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

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 ${\Theta }_{\mathrm{gau}}^{\times \mu }$-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 ${\Theta }_{\mathrm{gau}}^{\times \mu }$-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!] ${\Theta }_{\mathrm{gau}}^{\times \mu }$-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 ${\Theta }_{\mathrm{gau}}^{\times \mu }$-link is the study of "conjugate synchronization" via the ${\mathbb{F}}_l^{⋊\pm }$-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 ${\Theta }_{\mathrm{gau}}^{\times \mu }$-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.