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Bogomolov’s proof of the geometric version of the Szpiro Conjecture from the point of view of inter-universal Teichmüller theory

Research in the Mathematical Sciences20163:6

https://doi.org/10.1186/s40687-016-0057-x

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 inter-universal 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 half-plane. 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 inter-universal Teichmüller theory.

Keywords

Bogomolov’s proofSzpiro ConjectureHyperbolic geometrySymplectic geometryUpper half-planeTheta functionGaussian distributionInter-universal Teichmüller theoryMultiradial representationIndeterminacies

1 Background

Certain aspects of the inter-universal Teichmüller theory developed in [69]—namely
  1. (IU1)

    the geometry of \(\Theta ^{\pm {{{\text {ell}}}}}{} \mathbf{NF}\)-Hodge theaters (cf. [6, Definition 6.13]; [6, Remark 6.12.3]),

     
  2. (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

     
  3. (IU3)

    the estimates of log-volumes of certain subsets of log-shells that give rise to diophantine inequalities (cf. [9, §1, §2]; [8, Remark 3.10.1, (iii)]) such as the Szpiro Conjecture

     
—are substantially reminiscent of the theory surrounding Bogomolov’s proof of the geometric version of the Szpiro Conjecture, as discussed in [1, 10]. Put another way, these aspects of inter-universal Teichmüller theory may be thought of as arithmetic analogues of the geometric theory surrounding Bogomolov’s proof. Alternatively, Bogomolov’s proof may be thought of as a sort of useful elementary guide, or blueprint [perhaps even a sort of Rosetta stone!], for understanding substantial portions of inter-universal Teichmüller theory. The author would like to express his gratitude to Ivan Fesenko for bringing to his attention, via numerous discussions in person, e-mails, and skype conversations between December 2014 and January 2015, the possibility of the existence of such fascinating connections between Bogomolov’s proof and inter-universal Teichmüller theory.

After reviewing, in Sects. 24, the theory surrounding Bogomolov’s proof from a point of view that is somewhat closer to inter-universal 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 inter-universal 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 half-plane, 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 inter-universal 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].

