Inter-universal Teichmüller Theory I: Construction of Hodge Theaters

Abstract

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 – which we refer to as "inter-universal Teichmüller theory" – by applying the theory of semi-graphs of anabelioids, Frobenioids, the étale theta function, and log-shells developed in earlier papers by the author. We begin by fixing what we call "initial $\Theta$-data", which consists of an elliptic curve $E_F$ over a number field $F$, and a prime number $l\ge 5$, as well as some other technical data satisfying certain technical properties. This data determines various hyperbolic orbicurves that are related via finite étale coverings to the once-punctured elliptic curve $X_F$ determined by $E_F$. These finite étale coverings admit various symmetry properties arising from the additive and multiplicative structures on the ring ${\mathbb{F}}_l=\mathbb{Z}/l\mathbb{Z}$ acting on the $l$-torsion points of the elliptic curve. We then construct "${\Theta }^{\pm \mathrm{ell}}\mathrm{NF}$-Hodge theaters" associated to the given $\Theta$-data. These ${\Theta }^{\pm \mathrm{ell}}\mathrm{NF}$-Hodge theaters may be thought of as miniature models of conventional scheme theory in which the two underlying combinatorial dimensions of a number field – which may be thought of as corresponding to the additive and multiplicative structures of a ring or, alternatively, to the group of units and value group of a local field associated to the number field – are, in some sense, "dismantled" or "disentangled" from one another. All ${\Theta }^{\pm \mathrm{ell}}\mathrm{NF}$-Hodge theaters are isomorphic to one another, but may also be related to one another by means of a "$\Theta$-link", which relates certain Frobenioid-theoretic portions of one ${\Theta }^{\pm \mathrm{ell}}\mathrm{NF}$-Hodge theater to another in a fashion that is not compatible with the respective conventional ring/scheme theory structures. In particular, it is a highly nontrivial problem to relate the ring structures on either side of the $\Theta$-link to one another. This will be achieved, up to certain "relatively mild indeterminacies", in future papers in the series by applying the absolute anabelian geometry developed in earlier papers by the author. The resulting description of an "alien ring structure" [associated, say, to the domain of the $\Theta$-link] in terms of a given ring structure [associated, say, to the codomain of the $\Theta$-link] will be applied in the final paper of the series to obtain results in diophantine geometry. Finally, we discuss certain technical results concerning profinite conjugates of decomposition and inertia groups in the tempered fundamental group of a $p$-adic hyperbolic curve that will be of use in the development of the theory of the present series of papers, but are also of independent interest.