CLASSICAL ROOTS OF INTER-UNIVERSALTEICHMÜLLER THEORY

Shinichi Mochizuki (RIMS, Kyoto University)
November 2020

http://www.kurims.kyoto-u.ac.jp/~motizuki“Travel and Lectures”§1. Isogeny invariance of heights of elliptic curves§2. Crystals and Hodge filtrations§3. Complex Teichmüller theory§4. Theta function on the upper half-plane

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

Overview
Analogy with étale cohomology, Weil conjectures←→ classical singular (co)homology of topological spaces· Isogeny invariance of heights of elliptic curves (Faltings, 1983)· Crystals and Hodge filtrations (Grothendieck, late 1960’s)· Complex Teichmüller theory (Teichmüller, 1930’s)· Theta function on the upper half-plane (Jacobi, 19-th century)CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

3§1. Isogeny invariance of heights of elliptic curves(cf. [Alien], §2.3, §2.4)
We consider elliptic curves.
For l a prime number, the module of l-torsion points associated to adefTate curve E = Gm /q Z — over, say, C or a p-adic field — fits into anatural exact sequence:
0 −→ µl −→ E[l] −→ Z/lZ −→ 0.
Thus, one has canonical objects as follows:a “multiplicative subspace” µl ⊆ E[l] and “generators” ±1 ∈ Z/lZ.

4SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

In the following, we fix an elliptic curve E over a number ﬁeld F anda prime number l ≥ 5 such that E has stable reduction at all finite
places of F .
Then, in general, E[l] does not admit
a global “multiplicative subspace” and “generators”
that coincide with the above canonical “multiplicative subspace” and “generators” at all ﬁnite places where E has bad multiplicative reduction!

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

5Nevertheless, suppose (!!) that such global objects do in fact exist.
Then, if we denote byE → E∗

the isogeny obtained by forming the quotient of E by the“global multiplicative subspace”,
then, at each finite prime of bad multiplicative reduction, the respective
q-parameters satisfy the following relation:lqE= qE ∗ .

If we write log(qE ), log(qE ∗ ) for the arithmetic degrees ∈ R determinedby these q-parameters, then the above relation takes on the following form:l · log(qE ) = log(qE ∗ ) ∈ R.

6SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

On the other hand, if we consider the respective heights of the elliptic
curves by htE , htE ∗ ∈ R — i.e, roughly speaking, arithmetic degrees ofarithmetic line bundles on F
⊗2,
ωE

⊗2
ωE
∗

associated to the sheaves of square diﬀerentials — then we may conclude — cf. the discriminant mod. form, regarded as a section of the⊗12ample line bundle “ωM” on the compactiﬁed moduli stack Mell ofellelliptic curves! — thatht(−) ≈ 16 · log(q(−) )(where “≈” means “up to a discrepancy bounded by a constant”).

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

7

Moreover, by the famous computation concerning diﬀerentials due toFaltings (1983), one knows that:htE ∗ ≈ htE + log(l).
Thus, (by ignoring certain subtleties at archimedean places of F ) we conclude thatl · htE . htE + log(l),i.e.,htE .

1l−1 · log(l). constant— that is to say, that the height htE of the elliptic curve E can be boundedfrom above, and hence (under suitable hypotheses) that there are onlyﬁnitely many isomorphism classes of elliptic curves E that admit a “globalmultiplicative subspace”.

8SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

Key point:
lConsider distinct elliptic curves E, E ∗ such that qE= qE ∗ (!), butwhich (up to negligible discrepancies) share — i.e., “∧”! — a commonωE ≈ ωE ∗ .

One way to understand IUT, esp. Hodge theaters of [IUTchI]:
Apparatus to generalize the above argument — by focusing on the above
key point! — to the case of general elliptic curves for which “globalmultiplicative subspaces”, etc. do not necessarily exist.

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY§2. Crystals and Hodge ﬁltrations(cf. [Alien], §3.1, (iv), (v))
Let X: a smooth, proper, connected algebraic curve over C,
E: a vector bundle on X.
Consider the two projections:p1←− X × XXp2−→ X

Then in general, there exists a vector bundle F on X × X such that(F∼= p∗1 E)(∨)

F∼= p∗2 E ,but there does not exist a vector bundle F on X × X such that()F∼
= p∗1 E∧

(which would imply that E is trivial!).

()F∼= p∗2 E

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

Consider the ﬁrst inﬁnitesimal neighborhood of the diagonal
X = V (I) ,→ X × X,
defi.e., Xinf = V (I 2 ) ⊆ X × X:
“moduli space of pairs of points of X (cf. X × X!) that areinﬁnitesimally close to one another”.
Grothendieck definition of a connection on E:p∗1 E|Xinf∼→p∗2 E|Xinf ,i.e.,“isomorphism between the ﬁbers of E at pairs of points of X (cf.
∼p∗1 E → p∗2 E on X × X!) that are inﬁnitesimally close to oneanother”.

