INTER-UNIVERSAL TEICHMULLER¨ THEORY II: HODGE-ARAKELOV-THEORETIC EVALUATION Shinichi Mochizuki December 2020 Abstract. In the present paper, which is the second in a series of four pa- pers, we study the Kummer theory surrounding the Hodge-Arakelov-theoretic eval- uation — 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 [strictly speaking, shifted by a suitable 2-torsion point], for l ≥ 5 a prime number. In the first paper of the series, we studied “miniature models of conventional scheme theory”, which we referred to as Θ±ellNF-Hodge theaters, that were associated to certain data, called initial Θ-data, that includes an elliptic curve EF over a number field F , together with a prime num- ber l ≥ 5. The underlying Θ-Hodge theaters of these Θ±ellNF-Hodge theaters were glued to one another by means of “Θ-links”, that identify the [reciprocal of the l-th ±ell root of the] theta function at primes of bad reduction of EF in one Θ NF-Hodge theater with [2l-th roots of] the q-parameter at primes of bad reduction of EF in an- other Θ±ellNF-Hodge theater. The theory developed in the present paper allows one ×μ to construct certain new versions of this “Θ-link”. One such new version is the Θgau- link, which is similar to the Θ-link, but involves the theta values at l-torsion points, rather than the theta function itself. One important aspect of the constructions ×μ that underlie the Θgau-link is the study of multiradiality properties, i.e., properties of the “arithmetic holomorphic structure” — or, more concretely, the ring/scheme structure — arising from one Θ±ellNF-Hodge theater that may be formulated in such a way as to make sense from the point of the arithmetic holomorphic structure of another Θ±ellNF-Hodge theater which is related to the original Θ±ellNF-Hodge ×μ theater by means of the [non-scheme-theoretic!] Θgau-link. For instance, certain of the various rigidity properties of the ´etale theta function studied in an earlier paper by the author may be intepreted as multiradiality properties in the context of the theory of the present series of papers. Another important aspect of the constructions ×μ that underlie the Θgau-link is the study of “conjugate synchronization” via the ± ±ell Fl -symmetry of a Θ NF-Hodge theater. Conjugate synchronization refers to a certain system of isomorphisms — which are free of any conjugacy indeterminacies! — between copies of local absolute Galois groups at the various l-torsion points at which the theta function is evaluated. Conjugate synchronization plays an impor- tant role in the Kummer theory surrounding the evaluation of the theta function at l-torsion points and is applied in the study of coricity properties of [i.e., the study of ×μ objects left invariant by] the Θgau-link. Global aspects of conjugate synchronization require the resolution, via results obtained in the first paper of the series, of certain technicalities involving profinite conjugates of tempered cuspidal inertia groups. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI Contents: Introduction §1. Multiradial Mono-theta Environments §2. Galois-theoretic Theta Evaluation §3. Tempered Gaussian Frobenioids §4. Global Gaussian Frobenioids 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 EF over a number field F , together with a prime number l ≥ 5. In the present paper, which forms the sec- ond paper of the series, we study the Kummer theory surrounding the Hodge- Arakelov-theoretic evaluation [cf. Fig. I.1 below] — i.e., evaluation in the style of the scheme-theoretic Hodge-Arakelov theory of [HASurI], [HASurII] — of the reciprocal of the l-th root of the theta function 1 √ 2 −1 2 − def 1 (m+ 1 ) n 1 (n+ 1 ) n+ 1 l − · 2 2 · − · 2 2 · 2 Θv = 1 qv ( 1) qv Uv m∈Z n∈Z [cf. [EtTh], Proposition 1.4; [IUTchI], Example 3.2, (ii)] at l-torsion points [strictly speaking, shifted by a suitable 2-torsion point] in the context of the theory of Θ±ellNF-Hodge theaters developed in [IUTchI]. Here, relative to the notation bad of [IUTchI], §I1, v ∈ V ; qv denotes the q-parameter at v of the given elliptic curve EF over a number field F; Uv denotes the standard multiplicative coordinate on the Tate curve obtained by localizing EF at v.Letq be a 2l-th root of qv. v Then these “theta values at l-torsion points” will, up to a factor given by a 2l-th root of unity, turn out to be of the form [cf. Remark 2.5.1, (i)] j2 q v [Frobenius-like!] Kummer [´etale-like!] Frobenioid-theoretic Galois-theoretic ´etale ......... theta function theta function evalu- ⇓ ation evalu- ⇓ ation [Frobenius-like!] Kummer [´etale-like!] Frobenioid-theoretic Galois-theoretic ......... theta values theta values Fig. I.1: The Kummer theory surrounding