We denote by \({\mathcal M}\) the complex analytic moduli stack of elliptic curves (i.e., one-dimensional complex tori). Let
$$\begin{aligned} {\widetilde{\mathcal M}}\ \rightarrow \ {\mathcal M}\end{aligned}$$
be a universal covering of \({\mathcal M}\). Thus, \({\widetilde{\mathcal M}}\) is non-canonically isomorphic to the upper half-plane \({\mathfrak {H}}\). In the following, we shall denote by a subscript \({\widetilde{\mathcal M}}\) the result of restricting to \({\widetilde{\mathcal M}}\) objects over \({\mathcal M}\) that are denoted by a subscript \({\mathcal M}\).
Write
$$\begin{aligned} \omega _{\mathcal M}\ \rightarrow \ {\mathcal M}\end{aligned}$$
for the [geometric!] line bundle determined by the cotangent space at the origin of the tautological family of elliptic curves over \({\mathcal M}\); \(\omega ^\times _{\mathcal M}\subseteq \omega _{\mathcal M}\) for the complement of the zero section in \(\omega _{\mathcal M}\); \( {\mathcal E} _{\mathcal M}\) for the local system over \({\mathcal M}\) determined by the first singular cohomology modules with coefficients in \( {\mathbb R} \) of the fibers over \({\mathcal M}\) of the tautological family of elliptic curves over \({\mathcal M}\); \( {\mathcal E} ^\times _{\mathcal M}\subseteq {\mathcal E} _{\mathcal M}\) for the complement of the zero section in \( {\mathcal E} _{\mathcal M}\). Thus, if we think of bundles as geometric spaces/stacks, then there is a natural embedding
$$\begin{aligned} \omega _{\mathcal M}\ \hookrightarrow \ {\mathcal E} _{\mathcal M}\otimes _ {\mathbb R} {\mathbb C} \end{aligned}$$
(cf. the inclusion “\(\omega \hookrightarrow {\mathcal E} \)” of [6, Remark 4.3.3, (ii)]). Moreover, this natural embedding, together with the natural symplectic form
$$\begin{aligned} \langle \ \text {-}\ ,\ \text {-}\ \rangle _ {\mathcal E} \end{aligned}$$
on \( {\mathcal E} _{\mathcal M}\) [i.e., determined by the cup product on the singular cohomology of fibers over \({\mathcal M}\), together with the orientation that arises from the complex holomorphic structure on these fibers], gives rise to a natural metric (cf. the discussion of [6, Remark 4.3.3, (ii)]) on \(\omega _{\mathcal M}\). Write
$$\begin{aligned} \left( \omega _{\mathcal M}\ \supseteq \ \omega ^\times _{\mathcal M}\ \supseteq \right) \quad \omega ^\measuredangle _{\mathcal M}\ \rightarrow \ {\mathcal M}\end{aligned}$$
for the \({\mathbb S}^1\)-bundle over \({\mathcal M}\) determined by the points of \(\omega _{\mathcal M}\) of modulus one with respect to this natural metric.
Next, observe that the natural section \(\tfrac{1}{2}\cdot {{\text {tr}}}(-): {\mathbb C} \rightarrow {\mathbb R} \) (i.e., one-half the trace map of the Galois extension \( {\mathbb C} {/} {\mathbb R} \)) of the natural inclusion \( {\mathbb R} \hookrightarrow {\mathbb C} \) determines a section \( {\mathcal E} _{\mathcal M}\otimes _ {\mathbb R} {\mathbb C} \rightarrow {\mathcal E} _{\mathcal M}\) of the natural inclusion \( {\mathcal E} _{\mathcal M} \hookrightarrow {\mathcal E} _{\mathcal M}\otimes _ {\mathbb R} {\mathbb C} \) whose restriction to \(\omega _{\mathcal M}\) determines bijections
$$\begin{aligned} \omega _{\mathcal M}\ \ {\overset{\sim }{\rightarrow }{}}\ \ {\mathcal E} _{\mathcal M},\quad \omega ^\times _{\mathcal M}\ \ {\overset{\sim }{\rightarrow }{}}\ \ {\mathcal E} ^\times _{\mathcal M}\end{aligned}$$
(i.e., of geometric bundles over \({\mathcal M}\)). Thus, at the level of fibers, the bijection \(\omega _{\mathcal M}\ {\overset{\sim }{\rightarrow }{}}\ {\mathcal E} _{\mathcal M}\) may be thought of as a (non-canonical) copy of the natural bijection \( {\mathbb C} \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} ^2\).
Next, let us write \(E\) for the fiber (which is non-canonically isomorphic to \( {\mathbb R} ^2\)) of the local system \( {\mathcal E} _{\mathcal M}\) relative to some basepoint corresponding to a cusp
of \({\widetilde{\mathcal M}}\), \(E_ {\mathbb C} \buildrel {{{\text {def}}}} \over = E\otimes _ {\mathbb R} {\mathbb C} \), \(\textit{SL}(E)\) for the group of \( {\mathbb R} \)-linear automorphisms of \(E\) that preserve the natural symplectic form \(\langle \ \text {-}\ ,\ \text {-}\ \rangle _E\buildrel {{{\text {def}}}} \over = \langle \ \text {-}\ ,\ \text {-}\ \rangle _ {\mathcal E} |_E\) on \(E\) [so \(\textit{SL}(E)\) is non-canonically isomorphic to \(\textit{SL}_2( {\mathbb R} )\)]. Now since \({\widetilde{\mathcal M}}\) is contractible, the local systems \( {\mathcal E} _{\widetilde{\mathcal M}}\), \( {\mathcal E} ^\times _{\widetilde{\mathcal M}}\) over \({\widetilde{\mathcal M}}\) are trivial. In particular, we obtain natural projection maps
$$\begin{aligned} {\mathcal E} _{\widetilde{\mathcal M}}\ \twoheadrightarrow \ E,\quad {\mathcal E} ^\times _{\widetilde{\mathcal M}}\ \twoheadrightarrow \ E^\times \ \twoheadrightarrow \ E^\angle \ \twoheadrightarrow \ E^{|\angle |}\end{aligned}$$
—where we write
$$\begin{aligned} E^\times \buildrel {{{\text {def}}}} \over = E{\setminus }\{(0,0\}, \quad E^\angle \buildrel {{{\text {def}}}} \over = E^\times / {\mathbb R} _{>0}\end{aligned}$$
[so \(E^\times \), \(E^\angle \) are non-canonically isomorphic to \( {\mathbb R} ^{2\times }\buildrel {{{\text {def}}}} \over = {\mathbb R} ^2{\setminus }\{(0,0)\}\), \( {\mathbb R} ^{2\angle }\buildrel {{{\text {def}}}} \over = {\mathbb R} ^{2\times }{/} {\mathbb R} _{>0}\cong {\mathbb S}^1\), respectively] and
$$\begin{aligned} E^\angle \twoheadrightarrow E^{|\angle |}\buildrel {{{\text {def}}}} \over = E^\angle {/}\{\pm 1\} \end{aligned}$$
for the finite étale covering of degree 2 determined by forming the quotient by the action of ±1\(\in \textit{SL}(E)\).
Next, let us observe that over each point \({\widetilde{\mathcal M}}\), the composite
$$\begin{aligned} \omega ^\measuredangle _{\widetilde{\mathcal M}}\ \subseteq \ \omega ^\times _{\widetilde{\mathcal M}}\ \ {\overset{\sim }{\rightarrow }{}}\ \ {\mathcal E} ^\times _{\widetilde{\mathcal M}}\ \twoheadrightarrow \ E^\times \ \twoheadrightarrow \ E^\angle \end{aligned}$$
induces a homeomorphism between the fiber of \(\omega ^\measuredangle _{\widetilde{\mathcal M}}\) [over the given point of \({\widetilde{\mathcal M}}\)] and \(E^\angle \). In particular, for each point of \({\widetilde{\mathcal M}}\), the metric on this fiber of \(\omega ^\measuredangle _{\widetilde{\mathcal M}}\) determines a metric on \(E^\angle \) (i.e., which depends on the point of \({\widetilde{\mathcal M}}\) under consideration!). On the other hand, one verifies immediately that such metrics on \(E^\angle \) always satisfy the following property: Let
$$\begin{aligned} {\overline{D}}^\angle \ \subseteq \ E^\angle \end{aligned}$$
be a fundamental domain for the action of ±1 on \(E^\angle \), i.e., the closure of some open subset \(D^\angle \subseteq E^\angle \) such that \(D^\angle \) maps injectively to \(E^{|\angle |}\), while \({\overline{D}}^\angle \) maps surjectively to \(E^{|\angle |}\). Thus, \(\pm {\overline{D}}^\angle \) (i.e., the \(\{\pm 1\}\)-orbit of \({\overline{D}}^\angle \)) is equal to \(E^\angle \). Then the volume of \({\overline{D}}^\angle \) relative to metrics on \(E^\angle \) of the sort just discussed is always equal to \(\pi \), while the volume of \(\pm {\overline{D}}^\angle \) (i.e., \(E^\angle \)) relative to such a metric is always equal to \(2\pi \).
Over each point of \({\widetilde{\mathcal M}}\), the composite \(\omega ^\times _{\widetilde{\mathcal M}}\ {\overset{\sim }{\rightarrow }{}}\ {\mathcal E} ^\times _{\widetilde{\mathcal M}}\twoheadrightarrow E^\times \) corresponds (non-canonically) to a copy of the natural bijection \( {\mathbb C} ^\times \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} ^{2\times }\) that arises from the complex structure on \(E\) determined by the point of \({\widetilde{\mathcal M}}\). Moreover, this assignment of complex structures, or, alternatively, points of the one-dimensional complex projective space \({\mathbb P}(E_ {\mathbb C} )\), to points of \({\widetilde{\mathcal M}}\) determines a natural embedding
$$\begin{aligned} {\widetilde{\mathcal M}}\ \hookrightarrow \ {\mathbb P}(E_ {\mathbb C} ) \end{aligned}$$
—i.e., a copy of the usual embedding of the upper half-plane into the complex projective line—hence also natural actions of \(\textit{SL}(E)\) on \({\widetilde{\mathcal M}}\) and \( {\mathcal E} _{\widetilde{\mathcal M}}\) that are uniquely determined by the property that they be compatible, relative to this natural embedding and the projection \( {\mathcal E} _{\widetilde{\mathcal M}}\twoheadrightarrow E\), with the natural actions of \(\textit{SL}(E)\) on \({\mathbb P}(E_ {\mathbb C} )\) and \(E\). One verifies immediately that these natural actions also determine compatible natural actions of \(\textit{SL}(E)\) on \(\omega ^\measuredangle _{\widetilde{\mathcal M}}\subseteq \omega ^\times _{\widetilde{\mathcal M}}\ {\overset{\sim }{\rightarrow }{}}\ {\mathcal E} ^\times _{\widetilde{\mathcal M}}\) and that the natural action of \(\textit{SL}(E)\) on \(\omega ^\measuredangle _{\widetilde{\mathcal M}}\) determines a structure of \(\textit{SL}(E)\) -torsor on \(\omega ^\measuredangle _{\widetilde{\mathcal M}}\). Also, we observe that the natural embedding of the above display allows one to regard \(E^{|\angle |}\) as the “boundary” \(\partial {\widetilde{\mathcal M}}\) of \({\widetilde{\mathcal M}}\), i.e., the boundary of the upper half-plane.
Let \({\widetilde{\textit{SL}}}(E)\), \((\omega ^\measuredangle _{\mathcal M})^\sim \), \((\omega ^\times _{\mathcal M})^\sim \), \(( {\mathcal E} ^\times _{\mathcal M})^\sim \), \((E^\times )^\sim \), \((E^\angle )^\sim \) be compatible universal coverings of \(\textit{SL}(E)\), \(\omega ^\measuredangle _{\widetilde{\mathcal M}}\), \(\omega ^\times _{\widetilde{\mathcal M}}\), \( {\mathcal E} ^\times _{\widetilde{\mathcal M}}\), \(E^\times \), \(E^\angle \), respectively. Thus, \({\widetilde{\textit{SL}}}(E)\) admits a natural Lie group structure, together with a natural surjection of Lie groups \({\widetilde{\textit{SL}}}(E)\twoheadrightarrow \textit{SL}(E)\), whose kernel admits a natural generator
$$\begin{aligned} {\widetilde{\tau }}^\measuredangle \ \in \ {{\text {Ker}}}\left( {\widetilde{\textit{SL}}}(E)\twoheadrightarrow \textit{SL}(E)\right) \end{aligned}$$
determined by the clockwise orientation that arises from the complex structure on the fibers of \(\omega ^\times _{\mathcal M}\) over \({\mathcal M}\)]. This natural generator determines a natural isomorphism \( {\mathbb Z} \ {\overset{\sim }{\rightarrow }{}}\ {{\text {Ker}}}({\widetilde{\textit{SL}}}(E)\twoheadrightarrow \textit{SL}(E))\).
Next, observe that the natural actions of \(\textit{SL}(E)\) on \(\omega ^\measuredangle _{\widetilde{\mathcal M}}\), \(\omega ^\times _{\widetilde{\mathcal M}}\), \( {\mathcal E} ^\times _{\widetilde{\mathcal M}}\), \(E^\times \), \(E^\angle \) lift uniquely to compatible natural actions of \({\widetilde{\textit{SL}}}(E)\) on the respective universal coverings \((\omega ^\measuredangle _{\mathcal M})^\sim \), \((\omega ^\times _{\mathcal M})^\sim \), \(( {\mathcal E} ^\times _{\mathcal M})^\sim \), \((E^\times )^\sim \), \((E^\angle )^\sim \). In particular, the natural generator \({\widetilde{\tau }}^\measuredangle \) of \( {\mathbb Z} ={{\text {Ker}}}({\widetilde{\textit{SL}}}(E)\twoheadrightarrow \textit{SL}(E))\) determines a natural generator \({\widetilde{\tau }}^\angle \) of the group \({{\text {Aut}}}( (E^\angle )^\sim /E^\angle )\) of covering transformations of \((E^\angle )^\sim \twoheadrightarrow E^\angle \) and hence, taking into account the composite covering \((E^\angle )^\sim \twoheadrightarrow E^\angle \twoheadrightarrow E^{|\angle |}\), a natural \({{\text {Aut}}}_{\pi }( {\mathbb R} )\) -orbit of homeomorphisms [i.e., a “homeomorphism that is well defined up to composition with an element of \({{\text {Aut}}}_{\pi }( {\mathbb R} )\)”]
$$\begin{aligned} \left( E^\angle \right) ^\sim \ \ {\overset{\sim }{\rightarrow }{}}\ \ {\mathbb R} \quad \left( \curvearrowleft {{\text {Aut}}}_{\pi }( {\mathbb R} )\right) \end{aligned}$$
—where we write \({{\text {Aut}}}_{\pi }( {\mathbb R} )\) for the group of self-homeomorphisms \( {\mathbb R} \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} \) that commute with translation by \(\pi \). Here, the group of covering transformations of the covering \((E^\angle )^\sim \twoheadrightarrow E^\angle \) is generated by the transformation \({\widetilde{\tau }}^\angle \), which corresponds to translation by \(2\pi \); the group of covering transformations of the composite covering \((E^\angle )^\sim \twoheadrightarrow E^\angle \twoheadrightarrow E^{|\angle |}\) admits a generator \({\widetilde{\tau }}^{|\angle |}\) that satisfies the relation
and corresponds to translation by \(\pi \) (cf. the transformation “\(z(-)\)” of [10, Lemma 3.5]). Moreover, \({\widetilde{\tau }}^{|\angle |}\) arises from an element \({\widetilde{\tau }}^{|\measuredangle |}\in {\widetilde{\textit{SL}}}(E)\) that lifts −1 \(\in \textit{SL}(E)\) and satisfies the relation \(({\widetilde{\tau }}^{|\measuredangle |})^2={\widetilde{\tau }}^\measuredangle \). The geometry discussed so far is summarized in the commutative diagram of Fig. 1.
Fig. 1

The geometry surrounding Bogomolov’s proof

3 Fundamental groups in Bogomolov’s proof

Next, we discuss the various fundamental groups that appear in Bogomolov’s proof.

