Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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2 SHINICHI MOCHIZUKI §2. Complements on Tempered Coverings §3. Chains of Θ-Hodge Theaters §4. Multiplicative Combinatorial Teichmüller Theory §5. ΘNF-Hodge Theaters §6. Additive Combinatorial Teichmüller Theory Introduction §I1. Summary of Main Results §I2. Gluing Together Models of Conventional Scheme Theory §I3. Basepoints and Inter-universality §I4. Relation to Complex and p-adic Teichmüller Theory §I5. Other Galois-theoretic Approaches to Diophantine Geometry Acknowledgements §I1. Summary of Main Results The present paper is the first in a series of four papers, the goal of which is to establish an arithmetic version of Teichmüller theory for number fields equipped with an elliptic curve, by applying the theory of semi-graphs of anabelioids, Frobenioids, theétale theta function, andlog-shells developedin[SemiAnbd], [FrdI], [FrdII], [EtTh], and [AbsTopIII]. Unlike many mathematical papers, which are devoted to verifying properties of mathematical objects that are either wellknown or easily constructed from well-known mathematical objects, in the present series of papers, most of our efforts will be devoted to constructing new mathematical objects. It is only in the final portion of the third paper in the series, i.e., [IUTchIII], that we turn to the task of proving properties of interest concerning the mathematical objects constructed. In the fourth paper of the series, i.e., [IUTchIV], we show that these properties may be combined with certain elementary computations to obtain diophantine results concerning elliptic curves over number fields. The starting point of our constructions is a collection of initial Θ-data [cf. Definition 3.1]. Roughly speaking, this data consists, essentially, of · an elliptic curve E F over a number field F , · an algebraic closure F of F , · a prime number l ≥ 5, · a collection of valuations V of a certain subfield K ⊆ F ,and · a collection of valuations V bad mod of a certain subfield F mod ⊆ F that satisfy certain technical conditions — we refer to Definition 3.1 for more details. Here, we write F mod ⊆ F for the field of moduli of E F , K ⊆ F for the extension field of F determined by the l-torsion points of E F , X F ⊆ E F for the once-punctured elliptic curve obtained by removing the origin from E F ,andX F → C F for the hyperbolic orbicurve obtained by forming the stack-theoretic quotient of X F by the natural action of {±1}. Then F is assumed to be Galois over F mod , Gal(K/F)
INTER-UNIVERSAL TEICHMÜLLER THEORY I 3 is assumed to be isomorphic to a subgroup of GL 2 (F l )thatcontains SL 2 (F l ), E F is assumed to have stable reduction at all of the nonarchimedean valuations of F , def C K = C F × F K is assumed to be a K-core [cf. [CanLift], Remark 2.1.1], V is assumed to be a collection of valuations of K such that the natural inclusion F mod ⊆ F ⊆ K induces a bijection V → ∼ V mod between V and the set V mod of all valuations of the number field F mod ,and V bad mod ⊆ V mod is assumed to be some nonempty set of nonarchimedean valuations of odd residue characteristic over which E F has bad [i.e., multiplicative] reduction — i.e., roughly speaking, the subset of the set of valuations where E F has bad multiplicative reduction that will be “of interest” to us in the context of the theory of the present series of papers. Then we shall write V bad def = V bad mod × V mod V ⊆ V, V good def mod = V mod \ V bad mod , V good def = V\V bad . Also, we shall apply the superscripts “non” and “arc” to V, V mod to denote the subsets of nonarchimedean and archimedean valuations, respectively. This data determines, up to K-isomorphism [cf. Remark 3.1.3], a finite étale covering C K → C K of degree l such that the base-changed covering X K def def = C K × CF X F → X K = X F × F K arises from a rank one quotient E K [l] ↠ Q ( ∼ = Z/lZ) of the module E K [l]ofl-torsion points of E K (K) which,atv ∈ V bad , restricts to the quotient arising from coverings of the dual graph of the special fiber. Moreover, the above data also determines a cusp ɛ of C K which, at v ∈ V bad , corresponds to the canonical generator, upto±1, of Q [i.e., the generator determined by the unique loop of the dual graph of the special fiber]. Furthermore, at v ∈ V bad , one obtains a natural finite étale covering of degree l X v → X v def = X K × K K v (→ C v def = C K × K K v ) by extracting l-th roots of the theta function; at v ∈ V good , one obtains a natural finite étale covering of degree l X−→ v → X v def = X K × K K v (→ C v def = C K × K K v ) determined by ɛ. More details on the structure of the coverings C K , X K , X v [for v ∈ V bad ], −→v X [for v ∈ V good ] may be found in [EtTh], §2, as well as in §1 ofthe present paper. In this situation, the objects l def = (l − 1)/2; l ± def = (l +1)/2; F l def = F × l /{±1}; F⋊± l def = F l ⋊ {±1}
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conjugation by which maps φ NF IN
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(v) Write INTER-UNIVERSAL TEICHMÜL
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Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

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