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.