Recall that the 12th tensor power \(\omega _{\mathcal M}^{\otimes 12}\) of the line bundle \(\omega _{\mathcal M}\) admits a natural section; namely, the so-called discriminant modular form, which is nonzero over \({\mathcal M}\), hence determines a section of \(\omega ^{\times \otimes 12}_{\mathcal M}\) (i.e., the complement of the zero section of \(\omega _{\mathcal M}^{\otimes 12}\)). Thus, by raising sections of \(\omega ^\times _{\mathcal M}\) to the 12th power and then applying the trivialization determined by the discriminant modular form, we obtain natural holomorphic surjections
$$\begin{aligned} \omega ^\times _{\mathcal M}\ \twoheadrightarrow \ \omega ^{\times \otimes 12}_{\mathcal M}\ \twoheadrightarrow \ {\mathbb C} ^\times \end{aligned}$$
—where we note that the first surjection \(\omega ^\times _{\mathcal M}\twoheadrightarrow \omega ^{\times \otimes 12}_{\mathcal M}\), as well as the pull-back \(\omega ^\times _{\widetilde{\mathcal M}}\twoheadrightarrow \omega ^{\times \otimes 12}_{\widetilde{\mathcal M}}\) of this surjection to \({\widetilde{\mathcal M}}\), is in fact a finite étale covering of complex analytic stacks. Thus, the universal covering \((\omega ^\times _{\mathcal M})^\sim \) over \(\omega ^\times _{\mathcal M}\) may be regarded as a universal covering \((\omega ^{\times \otimes 12}_{\mathcal M})^\sim \buildrel {{{\text {def}}}} \over = (\omega ^\times _{\mathcal M})^\sim \) of \(\omega ^{\times \otimes 12}_{\mathcal M}\). In particular, if we regard \( {\mathbb C} \) as a universal covering of \( {\mathbb C} ^\times \) via the exponential map \({{\text {exp}}}: {\mathbb C} \twoheadrightarrow {\mathbb C} ^\times \), then the surjection \(\omega ^{\times \otimes 12}_{\mathcal M}\twoheadrightarrow {\mathbb C} ^\times \) determined by the discriminant modular form lifts to a surjection
$$\begin{aligned} \left( \omega ^\times _{\mathcal M}\right) ^\sim \ =\ \left( \omega ^{\times \otimes 12}_{\mathcal M}\right) ^\sim \ \twoheadrightarrow \ {\mathbb C} \end{aligned}$$
of universal coverings that is well defined up to composition with a covering transformation of the universal covering \({{\text {exp}}}: {\mathbb C} \twoheadrightarrow {\mathbb C} ^\times \).
Next, let us recall that the \( {\mathbb R} \)-vector space \(E\) is equipped with a natural \( {\mathbb Z} \)-lattice
$$\begin{aligned} E_ {\mathbb Z} \ \subseteq \ E\end{aligned}$$
(i.e., determined by the singular cohomology with coefficients in \( {\mathbb Z} \)). The set of elements of \(\textit{SL}(E)\) that stabilize \(E_ {\mathbb Z} \subseteq E\) determines a subgroup \(\textit{SL}(E_ {\mathbb Z} )\subseteq \textit{SL}(E)\) [so \(\textit{SL}(E_ {\mathbb Z} )\) is non-canonically isomorphic to \(\textit{SL}_2( {\mathbb Z} )\)], hence also a subgroup \({\widetilde{\textit{SL}}}(E_ {\mathbb Z} )\buildrel {{{\text {def}}}} \over = {\widetilde{\textit{SL}}}(E)\times _{\textit{SL}(E)}\textit{SL}(E_ {\mathbb Z} )\). Thus, \(\textit{SL}(E)\supseteq \textit{SL}(E_ {\mathbb Z} )\) admits a natural action on \(\omega ^\times _{\widetilde{\mathcal M}}\); \({\widetilde{\textit{SL}}}(E)\supseteq {\widetilde{\textit{SL}}}(E_ {\mathbb Z} )\) admits a natural action on \((\omega ^\times _{\mathcal M})^\sim \). Moreover, one verifies immediately that the latter natural action determines a natural isomorphism
$$\begin{aligned} {\widetilde{\textit{SL}}}(E_ {\mathbb Z} )\ \ {\overset{\sim }{\rightarrow }{}}\ \ \pi _1\left( \omega ^\times _{\mathcal M}\right) \end{aligned}$$
with the group of covering transformations of \((\omega ^\times _{\mathcal M})^\sim \) over \(\omega ^\times _{\mathcal M}\), i.e., with the fundamental group [relative to the basepoint corresponding to the universal covering \((\omega ^\times _{\mathcal M})^\sim \)] \(\pi _1(\omega ^\times _{\mathcal M})\).
In particular, if we use the generator −2\(\pi i\in {\mathbb C} \) to identify \(\pi _1( {\mathbb C} ^\times )\) with \( {\mathbb Z} \), then one verifies easily (by considering the complex elliptic curves that admit automorphisms of order >2) that we obtain a natural surjective homomorphism
$$\begin{aligned} \chi :\ {\widetilde{\textit{SL}}}(E_ {\mathbb Z} )\ =\ \pi _1\left( \omega ^\times _{\mathcal M}\right) \ \twoheadrightarrow \ \pi _1\left( {\mathbb C} ^\times \right) \ \ {\overset{\sim }{\rightarrow }{}}\ \ {\mathbb Z} \end{aligned}$$
whose restriction to \( {\mathbb Z} \ {\overset{\sim }{\rightarrow }{}}\ {{\text {Ker}}}({\widetilde{\textit{SL}}}(E_ {\mathbb Z} )\twoheadrightarrow \textit{SL}(E_ {\mathbb Z} ))\) is the homomorphism \( {\mathbb Z} \rightarrow {\mathbb Z} \) given by multiplication by 12, i.e.,
$$\begin{aligned} \chi \left( {\widetilde{\tau }}^\measuredangle \right) =12, \quad \chi \left( {\widetilde{\tau }}^{|\measuredangle |}\right) =6 \end{aligned}$$
(cf. the final portion of Sect. 2).
Finally, we recall that in Bogomolov’s proof, one considers a family of elliptic curves (i.e., one-dimensional complex tori)
$$\begin{aligned} X\ \rightarrow \ S\quad \left( \subseteq \ {\overline{S}}\right) \end{aligned}$$
over a hyperbolic Riemann surface S of finite type (gr) (so \(2g-2+r>0\)) that has stable bad reduction at every point at infinity (i.e., point \(\in {\overline{S}}{\setminus } S\)) of some compact Riemann surface \({\overline{S}}\) that compactifies S. Such a family determines a classifying morphism \(S\rightarrow {\mathcal M}\). The above discussion is summarized in the commutative diagrams and exact sequences of Figs. 2 and 3.
Fig. 2

Exact sequences related to Bogomolov’s proof

Fig. 3

Fundamental groups related to 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.

