarrow \ {\mathbb C} \) 
Geometry of \(\Theta ^{\pm {{{\text {ell}}}}}{} \mathbf{NF}\) Hodge theaters  Symplectic geometry of classical upper halfplane 
Gaussians/theta functions, i.e., exponentiations of \(\langle \ \text {}\ ,\ \text {}\ \rangle _E\)  Natural symplectic form \(\langle \ \text {}\ ,\ \text {}\ \rangle _E\) 
Arbitrary iterates of \({\mathfrak {log}}\) link  Single application of logarithm, i.e., \((E^\angle )^\sim \ \twoheadrightarrow \ E^\angle \) 
\(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\) link relates multiradial representation via monoanalytic logshells to conventional theory of arithmetic line bundles on number fields  Discriminant modular form “\(\chi \)” relates symplectic geometry “\(\textit{SL}(E)\ \curvearrowright \ \omega ^\measuredangle _{\widetilde{\mathcal M}}\)” to conventional algebraic theory of line bundles/divisors on \({\mathcal M}\) 
Multiradial representation, interuniversality  Nonholomorphic, real analytic nature of symplectic geometry 
 (GE1)
The multiradiality apparatus of interuniversal Teichmüller theory depends, in an essential way, on the supplementary geometric dimension constituted by the “geometric containers” (cf. [8, Remark 2.3.3, (i), (ii)]) furnished by theta functions and algebraic rational functions, which give rise, via Galois evaluation, to the theta values and elements of number fields that act directly on processions of monoanalytic logshells. That is to say, this multiradiality apparatus would collapse if one attempted to work with these theta values and elements of number fields directly. This state of affairs is substantially reminiscent of the fact that, in Bogomolov’s proof, it does not suffice to work directly with actions of (unipotent or toral/nonunipotent) elements of \(\textit{SL}(E)\ (\cong SL_2( {\mathbb R} ))\) on \(E^\angle \); that is to say, it is of essential importance that one work with liftings to \({\widetilde{\textit{SL}}}(E)\) of these elements of \(\textit{SL}(E)\), i.e., to make use of the supplementary geometric dimension constituted by the bundle \(\omega ^\times _{\mathcal M}\rightarrow {\mathcal M}\).
 (GE2)
The fact that the theory of Galois evaluation surrounding theta values plays a somewhat more central, prominent role in interuniversal Teichmüller theory (cf. [7, §1, §2, §3]; [8, §2]) than the theory of Galois evaluation surrounding number fields is reminiscent of the fact that the original exposition of Bogomolov’s proof in [1] essentially treats only the case of genus zero, i.e., in effect, only the central estimate of (B1), thus allowing one to ignore the estimates concerning commutators of (B2). It is only in the later exposition of [10] that one can find a detailed treatment of the estimates of (B2).
The content of the above discussion is summarized in Table 2. Also, certain aspects of our discussion—which, roughly speaking, concern the respective “estimation apparatuses” that occur in the two theories—are illustrated in Figs. 5 and 6. Here, we note that the mathematical content of Fig. 6 is essentially identical to the mathematical content of [8, Fig. I.6] (cf. also [6, Fig. I1.3]).
Declarations
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