Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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86 SHINICHI MOCHIZUKI “arithmetic collections of local analytic sections” of Spec(K) → Spec(F ) — cf. Fig. 4.1 below, where each “·−·−...−·−·” represents a [D-]prime-strip. In fact, if, for the sake of brevity, we abbreviate the phrase “collection of local analytic” by the term “local-analytic”, then each of these sections may be thought of as yielding not only an “arithmetic local-analytic global multiplicative subspace”, but also an “arithmetic local-analytic global canonical generator” [i.e., up to multiplication by ±1, of the quotient of the module of l-torsion points of the elliptic curve in question by the “arithmetic local-analytic global multiplicative subspace”]. We refer to Remark 4.9.1, (i), below, for more on this point of view. ·−·−·−...−·−·−· ·−·−·−...−·−·−· ... K ·−·−·−...−·−·−· ·−·−·−...−·−·−· Gal(K/F) ⊆ GL 2 (F l ) ⏐ ↓ ·−·−·−...−·−·−· F Fig. 4.1: Prime-strips as “sections” of Spec(K) → Spec(F ) (ii) The way in which these “arithmetic local-analytic sections” constituted by the [D-]prime-strips fail to be [globally] “arithmetically holomorphic” may be understood from several closely related points of view. The first point of view was already noted above in (i) — namely: (a) these sections fail to extend to ring homomorphisms K → F . The second point of view involves the classical phenomenon of decomposition of primes in extensions of number fields. The decomposition of primes in extensions of number fields may be represented by a tree, as in Fig. 4.2, below. If one thinks of the tree in large parentheses of Fig. 4.2 as representing the decomposition of primes over a prime v of F in extensions of F [such as K!], then the “arithmetic local-analytic sections” constituted by the D-prime-strips may be thought of as ⎛ ⎞ ... ⎛ ⎞ ... \|/ ... ... ⎜ ⎝ \|/ ⎟ ⎠ v ′ v ′′ v ′′′ ⊇ v ′ ⎜ ⎝ \ | / ⎟ ⎠ v Fig. 4.2: Prime decomposition trees
INTER-UNIVERSAL TEICHMÜLLER THEORY I 87 (b) an isomorphism, or identification, between v [i.e., a prime of F ]and v ′ [i.e., a prime of K] which [manifestly — cf., e.g., [NSW], Theorem 12.2.5] fails to extend to an isomorphism between the respective prime decomposition trees over v and v ′ . If one thinks of the relation “∈” between sets in axiomatic set theory as determining a “tree”, then the point of view of (b) is reminiscent of the point of view of [IUTchIV], §3, where one is concerned with constructing some sort of artificial solution to the “membership equation a ∈ a” [cf. the discussion of [IUTchIV], Remark 3.3.1, (i)]. The third point of view consists of the observation that although the “arithmetic local-analytic sections” constituted by the D-prime-strips involve isomorphisms of the various local absolute Galois groups, (c) these isomorphisms of local absolute Galois groups fail to extend to a section of global absolute Galois groups G F ↠ G K [i.e., a section of the natural inclusion G K ↩→ G F ]. Here, we note that in fact, by the Neukirch-Uchida theorem [cf. [NSW], Chapter XII, §2], one may think of (a) and (c) as essentially equivalent. Moreover, (b) is closely related to this equivalence, in the sense that the proof [cf., e.g., [NSW], Chapter XII, §2] of the Neukirch-Uchida theorem depends in an essential fashion on a careful analysis of the prime decomposition trees of the number fields involved. (iii) In some sense, understanding more precisely the content of the failure of these “arithmetic local-analytic sections” constituted by the D-prime-strips to be “arithmetically holomorphic” is a central theme of the theory of the present series of papers — a theme which is very much in line with the spirit of classical complex Teichmüller theory. Remark 4.3.2. The incompatibility of the “arithmetic local-analytic sections” of Remark 4.3.1, (i), with global prime distributions and global absolute Galois groups [cf. the discussion of Remark 4.3.1, (ii)] is precisely the technical obstacle that will necessitate the application — in [IUTchIII] — of the absolute p-adic monoanabelian geometry developed in [AbsTopIII], in the form of “panalocalization along the various prime-strips” [cf. [IUTchIII] for more details]. Indeed, the mono-anabelian theory developed in [AbsTopIII] represents the culmination of earlier research of the author during the years 2000 to 2007 concerning absolute p-adic anabelian geometry — research that was motivated precisely by the goal of developing a geometry that would allow one to work with the “arithmetic local-analytic sections” constituted by the prime-strips, so as to overcome the principal technical obstruction to applying the Hodge-Arakelov theory of [HASurI], [HASurII] [cf. Remark 4.3.1, (i)].
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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