Write
$$\begin{aligned} {{\text {Aut}}}_{\pi }\left( {\mathbb R} _{\ge 0}\right) \end{aligned}$$
for the group of self-homeomorphisms \( {\mathbb R} _{\ge 0}\ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} _{\ge 0}\) that stabilize and restrict to the identity on the subset \(\pi \cdot {\mathbb N} \subseteq {\mathbb R} _{\ge 0}\) and \( {\mathbb R} _{|\pi |}\) for the set of \({{\text {Aut}}}_{\pi }( {\mathbb R} _{\ge 0})\) -orbits of \( {\mathbb R} _{\ge 0}\) [relative to the natural action of \({{\text {Aut}}}_{\pi }( {\mathbb R} _{\ge 0})\) on \( {\mathbb R} _{\ge 0}\)]. Thus, one verifies easily that
$$\begin{aligned} {\mathbb R} _{|\pi |}= \left( \bigcup _{n\in {\mathbb N} }\{[n\cdot \pi ]\}\right) \bigcup \left( \bigcup _{m\in {\mathbb N} }\left\{ [(m\cdot \pi ,(m+1)\cdot \pi )]\right\} \ \right) \end{aligned}$$
—where we use the notation “\([-]\)” to denote the element in \( {\mathbb R} _{|\pi |}\) determined by an element or non-empty subset of \( {\mathbb R} _{\ge 0}\) that lies in a single \({{\text {Aut}}}_{\pi }( {\mathbb R} _{\ge 0})\)-orbit. In particular, we observe that the natural order relation on \( {\mathbb R} _{\ge 0}\) induces a natural order relation on \( {\mathbb R} _{|\pi |}\).
For \({\widetilde{\zeta }}\in {\widetilde{\textit{SL}}}(E)\), write
$$\begin{aligned} \left. \delta \left( {\widetilde{\zeta }}\right) \buildrel {{{\text {def}}}} \over = \left\{ \left[ \left| {\widetilde{\zeta }}(e)-e\right| \right] \right| \ e\in \left( E^\angle \right) ^\sim \right\} \ \subseteq \ {\mathbb R} _{|\pi |}\end{aligned}$$
—where the absolute value of differences of elements of \((E^\angle )^\sim \) is computed with respect to some fixed choice of a homeomorphism \((E^\angle )^\sim \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} \) that belongs to the natural \({{\text {Aut}}}_{\pi }( {\mathbb R} )\)-orbit of homeomorphisms discussed in Sect. 2, and we observe that it follows immediately from the definition of \( {\mathbb R} _{|\pi |}\) that the subset \(\delta ({\widetilde{\zeta }})\subseteq {\mathbb R} _{|\pi |}\) is in fact independent of this fixed choice of homeomorphism.
Since (one verifies easily, from the connectedness of the Lie group \({\widetilde{\textit{SL}}}(E)\), that) \({\widetilde{\tau }}^{|\measuredangle |}\) belongs to the center of the group \({\widetilde{\textit{SL}}}(E)\), it follows immediately [from the definition of \( {\mathbb R} _{|\pi |}\), by considering translates of \(e\in (E^\angle )^\sim \) by iterates of \({\widetilde{\tau }}^{|\measuredangle |}\)] that the set \(\delta ({\widetilde{\zeta }})\) is finite, hence admits a maximal element
$$\begin{aligned} \delta ^{\sup }\left( {\widetilde{\zeta }}\right) \ \buildrel {{{\text {def}}}} \over = \ \sup \left( \delta \left( {\widetilde{\zeta }}\right) \right) \end{aligned}$$
(cf. the length \(\ell (-)\) of the discussion preceding [10, Lemma 3.7]). Thus,
$$\begin{aligned} \delta \left( \left( {\widetilde{\tau }}^{|\measuredangle |}\right) ^n\right) = \left\{ \left[ |n|\cdot \pi \right] \right\} , \quad \delta ^{\sup }\left( \left( {\widetilde{\tau }}^{|\measuredangle |}\right) ^n\right) = \left[ |n|\cdot \pi \right] \end{aligned}$$
for \(n\in {\mathbb Z} \) (cf. the discussion preceding [10, Lemma 3.7]). We shall say that \({\widetilde{\zeta }}\in {\widetilde{\textit{SL}}}(E)\) is minimal if \(\delta ^{\sup }({\widetilde{\zeta }})\) determines a minimal element of the set \(\{\delta ^{\sup }({\widetilde{\zeta }}\cdot ({\widetilde{\tau }}^\measuredangle )^n)\}_{n\in {\mathbb Z} }\).
Next, observe that the cusp\(\infty \)” discussed in Sect. 2 may be thought of as a choice of some rank one submodule \(E_\infty \subseteq E_ {\mathbb Z} \) for which there exists a rank one submodule \(E_0\subseteq E_ {\mathbb Z} \)—which may be thought of as a cusp “0”—such that the resulting natural inclusions determine an isomorphism
$$\begin{aligned} E_\infty \oplus E_0\ \ {\overset{\sim }{\rightarrow }{}}\ \ E_ {\mathbb Z} \end{aligned}$$
of \( {\mathbb Z} \)-modules. Note that since \(E_\infty \) and \(E_0\) are free \( {\mathbb Z} \) -modules of rank one, it follows (from the fact that the automorphism group of the group \( {\mathbb Z} \) is of order two!) that there exist natural isomorphisms \(E_\infty ^{\otimes 2} \ {\overset{\sim }{\rightarrow }{}}\ E_0^{\otimes 2} \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb Z} \). On the other hand, the natural symplectic form \(\langle \ \text {-}\ ,\ \text {-}\ \rangle _{E_ {\mathbb Z} }\buildrel {{{\text {def}}}} \over = \langle \ \text {-}\ ,\ \text {-}\ \rangle _E|_{E_ {\mathbb Z} }\) on \(E_ {\mathbb Z} \) determines an isomorphism of \(E_\infty \) with the dual of \(E_0\), hence (by applying the natural isomorphism \(E_0^{\otimes 2}\ {\overset{\sim }{\rightarrow }{}}\ {\mathbb Z} \)) a natural isomorphism \(E_\infty \ {\overset{\sim }{\rightarrow }{}}\ E_0\).
This natural isomorphism \(E_\infty \ {\overset{\sim }{\rightarrow }{}}\ E_0\) determines a non-trivial unipotent automorphism \(\tau _\infty \in \textit{SL}(E_ {\mathbb Z} )\) of \(E_ {\mathbb Z} =E_\infty \oplus E_0\) that fixes \(E_\infty \subseteq E_ {\mathbb Z} \)—i.e., which may be thought of, relative to natural isomorphism \(E_\infty \ {\overset{\sim }{\rightarrow }{}}\ E_0\), as the matrix \({(\begin{array}{cc}1&{}\quad 1\\ 0&{}\quad 1\end{array})}\)—as well as an \(\textit{SL}(E_ {\mathbb Z} )\)-conjugate unipotent automorphism \(\tau _0\in \textit{SL}(E_ {\mathbb Z} )\)—i.e., which may be thought of, relative to natural isomorphism \(E_\infty \ {\overset{\sim }{\rightarrow }{}}\ E_0\), as the matrix \({(\begin{array}{cc}1&{}\quad 0\\ -1&{}\quad 1\end{array}})\). Thus, the product
$$\begin{aligned} \tau _\infty \cdot \tau _0\ =\left( \begin{array}{cc}1&{}\quad 1\\ 0&{}\quad 1\end{array}\right) \cdot \left( \begin{array}{cc}1&{}\quad 0\\ -1&{}\quad 1\end{array}\right) =\left( \begin{array}{cc}0&{}\quad 1\\ -1&{}\quad 1\end{array}\right) \in \ \textit{SL}(E_ {\mathbb Z} )\end{aligned}$$
lifts, relative to a suitable homeomorphism \((E^\angle )^\sim \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} \) that belongs to the natural \({{\text {Aut}}}_{\pi }( {\mathbb R} )\) -orbit of homeomorphisms discussed in Sect. 2, to an element \({\widetilde{\tau }}_\theta \in {\widetilde{\textit{SL}}}(E_ {\mathbb Z} )\) that induces the automorphism of \( {\mathbb R} \) given by translation by \(\theta \) for some \(\theta \in {\mathbb R} \) such that \(|\theta |=\tfrac{1}{3}\pi \).
The key observations that underlie Bogomolov’s proof may be summarized as follows (cf. [10, Lemmas 3.6, 3.7]):
  1. (B1)
    Every unipotent element \(\tau \in \textit{SL}(E)\) lifts uniquely to an element
    $$\begin{aligned} {\widetilde{\tau }}\ \in \ {\widetilde{\textit{SL}}}(E)\end{aligned}$$
    that 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
    $$\begin{aligned} \delta ^{\sup }\left( {\widetilde{\tau }}\right) < [\pi ]. \end{aligned}$$
     
  2. (B2)
    Every commutator \([{\widetilde{\alpha }},{\widetilde{\beta }}]\in {\widetilde{\textit{SL}}}(E)\) of elements \({\widetilde{\alpha }},{\widetilde{\beta }}\in {\widetilde{\textit{SL}}}(E)\) satisfies
    $$\begin{aligned} \delta ^{\sup }\left( \left[ {\widetilde{\alpha }},{\widetilde{\beta }}\right] \right) < [2\pi ]. \end{aligned}$$
     
  3. (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). Then
    $$\begin{aligned} {\widetilde{\tau }}_\infty \cdot {\widetilde{\tau }}_0={\widetilde{\tau }}_\theta ,\quad \hbox {and} \quad \theta =\tfrac{1}{3}\pi >0. \end{aligned}$$
    In particular,
    $$\begin{aligned} \left( {\widetilde{\tau }}_\infty \cdot {\widetilde{\tau }}_0\right) ^3={\widetilde{\tau }}^{|\measuredangle |},\quad \chi \left( {\widetilde{\tau }}_\infty \right) =\chi \left( {\widetilde{\tau }}_0\right) =1, \chi \left( {\widetilde{\tau }}^\measuredangle \right) =2\cdot \chi ({\widetilde{\tau }}^{|\measuredangle |})=12. \end{aligned}$$
     
