RIMS-1758
INTER-UNIVERSAL TEICHM
¨
ULLER THEORY III:
CANONICAL SPLITTINGS OF THE LOG-THETA-LATTICE
By
Shinichi MOCHIZUKI
August 2012
RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES
KYOTO UNIVERSITY, Kyoto, Japan
INTER-UNIVERSAL TEICHM
¨
ULLER THEORY III:
CANONICAL SPLITTINGS OF THE LOG-THETA-LATTICE
Shinichi Mochizuki
August 2012
Abstract. In the present paper, which is the third in a series of
four papers, we study the theory surrounding the log-theta-lattice,ahighly non-
commutative two-dimensional diagram of “miniature models of conventional scheme
theory”, called Θ
±ell
NF-Hodge theaters, that were associated, in the first paper of
the series, to certain data, called initial Θ-data, that includes an elliptic curve E
F
over a number field F , together with a prime number l ≥ 5. The horizontal arrows
of the log-theta-lattice are defined as certain versions of the “Θ-link” that was con-
structed, in the second paper of the series, by applying the theory of Hodge-Arakelov-
theoretic evaluation — i.e., evaluation in the style of the scheme-theoretic Hodge-
Arakelov theory established by the author in previous papers — of the [reciprocal
of the l-th root of the] theta function at
l-torsion points. In the present pa-
per, we study the theory surrounding the log-link between Θ
±ell
NF-Hodge theaters.
The log-link is obtained, roughly speaking, by applying, at each [say, for simplicity,
nonarchimedean] valuation of the number field under consideration, the local p-adic
logarithm. The significance of the log-link lies in the fact it allows one to construct
log-shells, i.e., roughly speaking, slightly adjusted forms of the image of the local
units at the valuation under consideration via the local p-adic logarithm. The theory
of log-shells was studied extensively in a previous paper of the author. The vertical
arrows of the log-theta-lattice are given by the log-link. Consideration of various
properties of the log-theta-lattice leads naturally to the establishment of multiradial
algorithms for constructing “splitting monoids of logarithmic Gaussian pro-
cession monoids”. Here, we recall that “multiradial algorithms” are algorithms
that make sense from the point of view of an “alien arithmetic holomorphic
structure”, i.e., the ring/scheme structure of a Θ
±ell
NF-Hodge theater related to
agivenΘ
±ell
NF-Hodge theater by means of a non-ring/scheme-theoretic horizontal
arrow of the log-theta-lattice. These logarithmic Gaussian procession monoids, or
LGP-monoids, for short, may be thought of as the log-shell-theoretic versions of
the Gaussian monoids that were studied in the second paper of the series. Finally,
by applying these multiradial algorithms for splitting monoids of LGP-monoids, we
obtain estimates for the log-volume of these LGP-monoids. These estimates will
be applied to verify various diophantine results in the fourth paper of the series.
Contents:
Introduction
§0. Notations and Conventions
§1. The Log-theta-lattice
§2. Multiradial Theta Monoids
§3. Multiradial Logarithmic Gaussian Procession Monoids
Typeset by A
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1
2 SHINICHI MOCHIZUKI
Introduction
In the following discussion, we shall continue to use the notation of the In-
troduction to the first paper of the present series of papers [cf. [IUTchI], §I1]. In
particular, we assume that are given an elliptic curve E
F
over a number field F ,
together with a prime number l ≥ 5. In the first paper of the series, we introduced
and studied the basic properties of Θ
±ell
NF-Hodge theaters, which may be thought
of as miniature models of the conventional scheme theory surrounding the given
elliptic curve E
F
over the number field F . In the present paper, which forms the
third paper of the series, we study the theory surrounding the log-link between
Θ
±ell
NF-Hodge theaters. The log-link induces an isomorphism between the under-
lying D-Θ
±ell
NF-Hodge theaters and, roughly speaking, is obtained by applying, at
each [say, for simplicity, nonarchimedean] valuation v
∈ V,thelocal p
v
-adic loga-
rithm to the local units [cf. Proposition 1.3, (i)]. The significance of the log-link lies
in the fact it allows one to construct log-shells, i.e., roughly speaking, slightly ad-
justed forms of the image of the local units at v
∈ V via the local p
v
-adic logarithm.
