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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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INTER-UNIVERSAL TEICHMÜLLER THEORY I 23<br />

its usual topology], the two combinatorial dimensions <strong>of</strong> k may also be thought <strong>of</strong><br />

as corresponding to the<br />

· two underlying topological/real dimensions<br />

<strong>of</strong> k. Alternatively, in both the nonarchimedean and archimedean cases, one may<br />

think <strong>of</strong> the two underlying combinatorial dimensions <strong>of</strong> k as corresponding to the<br />

· group <strong>of</strong> units O × k and value group k× /O × k<br />

<strong>of</strong> k. Indeed, in the nonarchimedean case, local class field theory implies that this<br />

last point <strong>of</strong> view is consistent with the interpretation <strong>of</strong> the two underlying combinatorial<br />

dimensions via cohomological dimension; in the archimedean case, the<br />

consistency <strong>of</strong> this last point <strong>of</strong> view with the interpretation <strong>of</strong> the two underlying<br />

combinatorial dimensions via topological/real dimension is immediate from the<br />

definitions.<br />

This last interpretation in terms <strong>of</strong> groups <strong>of</strong> units and value groups is <strong>of</strong><br />

particular relevance in the context <strong>of</strong> the theory <strong>of</strong> the present series <strong>of</strong> papers.<br />

That is to say, one may think <strong>of</strong> the Θ-link<br />

† F ⊩ tht<br />

∼<br />

→<br />

‡ F ⊩ mod<br />

{ † Θ v<br />

↦→ ‡ q<br />

v<br />

} v∈V<br />

bad<br />

— which, as discussed in §I1, induces a full poly-isomorphism<br />

† F ⊢×<br />

mod<br />

∼<br />

→ ‡ F ⊢×<br />

mod<br />

{O × F v<br />

∼<br />

→ O<br />

×<br />

F v<br />

} v∈V<br />

bad<br />

—asasort<strong>of</strong>“Teichmüller deformation relative to a Θ-dilation”, i.e., a deformation<br />

<strong>of</strong> the ring structure <strong>of</strong> the number field equipped with an elliptic<br />

curve constituted by the given initial Θ-data in which one dilates the underlying<br />

combinatorial dimension corresponding to the local value groups relative to a “Θfactor”,<br />

while one leaves fixed, up to isomorphism, the underlying combinatorial dimension<br />

corresponding to the local groups <strong>of</strong> units [cf. Remark 3.9.3]. This point<br />

<strong>of</strong> view is reminiscent <strong>of</strong> the discussion in §I1 <strong>of</strong> “disentangling/dismantling”<br />

<strong>of</strong> various structures associated to number field.<br />

In [IUTchIII], we shall consider two-dimensional diagrams <strong>of</strong> Θ ±ell NF-<strong>Hodge</strong><br />

theaters which we shall refer to as log-theta-lattices. The two dimensions <strong>of</strong> such<br />

diagrams correspond precisely to the two underlying combinatorial dimensions <strong>of</strong><br />

a ring. Of these two dimensions, the “theta dimension” consists <strong>of</strong> the Frobeniuspicture<br />

associated to [more elaborate versions <strong>of</strong>] the Θ-link. Many<strong>of</strong>theimportant<br />

properties that involve this “theta dimension” are consequences <strong>of</strong> the theory<br />

<strong>of</strong> [FrdI], [FrdII], [EtTh]. On the other hand, the “log dimension” consists <strong>of</strong> iterated<br />

copies <strong>of</strong> the log-link, i.e., diagrams <strong>of</strong> the sort that are studied in [AbsTopIII].<br />

That is to say, whereas the “theta dimension” corresponds to “deformations <strong>of</strong> the<br />

arithmetic holomorphic structure” <strong>of</strong> the given number field equipped with an elliptic<br />

curve, this “log dimension” corresponds to “rotations <strong>of</strong> the two underlying

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