Observation (B1) follows immediately, in light of the various definitions involved, together with the fact that \({\widetilde{\tau }}^{|\measuredangle |}\) belongs to the center of the group \({\widetilde{\textit{SL}}}(E)\), from the fact \(\tau \) fixes the [distinct!] images in \(E^\angle \) of \(\pm v\in E\) for some nonzero \(v\in E\).
Next, let us write \(|\textit{SL}(E)|\buildrel {{{\text {def}}}} \over = \textit{SL}(E){/}\{\pm 1\}\). Then observe that since the generator \({\widetilde{\tau }}^{|\measuredangle |}\) of \({{\text {Ker}}}({\widetilde{\textit{SL}}}(E)\twoheadrightarrow \textit{SL}(E)\twoheadrightarrow |\textit{SL}(E)|)\) belongs to the center of \({\widetilde{\textit{SL}}}(E)\), it follows that every commutator \([{\widetilde{\alpha }},{\widetilde{\beta }}]\) as in observation (B2) is completely determined by the respective images \(|\alpha |,|\beta |\in |\textit{SL}(E)|\) of \({\widetilde{\alpha }},{\widetilde{\beta }}\in {\widetilde{\textit{SL}}}(E)\). Now recall (cf. the proof of Lemma 3.5 [10]) that it follows immediately from an elementary linear algebra argument—i.e., consideration of a solutionx” to the equation
$$\begin{aligned} {{\text {det}}}\left( \left( \begin{array}{cc}a&{}\quad b\\ c&{}\quad d\end{array}\right) - \left( \begin{array}{cc}1&{}\quad x\\ 0&{}\quad 1\end{array}\right) \right) = 0 \end{aligned}$$
associated to an element \({a\ \ b\atopwithdelims ()c\ \ d}\in \textit{SL}_2( {\mathbb R} )\) such that \(c\not =0\)—that every element of \(\textit{SL}(E)\) other than \(-1\in \textit{SL}(E)\) may be written as a product of two unipotent elements of \(\textit{SL}(E)\). In particular, we conclude that every commutator \([{\widetilde{\alpha }},{\widetilde{\beta }}] = ({\widetilde{\alpha }}\cdot {\widetilde{\beta }}\cdot {\widetilde{\alpha }}^{-1})\cdot {\widetilde{\beta }}^{-1}\) as in observation (B2) may be written as a product
$$\begin{aligned} {\widetilde{\tau }}_1\cdot {\widetilde{\tau }}_2\cdot {\widetilde{\tau }}^*_2\cdot {\widetilde{\tau }}^*_1 \end{aligned}$$
of four minimal liftings\({\widetilde{\tau }}\)” as in (B1) such that \({\widetilde{\tau }}^*_1\), \({\widetilde{\tau }}^*_2\) are \({\widetilde{\textit{SL}}}(E)\)-conjugate to \({\widetilde{\tau }}_1^{-1}\), \({\widetilde{\tau }}_2^{-1}\), respectively. On the other hand, it follows immediately from the fact that the action on \(E^\angle \) of any non-trivial (i.e., \(\not =1\)) unipotent element of \(\textit{SL}(E)\) has precisely two fixed points (i.e., precisely one \(\{\pm 1\}\)-orbit of fixed points) that, for \(i=1,2\), there exists an element \(\epsilon _i\in \{\pm 1\}\) such that, relative to the action of \({\widetilde{\textit{SL}}}(E)\) on \((E^\angle )^\sim \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} \), \({\widetilde{\tau }}_i^{\epsilon _i}\) maps every element \(x\in {\mathbb R} \) to an element \( {\mathbb R} \ni {\widetilde{\tau }}_i^{\epsilon _i}(x)\ge x\). [Indeed, consider the continuity properties of the map \( {\mathbb R} \ni x\mapsto {\widetilde{\tau }}_i(x)-x\in {\mathbb R} \), which is invariant with respect to translation by \(\pi \) in its domain!] Moreover, since any element of \({\widetilde{\textit{SL}}}(E)\) induces a self-homeomorphism of \((E^\angle )^\sim \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} \) that commutes with the action of \({\widetilde{\tau }}^{|\measuredangle |}\), hence is necessarily strictly monotone increasing, we conclude that, for \(i=1,2\), \(({\widetilde{\tau }}^*_i)^{\epsilon _i}\) maps every element \(x\in {\mathbb R} \) to an element \( {\mathbb R} \ni ({\widetilde{\tau }}^*_i)^{\epsilon _i}(x)\le x\). In particular, any computation of the displacements \(\in {\mathbb R} \) that occur as the result of applying the above product \({\widetilde{\tau }}_1\cdot {\widetilde{\tau }}_2\cdot {\widetilde{\tau }}^*_2\cdot {\widetilde{\tau }}^*_1\) to some element of \((E^\angle )^\sim \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} \) yields, in light of the estimates \(\delta ^{\sup }({\widetilde{\tau }}_1)= \delta ^{\sup }({\widetilde{\tau }}^*_1)< [\pi ]\), \(\delta ^{\sup }({\widetilde{\tau }}_2)=\delta ^{\sup }({\widetilde{\tau }}^*_2)< [\pi ]\) of (B1), a sum
$$\begin{aligned} \left( \left( \left( a^*_1+a^*_2\right) +a_2\right) +a_1\right) = \left( a_1+a^*_1\right) +\left( a_2+a^*_2\right) \ \in \ {\mathbb R} \end{aligned}$$
for suitable elements
$$\begin{aligned}&a_1 \in \epsilon _1\cdot [0,\pi ) \subseteq {\mathbb R} ;\quad a^*_1\ \in \ -\epsilon _1\cdot [0,\pi ) \subseteq {\mathbb R} ;\\&a_2 \in \epsilon _2\cdot [0,\pi ) \subseteq {\mathbb R} ;\quad a^*_2 \in \ -\epsilon _2\cdot [0,\pi ) \subseteq \ re. \end{aligned}$$
Thus, the estimate \(\delta ^{\sup }([{\widetilde{\alpha }},{\widetilde{\beta }}])< [2\pi ]\) of observation (B2) follows immediately from the estimates \(|a_1+a^*_1|<\pi \), \(|a_2+a^*_2|<\pi \).

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 \(\tau _\infty \cdot \tau _0\).

Next, recall the well-known presentation via generators \(\alpha ^S_1,\ldots ,\alpha ^S_g\), \(\beta ^S_1,\ldots ,\beta ^S_g\), \(\gamma ^S_1,\ldots ,\gamma ^S_r\) (where \(\gamma ^S_1,\dots ,\gamma ^S_r\) generate the respective inertia groups at the points at infinity \({\overline{S}}{\setminus } S\) of S) subject to the relation
$$\begin{aligned} \left[ \alpha ^S_1,\beta ^S_1\right] \cdot \ldots \cdot \left[ \alpha ^S_g,\beta ^S_g\right] \cdot \gamma ^S_1\cdot \ldots \cdot \gamma ^S_r\ =\ 1 \end{aligned}$$
of the fundamental group \(\Pi _S\) of the Riemann surface S. These generators map, via the outer homomorphism \(\Pi _S \rightarrow \Pi _{\mathcal M}\) induced by the classifying morphism of the family of elliptic curves under consideration, to elements \(\alpha _1,\ldots ,\alpha _g\), \(\beta _1,\ldots ,\beta _g\), \(\gamma _1,\ldots ,\gamma _r\) subject to the relation
$$\begin{aligned} \left[ \alpha _1,\beta _1\right] \cdot \ldots \cdot \left[ \alpha _g,\beta _g\right] \cdot \gamma _1\cdot \ldots \cdot \gamma _r\ =\ 1 \end{aligned}$$
of the fundamental group \(\Pi _{\mathcal M}= \textit{SL}(E_ {\mathbb Z} )\) (for a suitable choice of basepoint) of \({\mathcal M}\).
Next, let us choose liftings \({\widetilde{\alpha }}_1,\ldots ,{\widetilde{\alpha }}_g\), \({\widetilde{\beta }}_1,\ldots ,{\widetilde{\beta }}_g\), \({\widetilde{\gamma }}_1,\ldots ,{\widetilde{\gamma }}_r\) of \(\alpha _1,\ldots ,\alpha _g\), \(\beta _1,\ldots ,\beta _g\), \(\gamma _1,\ldots ,\gamma _r\) to elements of \({\widetilde{\textit{SL}}}(E_ {\mathbb Z} )\) such that \({\widetilde{\gamma }}_1,\ldots ,{\widetilde{\gamma }}_r\) are minimal liftings as in (B1). Thus, we obtain a relation
$$\begin{aligned} \left[ {\widetilde{\alpha }}_1,{\widetilde{\beta }}_1\right] \cdot \ldots \cdot \left[ {\widetilde{\alpha }}_g,{\widetilde{\beta }}_g\right] \cdot {\widetilde{\gamma }}_1\cdot \ldots \cdot {\widetilde{\gamma }}_r = \left( {\widetilde{\tau }}^\measuredangle \right) ^{n^\measuredangle }= \left( {\widetilde{\tau }}^{|\measuredangle |}\right) ^{2{n^\measuredangle }} \end{aligned}$$
in \({\widetilde{\textit{SL}}}(E_ {\mathbb Z} )\) for some \({n^\measuredangle }\in {\mathbb Z} \). The situation under consideration is summarized in Fig. 4.
Fig. 4

The setup of Bogomolov’s proof

Now it follows from the various definitions involved, together with the well-known theory of Tate curves, that, for \(i=1,\ldots ,r\),
$$\begin{aligned} \hbox {the element}~\gamma _i~\hbox {is an}~\textit{SL}(E_ {\mathbb Z} )\hbox {-conjugate}~ \hbox {of}~\tau _\infty ^{v_i} \end{aligned}$$
for some \(v_i\in {\mathbb N} \). Put another way, \(v_i\) is the order of the q -parameter of the Tate curve determined by the given family \(X\rightarrow S\) at the point at infinity corresponding to \(\gamma ^S_i\).
Thus, by applying \(\chi (-)\) to the above relation, we conclude (cf. the discussion preceding [10, Lemma 3.7]) from the equalities in the final portion of (B3) (together with the evident fact that commutators necessarily lie in the kernel of \(\chi (-)\)) that
  1. (B4)
    The “orders of q-parameters” \(v_1,\ldots ,v_r\) satisfy the equality
    $$\begin{aligned} \sum \limits _{i=1}^r\ v_i\ =\ 12{n^\measuredangle }\end{aligned}$$
    —where \({n^\measuredangle }\in {\mathbb Z} \) is the quantity defined in the above discussion.
     