The theory of log-shells was studied extensively in [AbsTopIII]. The introduction
of log-shells leads naturally to the construction of new versions — namely, the
Θ
×μ
LGP
-/Θ
×μ
lgp
-links [cf. Definition 3.8, (ii)] — of the Θ-/Θ
×μ
-/Θ
×μ
gau
-links studied
in [IUTchI], [IUTchII]. The resulting [highly non-commutative!] diagram of iterates
of the log- [i.e., the vertical arrows]andΘ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
×μ
LGP
-/Θ
×μ
lgp
-links [i.e., the
horizontal arrows] — which we refer to as the log-theta-lattice [cf. Definitions
1.4; 3.8, (iii), as well as Fig. I.1 below, in the case of the Θ
×μ
LGP
-link] — plays a
central role in theory of the present series of papers.
.
.
.
.
.
.
⏐
⏐
log
⏐
⏐
log
...
Θ
×μ
LGP
−→
n,m+1
HT
Θ
±ell
NF
Θ
×μ
LGP
−→
n+1,m+1
HT
Θ
±ell
NF
Θ
×μ
LGP
−→ ...
⏐
⏐
log
⏐
⏐
log
...
Θ
×μ
LGP
−→
n,m
HT
Θ
±ell
NF
Θ
×μ
LGP
−→
n+1,m
HT
Θ
±ell
NF
Θ
×μ
LGP
−→ ...
⏐
⏐
log
⏐
⏐
log
.
.
.
.
.
.
Fig. I.1: The [LGP-Gaussian] log-theta-lattice
Consideration of various properties of the log-theta-lattice leads naturally to
the establishment of multiradial algorithms for constructing “splitting monoids
of logarithmic Gaussian procession monoids” [cf. Theorem A below]. Here,
we recall that “multiradial algorithms” [cf. the discussion of [IUTchII], Introduc-
tion] are algorithms that make sense from the point of view of an “alien arithmetic
holomorphic structure”, i.e., the ring/scheme structure of a Θ
±ell
NF-Hodge
theater related to a given Θ
±ell
NF-Hodge theater by means of a non-ring/scheme-
theoretic Θ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
×μ
LGP
-/Θ
×μ
lgp
-link. These logarithmic Gaussian procession
INTER-UNIVERSAL TEICHM
¨
ULLER THEORY III 3
monoids, or LGP-monoids, for short, may be thought of as the log-shell-theoretic
versions of the Gaussian monoids that were studied in [IUTchII]. Finally, by apply-
ing these multiradial algorithms for splitting monoids of LGP-monoids, we obtain
estimates for the log-volume of these LGP-monoids [cf. Theorem B below].
These estimates will be applied to verify various diophantine results in [IUTchIV].
Recall [cf. [IUTchI], §I1] the notion of an F-prime-strip.AnF-prime-strip
consists of data indexed by the valuations v
∈ V; roughly speaking, the data at
each v
consists of a Frobenioid, i.e., in essence, a system of monoids over a base
category. For instance, at v
∈ V
bad
, this data may be thought of as an isomorphic
copy of the monoid with Galois action
Π
v
O
F
v
— where we recall that O
F
v
denotes the multiplicative monoid of nonzero integral
elements of the completion of an algebraic closure
F of F at a valuation lying over
v
[cf. [IUTchI], §I1, for more details]. The p
v
-adic logarithm log
v
: O
×
F
v
→ F
v
at v
then defines a natural Π
v
-equivariant isomorphism of topological modules
(O
×μ
F
v
⊗ Q
∼
→ ) O
×
F
v
⊗ Q
∼
→ F
v
— where we recall the notation “O
×μ
F
v
= O
×
F
v
/O
μ
F
v
” from the discussion of [IUTchI],
§1 — which allows one to equip O
×
F
v
⊗ Q with the field structure arising from the
field structure of
F
v
.Theportionatv of the log-link associated to an F-prime-strip
[cf. Definition 1.1, (iii); Proposition 1.2] may be thought of as the correspondence
Π
v
O
F
v
log
−→
Π
v
O
F
v
in which one thinks of the copy of “O
F
v
”ontheright as obtained from the field
structure induced by the p
v
-adic logarithm on the tensor product with Q of the
copy of the units “O
×
F
v
⊆O
F
v
”ontheleft. Since this correspondence induces an
isomorphism of topological groups between the copies of Π
v
on either side, one may
think of Π
v
as “immune to”/“neutral with respect to” — or, in the terminology
of the present series of papers, “coric” with respect to — the transformation
constituted by the log-link. This situation is studied in detail in [AbsTopIII], §3,
and reviewed in Proposition 1.2 of the present paper.