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

11In general, E does not admit a connection. The obstruction to the existence of a connection (cf. Weil!) on det(E) is a cohomology class inH 1 (X, ωX ),which is in fact equal to the ﬁrst Chern class of E, i.e., from the point ofview of de Rham cohomology, the degree of E:deg(E) ∈ Z.
Thus, if E is a line bundle, thenE admits a connection⇐⇒deg(E) = 0.

There also exists a logarithmic version of this discussion: by considering
logarithmic poles at a finite number of points of X(C).

12SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

Suppose that X is equipped with a log structure determined by a finite setof rX points of X(C). Write X log for the resulting log scheme, U ⊆ X for
the interior of X log .
Consider a (compactified) family of elliptic curves
f :E→X(i.e., a family of one-dimensional semi-abelian schemes over X with proper
fibers over U ⊆ X). Then the relative ﬁrst de Rham cohom. moduleof this family determines a rank two vector bundle on X
Edef

=R1 fDR,∗ OE

equipped with: Gauss-Manin (logarithmic!) connection ∇E anda rank one Hodge subbundle ωE ⊆ E s.t. ωE ⊗OX (E/ωE ) ∼= OX(cf. the bundle ωMell of §1!).

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

13Note: ωE does not admit a connection, i.e., in general, p∗1 ωE |Xinf is not isom.
to p∗2 ωE |Xinf ! But one can measure the extent to which ωE fails to admit
a connection by means of ∇E , i.e., by considering the (generically nonzero,OX -linear!) composite morphism:
ωE,→E

∇y E
log
E ⊗OX ωX

log
−1
ωE
⊗OX ωX.

The resulting Kodaira-Spencer morphismlog⊗2κE : ωE,→ ωX,⊗2yields a bound (“geometric Szpiro”) on the height deg(ωE) of f : E → X(cf. §1!):log⊗2deg(ωE) ≤ deg(ωX) = 2gX − 2 + rX .

14SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

Key point:p∗1 E ∼= p∗2 E serves as a common — i.e., “∧”! — container(cf. the common “ωE ≈ ωE ∗ ” of §1!) that is· suﬃciently large to house both p∗1 ωE ,→ p∗1 E and p∗2 ωE ,→ p∗2 E, but· suﬃciently small to yield a nontrivial estimate on the height ofthe family of elliptic curves f : E → X under consideration.
One way to understand IUT, esp. multiradial rep. of [IUTchIII]:Construction — by means of· absolute anabelian geometry and· the theory of the étale theta function— of a common container that is· suﬃciently large to house the incompatible ring structures onNeither side of the gluing constituted by the theta link qE
7→ qE , but· suﬃciently small to yield nontrivial estimate on the height ofthe elliptic curve over a number field under consideration.

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY15§3. Complex Teichmüller theory(cf. [Pano], §2; [Alien], §3.3, (ii))
Relative to a canonical coordinate z = x + iy — assoc’d to a square diﬀerential — on a Riemann surface, Teichmüller deformations given byhol. str.
yC∋
z7→ ζ = ξ + iη = λx + iyhol. str.
y∈C— where 1 < λ < ∞ is the dilation factor.
Key points:· non-hol. map, but common real analytic str. — i.e., “∧”!
· one hol. dim., but two underlying real dims., of which
one is dilated/deformed, while theother is left ﬁxed/undeformed!

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

Classical complex Teichmüller deformation:

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

17∼

Recall: the upper half-plane H ( → open unit disk D) may be regarded
as the moduli space of hol. strs. on R2 — cf. the bijection:
hol. str.
hol. str.
yy
C× \GL+ (R)/C×(λ 0 )0 1∼→[0, 1)7→λ−1
λ+1— where( )· λ ∈ R≥1 , and we regard λ0 10 as a dilation;· GL+ (R) denotes the group of 2 × 2 real matrices with determinant > 0;
· C× denotes the multiplicative( a b ) group of C, which we regard as a subgroup+of GL (R) via a + ib 7→ −b a , for a, b ∈ R s.t. (a, b) ̸= (0, 0).
Relative to GL+ (R) y H by linear fractional transformations, C× is thestabilizer of i ∈ H, so the above bijection just states that any w ∈ D may
be mapped to 0 ∈ D by a rotation ∈ C× , followed by a dilation.

18SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

The fundamental domain of the upper half-plane and the unit disk:(cf. https://www.mathsisfun.com/geometry/unit-circle.html ;http://www.math.tifr.res.in/~dprasad/mf2.pdf)

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

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Key point:
In the discussion of H: R2 (with standard orientation) serves as acommon — i.e., “∧”! — container for various hol. strs.
In summary:hol. str.
≈ C×

ydilation(λ 0)
0 1xhol. str.
≈ C×One way to understand IUT, esp. log-theta-lattice of [IUTchIII]:“inﬁnite H” portion (i.e., portion that is actually used) of log-theta-lattice:rotationsof arith.
hol. str.
≈ log

ydilationof ringstr. via
theta linkx

rotationsof arith.
hol. str.
≈ log

Here, arith. hol. str. ≈ ring str., which is not preserved by theta link
N7→ qE ”!
“qE20SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

The entire log-theta-lattice and the “inﬁnite H” portion that is actuallyused:
..
.
x
log
..
.
x
log
ΘΘΘΘΘ

Θ

..
.
x
log


..
.
x
log


•x
log


•x
log
...

