# Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation

### Shinichi Mochizuki

Kyoto University, Japan

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## Abstract

In the present paper, which is the second in a series of four papers, we study the **Kummer theory** surrounding the 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** [strictly speaking, shifted by a suitable 2-torsion point], for $l\ge 5$ a prime number. In the first paper of the series, we studied *"miniature models of conventional scheme theory"*, which we referred to as *$\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theaters*, that were associated to certain data, called *initial $\Theta$-data*, that includes an *elliptic curve $E_F$* over a *number field* $F$, together with a *prime number* $l\ge 5$. The underlying $\Theta$-Hodge theaters of these $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theaters were *glued* to one another by means of *"$\Theta$-links"*, that identify the [reciprocal of the $l$-th root of the] *theta function* at primes of bad reduction of $E_F$ in one $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theater with [$2l$-th roots of] the *$q$-parameter* at primes of bad reduction of $E_F$ in another $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theater. The theory developed in the present paper allows one to construct certain new versions of this "$\Theta$-link". One such new version is the **$\Theta^{\times\mu}_{\mathrm{gau}}$-link**, which is similar to the $\Theta$-link, but involves the *theta values at $l$-torsion points*, rather than the theta function itself. One important aspect of the constructions that underlie the $\Theta^{\times\mu}_{\mathrm{gau}}$-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* $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theater that may be formulated in such a way as to make sense from the point of view of the arithmetic holomorphic structure of *another* $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theater which is related to the original $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-Hodge theater by means of the [*non-scheme-theoretic*!] $\Theta^{\times\mu}_{\mathrm{gau}}$-link. For instance, certain of the various *rigidity* properties of the *étale theta function* studied in an earlier paper by the author may be interpreted as *multiradiality* properties in the context of the theory of the present series of papers. Another important aspect of the constructions that underlie the $\Theta^{\times\mu}_{\mathrm{gau}}$-link is the study of **"conjugate synchronization"** via the **$\mathbb{F}^{\rtimes\pm}_l$-symmetry** of a $\Theta^{\pm\mathrm{ell}}\mathrm{NF}$-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 important 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 objects left *invariant* by] the $\Theta^{\times\mu}_{\mathrm{gau}}$-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.

## Cite this article

Shinichi Mochizuki, Inter-universal Teichmüller Theory II: Hodge–Arakelov-Theoretic Evaluation. Publ. Res. Inst. Math. Sci. 57 (2021), no. 1, pp. 209–401

DOI 10.4171/PRIMS/57-1-2