By applying various results from absolute anabelian geometry,onemay
algorithmically reconstruct a copy of the data “Π
v
O
F
v
”fromΠ
v
.Moreover,
by applying Kummer theory, one obtains natural isomorphisms between this “coric
version” of the data “Π
v
O
F
v
” and the copies of this data that appear on
either side of the log-link. On the other hand, one verifies immediately that these
Kummer isomorphisms are not compatible with the coricity of the copy of the
data “Π
v
O
F
v
” algorithmically constructed from Π
v
. This phenomenon is, in
some sense, the central theme of the theory of [AbsTopIII], §3, and is reviewed in
Proposition 1.2, (iv), of the present paper.
4 SHINICHI MOCHIZUKI
The introduction of the log-link leads naturally to the construction of log-
shells at each v
∈ V. If, for simplicity, v ∈ V
bad
, then the log-shell at v is given,
roughly speaking, by the compact additive module
I
v
def
= p
−1
v
· log
v
(O
×
K
v
) ⊆ K
v
⊆ F
v
[cf. Definition 1.1, (i), (ii); Remark 1.2.2, (i), (ii)]. One has natural functorial algo-
rithms for constructing various versions — i.e., mono-analytic/holomorphic and
´etale-like/Frobenius-like —fromD
-/D-/F
-/F-prime-strips [cf. Proposition
1.2, (v), (vi), (vii), (viii), (ix)]. Although, as discussed above, the relevant Kummer
isomorphisms are not compatible with the log-link “at the level of elements”,the
log-shell I
v
at v satisfies the important property
O
K
v
⊆I
v
;log
v
(O
×
K
v
) ⊆I
v
— i.e., it contains the images of the Kummer isomorphisms associated to both the
domain and the codomain of the log-link [cf. Proposition 1.2, (v); Remark 1.2.2, (i),
(ii)]. In light of the compatibility of the log-link with log-volumes [cf. Propositions
1.2, (iii); 3.9, (iv)], this property will ultimately lead to upper bounds — i.e., as
opposed to “precise equalities” — in the computation of log-volumes in Corollary
3.12 [cf. Theorem B below]. Put another way, although iterates of the log-link
fail to be compatible with the various Kummer isomorphisms that arise, one may
nevertheless consider the entire diagram that results from considering such iterates
of the log-link and related Kummer isomorphisms [cf. Proposition 1.2, (x)]. We
shall refer to such diagrams
... →•→•→•→...
... ↓ ...
◦
— i.e., where the horizontal arrows correspond to the log-links [that is to say, to
the vertical arrows of the log-theta-lattice!]; the “•’s” correspond to the Frobenioid-
theoretic data within a Θ
±ell
NF-Hodge theater; the “◦” corresponds to the coric
version of this data [that is to say, in the terminology discussed below, verti-
cally coric data of the log-theta-lattice]; the vertical/diagonal arrows correspond
to the various Kummer isomorphisms —aslog-Kummer correspondences [cf.
Theorem 3.11, (ii); Theorem A, (ii), below]. Then the inclusions of the above
display may be interpreted as a sort of “upper semi-commutativity” of such
diagrams [cf. Remark 1.2.2, (iii)], which we shall also refer to as the “upper semi-
compatibility” of the log-link with the relevant Kummer isomorphisms —cf. the
discussion of the “indeterminacy” (Ind3) in Theorem 3.11, (ii).
By considering the log-links associated to the various F-prime-strips that occur
in a Θ
±ell
NF-Hodge theater, one obtains the notion of a log-link between Θ
±ell
NF-
Hodge theaters
†
HT
Θ
±ell
NF
log
−→
‡
HT
Θ
±ell
NF
[cf. Proposition 1.3, (i)]. As discussed above, by considering the iterates of the log-
[i.e., the vertical arrows]andΘ-/Θ
×μ
-/Θ
×μ
gau
-/Θ
×μ
LGP
-/Θ
×μ
lgp
-links [i.e., the horizontal