On the other hand, by applying \(\delta ^{\sup }(-)\) to the above relation, we conclude (cf. the discussion following the proof of [10, Lemma 3.7]) from the estimates of (B1) and (B2), the equality of (B4), and the equality \(\delta ^{\sup }( ({\widetilde{\tau }}^{|\measuredangle |})^n)\ =\ [\ |n|\cdot \pi \ ]\), for \(n\in {\mathbb Z} \), that
$$\begin{aligned} \left( \sum _{i=1}^r\ \pi \right) + \left( \sum _{j=1}^g\ 2\pi \right) > 2\pi \cdot {n^\measuredangle }= \frac{1}{6}\cdot \pi \cdot \sum _{i=1}^r\ v_i \end{aligned}$$
—i.e., that
  1. (B5)
    The “orders of q-parameters” \(v_1,\ldots ,v_r\) satisfy the estimate
    $$\begin{aligned} \frac{1}{6}\cdot \sum \limits _{i=1}^r\ v_i < 2g+r \end{aligned}$$
    —where (gr) is the type of the hyperbolic Riemann surface S.
     
Finally, we conclude (cf. the discussion following the proof of [10, Lemma 3.7]) the geometric version of the Szpiro inequality
$$\begin{aligned} \frac{1}{6}\cdot \sum \limits _{i=1}^r v_i \le 2g-2+r \end{aligned}$$
by applying (B5) (multiplied by a normalization factor \(\frac{1}{d}\)) to the families obtained from the given family \(X\rightarrow S\) by base-changing to finite étale Galois coverings of S of degree d and passing to the limit \(d\rightarrow \infty \).

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 inter-universal Teichmüller theory than the formulation of [1, 10].

We begin by considering the relationship between Bogomolov’s proof and (IU1), i.e., the theory of \(\Theta ^{\pm {{{\text {ell}}}}}{} \mathbf{NF}\)-Hodge theaters, as developed in [6]. First of all, Bogomolov’s proof clearly centers around the hyperbolic geometry of the upper half-plane. This aspect of Bogomolov’s proof is directly reminiscent of the detailed analogy discussed in [6, Remark 6.12.3]; [6, Fig. 6.4], between the structure of \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)-Hodge theaters and the classical geometry of the upper half-plane—cf., e.g., the discussion of the natural identification of \(E^{|\angle |}\) with the boundary \(\partial {\widetilde{\mathcal M}}\) of \({\widetilde{\mathcal M}}\) in Sect. 2; the discussion of the boundary of the upper half-plane in [6], Remark 6.12.3, (iii). In particular, one may think of
  • 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 \(\tau _\infty \), \(\tau _0\), \(\gamma _i\)

that appear in Bogomolov’s proof and of
  • the multiplicative \({\mathbb F}_l^{\divideontimes }\) -symmetry portion of a \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)-Hodge theater as corresponding to the toral/“typically non-unipotent” transformations \(\tau _\infty \cdot \tau _0\), \(\alpha _i\), \(\beta _i\)

that appear in Bogomolov’s proof, i.e., typically as products of two non-commuting unipotent transformations (cf. the proof of (B2)!). Here, we recall that the notation \({\mathbb F}_l^{\rtimes \pm }\) denotes the semi-direct product group \( {{\mathbb F}_l} \rtimes \{\pm 1\}\) (relative to the natural action of \(\{\pm 1\}\) on the underlying additive group of \( {{\mathbb F}_l} \)), while the notation \({\mathbb F}_l^{\divideontimes }\) denotes the quotient of the multiplicative group \( {{\mathbb F}^\times _l} \) by the action of \(\{\pm 1\}\).
One central aspect of the theory of \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)-Hodge theaters developed in [6] lies in the goal of somehow “simulating” a situation in which the module of l-torsion points of the given elliptic curve over a number field admits a “global multiplicative subspace” (cf. the discussion of [6, §I1]; [6, Remark 4.3.1]). One way to understand this sort of “simulated” situation is in terms of the one-dimensional additive geometry associated to a non-trivial unipotent transformation. That is to say, whereas, from an a priori point of view, the one-dimensional additive geometries associated to conjugate, non-commuting unipotent transformations are distinct and incompatible, the “simulation” under consideration may be understood as consisting of the establishment of some sort of geometry in which these distinct, incompatible one-dimensional additive geometries are somehow “identified” with one another as a single, unified one-dimensional additive geometry. This fundamental aspect of the theory of \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)-Hodge theaters in [6] is thus reminiscent of the
$$\begin{aligned} \mathbf{single, unified~one}\hbox {-}{} \mathbf{dimensional~objects}~E^\angle \left( \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb S}^1\right) ,\ \ \left( E^\angle \right) ^\sim \ \left( \ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} \right) \end{aligned}$$
in Bogomolov’s proof which admit natural actions by conjugate, non-commuting unipotent transformations \(\in \textit{SL}(E)\) (i.e., such as \(\tau _\infty \), \(\tau _0\)) and their minimal liftings to \({\widetilde{\textit{SL}}}(E)\) [i.e., such as \({\widetilde{\tau }}_\infty \), \({\widetilde{\tau }}_0\)—cf. (B1)].
The issue of simulation of a “global multiplicative subspace” as discussed in [6, Remark 4.3.1] is closely related to the application of absolute anabelian geometry as developed in [5], i.e., to the issue of establishing global arithmetic analogues for number fields of the classical theories of analytic continuation and Kähler metrics, constructed via the use of logarithms, on hyperbolic Riemann surfaces (cf. [6, Remarks 4.3.2, 4.3.3, 5.1.4]). These aspects of inter-universal Teichmüller theory are, in turn, closely related (cf. the discussion of [6, Remark 4.3.3]) to the application in [8] of the theory of log-shells [cf. (IU3)] as developed in [5] to the task of constructing multiradial mono-analytic containers, as discussed in the Introductions to [8, 9]. These multiradial mono-analytic containers play the crucial role of furnishing containers for the various objects of interest—i.e., the theta value and global number field portions of \(\Theta \) -pilot objects—that, although subject to various indeterminacies (cf. the discussion of the indeterminacies (Ind\({1}\)), (Ind\({2}\)), (Ind\({3}\)) in the Introduction to [8]), allow one to obtain the estimates (cf. [8, Remark 3.10.1, (iii)]) of these objects of interest as discussed in detail in [9, §1, §2] (cf., especially, the proof of [9, Theorem 1.10]). These aspects of inter-universal Teichmüller theory may be thought of as corresponding to the essential use of \((E^\angle )^\sim \ (\ {\overset{\sim }{\rightarrow }{}}\ {\mathbb R} )\) in Bogomolov’s proof, i.e., which is reminiscent of the log-shells that appear in inter-universal Teichmüller theory in many respects:
  1. (L1)

    The object \((\omega ^\measuredangle _{\mathcal M})^\sim \) that appears in Bogomolov’s proof may be thought of as corresponding to the holomorphic log-shells of inter-universal 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.

     
  2. (L2)

    Each fiber over \({\widetilde{\mathcal M}}\) of the “holomorphic log-shell” \((\omega ^\measuredangle _{\mathcal M})^\sim \) 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 mono-analytic log-shells of inter-universal Teichmüller theory.

     
  3. (L3)

    Just as in the case of the mono-analytic log-shells of inter-universal 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 inter-universal Teichmüller theory].

     
  4. (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 inter-universal Teichmüller theory, while the estimates of “typically non-unipotent” transformations given in (B2) may be thought of as corresponding to the estimates involving global number field portions of \(\Theta \)-pilot objects in inter-universal Teichmüller theory.

     
  5. (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 inter-universal Teichmüller theory are highly reminiscent of the crucial estimate of (B1) involving
    $$\begin{aligned} \hbox {the}~\mathbf{volume}~\pi ~\hbox {of a}~\mathbf{fundamental~domain}~ {\overline{D}}^\angle \end{aligned}$$
    for 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 inter-universal Teichmüller theory to establish conjugate synchronization, which plays an indispensable role in the construction of bi-coric mono-analytic log-shells (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.
     