−→

Θ• −→ • −→ . . .
xx
log
log



...

−→

Θ• −→ • −→ . . .
xx
log
log


...

−→

Θ

• −→ • −→ . . .
xx
log
log



•x
log
•
x
log


..
.

..
.

..
.

..
.

⊇Θ

• −→ •xx
log
log


CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY21§4. Theta function on the upper half-plane(cf. final portion of [Pano], §3; discussion surrounding [Pano], Fig. 4.2)
def

Recall the theta function on H ∋ z = x + iy, where q = e2πiz :+∞
∑1 2
def
q2n .
θ(q) =n=−∞def

Restricting to the imaginary axis (i.e., x = 0) yields, for t = y:+∞∑2def
θ(t) =e−πn t .
n=−∞Then the Jacobi identity holds:
θ(t) = t− 2 · θ(t−1 ).
1Here, we note thatdef ( 0 1 )−1 0

GL+ (R) ⊇ C× ∋ ι =maps z 7→ −z −1 , hence iy 7→ −iy −1 , i.e., t 7→ t−1 .

22SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

As one travels along the imag. axis via GL+ (R) ⊇ C× ∋(y 0)0 1: i 7→ iy:

When |q| → 0 ⇐⇒ y → +∞:θ(t) series terms are rapidly decreasing =⇒ easy to compute!
∧ (!)
When |q| → 1 ⇐⇒ y → +0:
θ(t) series terms not rapidly decreasing =⇒ diﬃcult to compute!
Note: “∧” makes sense precisely because one distinguishes the ι-conjugate
regions “|q| → 0 ⇐⇒ y → +∞” and “|q| → 1 ⇐⇒ y → +0”!
This situation parallels the Θ-link of IUT (cf. |q N | → 0 vs.

|q| ≈ 1!).

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

Jacobi identity θ(t) = t− 2 · θ(t−1 ) may be interpreted as follows:1θ(t) descends, up to a suitable factor t− 2 , to the quotient by ι.
1Comparison with IUT:
Jacobi identity←→ multiradial representation of IUT

the factor t− 2←→

indeterminacies of multirad. rep.

involution ι ∈ C×←→

log-link of IUT: rotat. of hol. str.

descent to quotient by ι←→descent to single hol. str./ring str.

123

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

Behavior of θ(t) series terms upon applying Jacobi identity:∞GaussianθGaussian０CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

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Proof of Jacobi identity: One computes θ(t−1 ) by using the fact that()22
11
1Fourier transform (e−t· ) ≈ π − 2 t− 2 · e− t ·
— a computation closely related to the computation of the Gaussian integral∫ +∞e−x dx = π 221−∞via polar coordinates!
This computation is essentially a consequence of the quadratic form inthe exponent of the Gaussian:e−t·“ ” .
2

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SHINICHI MOCHIZUKI (RIMS, KYOTO UNIVERSITY)

quad. form ≈ Chern class “2 ”=⇒theta group symmetries=⇒

rigidity properties ofétale theta function in IUT=⇒

Kummer theoryof étale theta function
compatible with log-link
12(cf. “t · 2t · ”
in above computation!)and multiradial rep. of IUT

CLASSICAL ROOTS OF INTER-UNIVERSAL TEICHMÜLLER THEORY

27References[IUTchI] S. Mochizuki, Inter-universal Teichmüller Theory I: Constructionof Hodge Theaters, RIMS Preprint 1756 (August 2012), to appear in Publ.
Res. Inst. Math. Sci.
[IUTchII] S. Mochizuki, Inter-universal Teichmüller Theory II: Hodge-Arakelov-theoretic Evaluation, RIMS Preprint 1757 (August 2012), to appear
in Publ. Res. Inst. Math. Sci.
[IUTchIII] S. Mochizuki, Inter-universal Teichmüller Theory III: Canonical
Splittings of the Log-theta-lattice, RIMS Preprint 1758 (August 2012), toappear in Publ. Res. Inst. Math. Sci.
[IUTchIV] S. Mochizuki, Inter-universal Teichmüller Theory IV: Log-volume
Computations and Set-theoretic Foundations, RIMS Preprint 1759 (August2012), to appear in Publ. Res. Inst. Math. Sci.
[Pano] S. Mochizuki, A Panoramic Overview of Inter-universal Teichmüller
Theory, Algebraic number theory and related topics 2012, RIMS Kōkyūroku
Bessatsu B51, Res. Inst. Math. Sci. (RIMS), Kyoto (2014), pp. 301-345.
[Alien] S. Mochizuki, The Mathematics of Mutually Alien Copies: fromGaussian Integrals to Interuniversal Teichmüller Theory, RIMS Preprint
1854 (July 2016).
Updated versions are available at the following webpage:
http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.html