Hodge-Arakelov-theoretic evaluation INTER-UNIVERSAL TEICHMULLER¨ THEORY II 3 —wherej ∈{0, 1,...,l def=(l − 1)/2},soj is uniquely determined by its image def j ∈|Fl| = Fl/{±1} = {0} Fl [cf. the notation of [IUTchI], §I1]. In order to understand the significance of Kummer theory in the context of Hodge-Arakelov-theoretic evaluation, it is important to recall the notions of “Frobenius-like” and “´etale-like” mathematical structures [cf. the discussion of [IUTchI], §I1]. In the present series of papers, the Frobenius-like structures consti- tuted by [the monoidal portions of] Frobenioids — i.e., more concretely, by various monoids — play the important role of allowing one to construct gluing isomor- phisms such as the Θ-link which lie outside the framework of conventional scheme/ring theory [cf. the discussion of [IUTchI], §I2]. Such gluing isomor- phisms give rise to Frobenius-pictures [cf. the discussion of [IUTchI], §I1]. On the other hand, the ´etale-like structures constituted by various Galois and arith- metic fundamental groups give rise to the canonical splittings of such Frobenius- pictures furnished by corresponding ´etale-pictures [cf. the discussion of [IUTchI], §I1]. In [IUTchIII], absolute anabelian geometry will be applied to these Galois and arithmetic fundamental groups to obtain descriptions of alien arithmetic holomorphic structures, i.e., arithmetic holomorphic structures that lie on the opposite side of a Θ-link from a given arithmetic holomorphic structure [cf. the discussion of [IUTchI], §I3]. Thus, in light of the equally crucial but substantially different roles played by Frobenius-like and ´etale-like structures in the present series of papers, it is of crucial importance to be able to relate corresponding Frobenius-like and ´etale-like versions of vari- ous objects to one another. This is the role played by Kummer theory. In particular, in the present paper, we shall study in detail the Kummer theory that relates Frobenius-like and ´etale- like versions of the theta function and its theta values at l-torsion points to one another [cf. Fig. I.1]. One important notion in the theory of the present paper is the notion of mul- tiradiality. To understand this notion, let us recall the ´etale-picture discussed in [IUTchI], §I1 [cf. [IUTchI], Fig. I1.6]. In the context of the present paper, we shall be especially interested in the ´etale-like version of the theta function and its ±ell theta values constructed in each D-Θ±ellNF-Hodge theater (−)HT D-Θ NF;thus, one can think of the ´etale-picture under consideration as consisting of the diagram given in Fig. I.2 below. As discussed earlier, we shall ultimately be interested in applying various absolute anabelian reconstruction algorithms to the various arith- metic fundamental groups that [implicitly] appear in such ´etale-pictures in order to obtain descriptions of alien holomorphic structures, i.e., descriptions of objects that arise on one “spoke” [i.e., “arrow emanating from the core”] that make sense from the point of view of another spoke. In this context, it is natural to classify the various algorithms applied to the arithmetic fundamental groups lying in a given spoke as follows [cf. Example 1.7]: · We shall refer to an algorithm as coric if it in fact only depends on input data arising from the mono-analytic core of the ´etale-picture, i.e., the data that is common to all spokes. 4 SHINICHI MOCHIZUKI · We shall refer to an algorithm as uniradial if it expresses the objects constructed from the given spoke in terms that only make sense within the given spoke. · We shall refer to an algorithm as multiradial if it expresses the objects constructed from the given spoke in terms of corically constructed objects, i.e., objects that make sense from the point of view of other spokes. Thus, multiradial algorithms are compatible with simultaneous execution at multiple spokes [cf. Example 1.7, (v); Remark 1.9.1], while uniradial algorithms may only be consistently executed at a single spoke. Ultimately, in the present series of papers, we shall be interested — relative to the goal of obtaining “descriptions of alien holomorphic structures” — in the establishment of multiradial algorithms for constructing the objects of interest, e.g., [in the context of the present paper] the ´etale-like versions of the theta functions and the corresponding theta values discussed above. Typically, in order to obtain such multiradial algorithms, i.e., algorithms that make sense from the point of view of other spokes, it is necessary to allow for some sort of “indeterminacy” in the descriptions that appear in the algorithms of the objects constructed from the given spoke.