  6. (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 inter-universal Teichmüller theory are highly reminiscent of the crucial estimate of (B2)—which arises from considering products of two non-commuting unipotent transformations, i.e., corresponding to “two distinct localizations”—involving
    $$\begin{aligned} \hbox {the}~\mathbf{volume}~2\pi ~\hbox {of the}~\{\pm 1\}\hbox {-}{} \mathbf{orbit} \pm {\overline{D}}^\angle ~\hbox {of a}~\mathbf{fundamental~domain}~{\overline{D}}^\angle \end{aligned}$$
    for the action of \(\{\pm 1\}\) on \(E^\angle \) (i.e., as opposed to the volume \(\pi \) of \({\overline{D}}^\angle \)!).
     
  7. (L7)

    The analytic continuation aspect (say, from “\(\infty \)” to “0”) of inter-universal 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 log-shell \((\omega ^\measuredangle _{\mathcal M})^\sim \),” which relates, in particular, the localizations at the cusps “\(\infty \)” and “0.”

     
Here, we note in passing that one way to understand certain aspects of the phenomena discussed in (L4)–(L6) is in terms of the following “general principle”: Let k be an algebraically closed field. Write \(k^\times \) for the multiplicative group of nonzero elements of k, \(\textit{PGL}_2(k)\buildrel {{{\text {def}}}} \over = GL_2(k)/k^\times \). Thus, by thinking in terms of fractional linear transformations, one may regard \(\textit{PGL}_2(k)\) as the group of k-automorphisms of the projective line \(P\buildrel {{{\text {def}}}} \over = {\mathbb P}^1_k\) over k. We shall say that an element of \(\textit{PGL}_2(k)\) is unipotent if it arises from a unipotent element of \(\textit{GL}_2(k)\). Let \(\xi \in \textit{PGL}_2(k)\) be a non-trivial element. Write \(P^\xi \) for the set of k-rational points of P that are fixed by \(\xi \). Then observe that
$$\begin{aligned} \xi ~\hbox {is}~\mathbf{unipotent}\Longleftrightarrow & {} P^\xi ~\hbox {is of}~\mathbf{cardinality~one};\\ \xi ~\hbox {is}~\mathbf{non}\text {-}{} \mathbf{unipotent}\Longleftrightarrow & {} P^\xi ~\hbox {is of}~\mathbf{cardinality~two}. \end{aligned}$$
That is to say,
   General principle:
  • A non-trivial 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 non-trivial non-unipotent 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 \(P^\xi \)).

Next, we recall that the suitability of the multiradial mono-analytic containers furnished by log-shells for explicit estimates (cf. [8, Remark 3.10.1, (iii)]) lies in sharp contrast to the precise, albeit somewhat tautological, nature of the correspondence [cf. (IU2)] concerning arithmetic degrees of objects of interest (i.e., 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)]). This precise correspondence is reminiscent of the precise, but relatively “superficial” [i.e., by comparison with the estimates (B1), (B2)], relationships concerning degrees [cf. (B4)] that arise from the homomorphism \(\chi \) [i.e., which is denoted “\({{\text {deg}}}\)” in [10]!]. On the other hand, the final estimate (B5) requires one to apply both the precise computation of (B4) and the non-trivial estimates of (B1), (B2). This state of affairs is highly reminiscent of the discussion surrounding [8, Fig. I.8], of two equivalent ways to compute log-volumes, i.e., the precise correspondence furnished by the \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\) -link and the non-trivial estimates via the multiradial mono-analytic containers furnished by the log-shells.
Finally, we observe that the complicated interplay between “Frobenius-like” and “étale-like” objects in inter-universal Teichmüller theory may be thought of as corresponding to the complicated interplay in Bogomolov’s proof between
  • 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

—i.e., which correspond to Frobenius-like objects in inter-universal Teichmüller theory–and
  • 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

—i.e., which correspond to étale-like objects in inter-universal Teichmüller theory.

The analogies discussed above are summarized in Table 1.

Table 1

Similarities between the two theories

Inter-universal 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/non-unipotent symmetries of upper half-plane

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 log-shells, 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 mono-analytic containers via log-shells 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 non-commuting unipotent transformations

Arithmetic degree computations via precise \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\) -link versus log-shell estimates

Degree computations via precise homomorphism \(\chi \) (B4) versus \(\delta ^{\sup }\) estimates (B1), (B2)

Frobenius-like versus étale-like objects

Complex holomorphic objects such as line bundles versus local systems, fundamental groups

6 Differences between the two theories

In a word, the most essential difference between inter-universal Teichmüller theory and Bogomolov’s proof appears to lie in the
$$\begin{aligned}&\mathbf{absence}~in~Bogomolov\hbox {'}s~proof~ of\\&\mathbf{Gaussian~distributions}~and~\mathbf{theta~ functions}, \end{aligned}$$
i.e., which play a central role in inter-universal Teichmüller theory.
In some sense, Bogomolov’s proof may be regarded as arising from the geometry surrounding the natural symplectic form
$$\begin{aligned} \langle \ \text {-}\ ,\ \text {-}\ \rangle _E\ \buildrel {{{\text {def}}}} \over = \ \langle \ \text {-}\ ,\ \text {-}\ \rangle _ {\mathcal E} |_E\end{aligned}$$
on the two-dimensional \( {\mathbb R} \)-vector space \(E\). The natural arithmetic analogue of this symplectic form is the Weil pairing on the torsion points—i.e., such as the l-torsion points that appear in inter-universal Teichmüller theory–of an elliptic curve over a number field.
On the other hand, one fundamental difference between this Weil pairing on torsion points and the symplectic form \(\langle \ \text {-}\ ,\ \text {-}\ \rangle _E\) is the following:
  • 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.,
    $$\begin{aligned} \exists \ {\mathbb R} \ \hookrightarrow \ {\mathbb C} \end{aligned}$$
    —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 l-torsion points is defined cannot be regarded as a subfield—i.e.,
    $$\begin{aligned} \not \exists \ {{\mathbb F}_l} \ \hookrightarrow \ {\mathbb Q} \end{aligned}$$
    —of the number field over which the (algebraic) elliptic curve under consideration is defined.
This phenomenon of compatibility/incompatibility of fields of definition is reminiscent of the “mysterious tensor products” that occur in p-adic Hodge theory, i.e., in which the “\({ {\mathbb Z} _p}\)” that acts on a p-adic Tate module is identified (despite its somewhat alien nature!) with the “\({ {\mathbb Z} _p}\)” that includes as a subring of the structure sheaf of the p-adic scheme under consideration (cf. the discussion of [3, Remark 3.7]; the final portion of [4, Remark 2.16.2]; [6, Remarks 4.3.1, 4.3.2]; [6, Remark 6.12.3, (i), (ii)]; [9, Remark 3.3.2]). Here, we observe further that the former\({ {\mathbb Z} _p}\),” as well as the fields of definition of the symplectic form \(\langle \ \text {-}\ ,\ \text {-}\ \rangle _E\) and the Weil pairing on torsion points, are, from the point of view of inter-universal Teichmüller theory, étale -like objects, whereas the latter\({ {\mathbb Z} _p}\),” as well as other instances of subrings of the structure sheaf of the scheme under consideration, are Frobenius-like objects. That is to say, the point of view of inter-universal Teichmüller theory may be summarized as follows:
  • Certain geometric aspects—i.e., aspects that, in effect, correspond to the geometry of the classical upper half-plane (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 inter-universal Teichmüller theory by applying various absolute anabelian algorithms to pass from étale-like to Frobenius-like objects, as well as various cyclotomic rigidity algorithms to pass, via Kummer theory, from Frobenius-like to étale-like objects.

Indeed, as discussed in [6, Remarks 4.3.1, 4.3.2], it is precisely this circle of ideas that forms the starting point for the construction of \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\)-Hodge theaters given in [6], by applying the absolute anabelian geometry of [5].
One way to understand the gap between fields of definition of first cohomology modules or modules of torsion points, on the one hand, and the field of definition of the given base scheme, on the other, is to think of elements of fields/rings of the former sort as objects that occur as exponents of regular functions on the base scheme, i.e., elements of rings that naturally contain fields/rings of the latter sort. For instance, this sort of situation may be seen at a very explicit level by consider the powers of the q -parameter that occur in the theory of Tate curves over p-adic fields (cf. the discussion of the final portion of [4, Remark 2.16.2]). From this point of view, the approach of inter-universal Teichmüller theory may be summarized as follows:
  • Certain function-theoretic 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 inter-universal 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.

Indeed, it is precisely as a result of such exponentiation operations that one is obliged to work, in inter-universal Teichmüller theory, with arbitrary iterates of the \({\mathfrak {log}}\)-link (cf. the theory of [5, 8]; the discussion of [8, Remark 1.2.2]) in order to relate and indeed identify, in effect, the function theory of exponentiated objects with the function theory of non-exponentiated objects. This situation differs somewhat from the single application of the logarithm constituted by the covering \((E^\angle )^\sim \twoheadrightarrow E^\angle \) in Bogomolov’s proof.
So far in the present Sect. 6, our discussion has centered around
  • the geometry of \(\Theta ^{\pm {{{\text {ell}}}}}\text {NF}\) -Hodge theaters (as discussed in [6, §4–§6]) and

  • the multiradial representation via mono-analytic log-shells (cf. [8, Theorem 3.11, (i), (ii)])

of inter-universal Teichmüller theory, which correspond, respectively, to the symplectic geometry of the upper half-plane (cf. §1) and the \(\delta ^{\sup }\) estimates (cf. (B1), (B2)) of Bogomolov’s proof.
On the other hand, the degree computations via the homomorphism \(\chi \), which arises, in essence, by considering the discriminant modular form, also play a key role [cf. (B4)] in Bogomolov’s proof. One may think of this aspect of Bogomolov’s proof as consisting of the application of the discriminant modular form to relate the symplectic geometry discussed in Sect. 2—cf., especially, the natural \(\textit{SL}(E)\) -torsor structure on \(\omega ^\measuredangle _{\widetilde{\mathcal M}}\)—to the conventional algebraic theory of line bundles and divisors on the algebraic stack \({\mathcal M}\). In particular, this aspect of Bogomolov’s proof is reminiscent of the \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\) -link, i.e., which serves to relate the Gaussian distributions (that is to say, exponentiated symplectic forms) that appear in the multiradial representation via mono-analytic log-shells to the conventional theory of arithmetic line bundles on the number field under consideration. We remark in passing that this state of affairs is reminiscent of the point of view discussed in [2, §1.2, §1.3.2], to the effect that the constructions of scheme-theoretic Hodge–Arakelov theory (i.e., which may be regarded as a sort of scheme-theoretic precursor of inter-universal Teichmüller theory) may be thought of as a sort of function-theoretic vector bundle version of the discriminant modular form. The \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\)-link is not compatible with the various ring/scheme structures—i.e., the “arithmetic holomorphic structures”—in its domain and codomain. In order to surmount this incompatibility, one must avail oneself of the theory of multiradiality developed in [7, 8]. The non-ring-theoretic nature of the resulting multiradial representation via mono-analytic log-shells—cf. [8, Theorem 3.11, (i), (ii)]; the discussion of inter-universality in [9, Remark 3.6.3, (i)]—of inter-universal Teichmüller theory may then be thought of as corresponding to the real analytic (i.e., non-holomorphic) nature of the symplectic geometry that appears in Bogomolov’s proof. In this context, we recall that
  1. (E1)

    one central feature of Bogomolov’s proof is the following fundamental difference between the crucial estimate (B1), which arises from the (non-holomorphic) 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 \({\widetilde{\tau }}\).

     
This central feature of Bogomolov’s proof is highly reminiscent of the situation in inter-universal Teichmüller theory in which
  1. (E2)

    although the multiradial representation of \(\Theta \) -pilot objects via mono-analytic log-shells in the domain of the \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\)-link is related, via the \(\Theta ^{\times {\varvec{\mu }}}_{{{\text {LGP}}}}\)-link, to q-pilot 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 q-pilot objects, for integers \(N\ge 2\)

     
(cf. the discussion of [8, Remark 3.12.1, (ii)]).
Thus, in summary, if, relative to the point of view of Bogomolov’s proof, one
  • 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,

then inter-universal Teichmüller theory bears numerous striking resemblances to Bogomolov’s proof. Put another way, the bridge furnished by inter-universal Teichmüller theory between the analogy discussed in detail at the beginning of Sect. 5
  1. (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 inter-universal Teichmüller theory, on the one hand,

     
and the analogy discussed extensively in (L1–L7)
  1. (A2)

    between the geometry surrounding \(E^\angle \) in Bogomolov’s proof and the holomorphic/mono-analytic log-shells—i.e., in essence, the local unit groups associated to various completions of a number field—that occur in inter-universal Teichmüller theory, on the other

     
—i.e., the bridge between l -torsion points and log-shells—may be understood as consisting of the following apparatus of inter-universal Teichmüller theory:
  1. (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 log-shells [cf. (A2)] that are constructed in a situation in which one considers arbitrary iterates of the \({\mathfrak {log}}\) -link (cf. [8, Fig. I.6]).

     
In the context of the analogies (A1), (A2), it is also of interest to observe that the multiradial containers that are ultimately used in inter-universal Teichmüller theory (cf. [8, Fig. I.6]; [8, Theorem A]) consist of processions of mono-analytic log-shells, i.e., collections of mono-analytic log-shells whose labels essentially correspond to the elements of \(|{\mathbb F}_l|\) [i.e., the quotient of the set \( {{\mathbb F}_l} \) by the natural action of \(\{\pm 1\}\)]. This observation is especially of interest in light of the following aspects of inter-universal Teichmüller theory:
  1. (P1)

    in inter-universal 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)]);

     
  2. (P2)

    at each non-archimedean 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 non-archimedean prime (cf. [7, Remark 4.7.3, (i)]);

     
  3. (P3)

    at each archimedean prime of the number field over which the elliptic curve under consideration is defined, a mono-analytic log-shell 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)]).

     
That is to say, if one thinks in terms of the correspondences
$$\begin{aligned} \mathbf{mono}\hbox {-}{} \mathbf{analytic}~\mathbf{log}\hbox {-}{} \mathbf{shells}&\quad \longleftrightarrow \quad E^\angle \left( \cong {\mathbb S}^1\right) , \\ \mathbf{procession~labels}~\in |{\mathbb F}_l|\ (\twoheadleftarrow {{\mathbb F}_l} \approx {\mathbb Z} )&\quad \longleftrightarrow \quad {\mathbb Z} \cdot \pi \ {\overset{\sim }{\rightarrow }{}}\ {{\text {Aut}}}\left( (E^\angle )^\sim {/}E^{|\angle |}\right) , \end{aligned}$$
then the collection of data constituted by a “procession of mono-analytic log-shells” is substantially reminiscent of the objects \((E^\angle )^\sim \ (\cong {\mathbb R} )\), \( {\mathbb R} _{|\pi |}\)—i.e., in essence, copies of \( {\mathbb R} \), \( {\mathbb R} _{\ge 0}\) that are subject to \({{\text {Aut}}}_{\pi }( {\mathbb R} )\) -, \({{\text {Aut}}}_{\pi }( {\mathbb R} _{\ge 0})\) -indeterminacies—that play a central role in Bogomolov’s proof.
Table 2

Contrasts between corresponding aspects of the two theories

Inter-universal 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 étale-like, Frobenius-like objects via absolute anabelian algorithms, Kummer theory/ cyclotomic rigidity algorithms

Confusion between étale-like, Frobenius-like objects via \( {\mathbb R} \ \hookrightarrow \ {\mathbb C} \)

Geometry of \(\Theta ^{\pm {{{\text {ell}}}}}{} \mathbf{NF}\) -Hodge theaters

Symplectic geometry of classical upper half-plane

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 mono-analytic log-shells 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, inter-universality

Non-holomorphic, real analytic nature of symplectic geometry

Fig. 5

The real analytic estimation apparatus of Bogomolov’s proof

Fig. 6

The multiradial estimation apparatus of inter-universal Teichmüller theory

Before concluding, we observe that, in the context of the above discussion of the technique of Galois evaluation [cf. (GE)], which plays an important role in inter-universal Teichmüller theory, it is also perhaps of interest to note the following further correspondences between the two theories:
  1. (GE1)

    The multiradiality apparatus of inter-universal 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 mono-analytic log-shells. 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/non-unipotent) 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 \(\omega ^\times _{\mathcal M}\rightarrow {\mathcal M}\).

     
  2. (GE2)

    The fact that the theory of Galois evaluation surrounding theta values plays a somewhat more central, prominent role in inter-universal 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).

     
We conclude by observing that the numerous striking resemblances discussed above are perhaps all the more striking in light of the complete independence of the development of inter-universal Teichmüller theory from developments surrounding Bogomolov’s proof: That is to say, the author was completely ignorant of Bogomolov’s proof during the development of inter-universal Teichmüller theory. Moreover, inter-universal Teichmüller theory arose not as a result of efforts to “generalize Bogomolov’s proof by substituting exponentiations of \(\langle \ \text {-}\ ,\ \text {-}\ \rangle _E\) for \(\langle \ \text {-}\ ,\ \text {-}\ \rangle _E\),” but rather as a result of efforts (cf. the discussion of [2, §1.5.1, §2.1]; [4, Remarks 1.6.2, 1.6.3]) to overcome obstacles to applying scheme-theoretic Hodge–Arakelov theory to diophantine geometry by developing some sort of arithmetic analogue of the classical functional equation of the theta function. That is to say, despite the fact that the starting point of such efforts, namely the classical functional equation of the theta function, was entirely absent from the theory surrounding Bogomolov’s proof, the theory, namely inter-universal Teichmüller theory, that ultimately arose from such efforts turned out, in hindsight, as discussed above, to be remarkably similar in numerous aspects to the theory surrounding Bogomolov’s proof.

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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Authors’ Affiliations

(1)
RIMS, Kyoto University

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