Construction of Arithmetic Teichmuller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12

Abstract

This paper deals with consequences of the existence of Arithmetic Teichmuller spaces established here and here. Theorem 9.2.1 provides a proof of a local version of Mochizuki's Corollary 3.12. Local means for a fixed p-adic field. There are several new innovations in this paper. Some of the main results are as follows. Theorem 3.5.1 shows that one can view the Tate parameter of Tate elliptic curve as a function on the arithmetic Teichmuller space of [Joshi, 2021a], [Joshi, 2022]. The next important point is the construction of Mochizuki's Θ gau -links and the set of such links, called Mochizuki's Ansatz in § 6. Theorem 6.9.1 establishes valuation scaling property satisfied by points of Mochizuki's Ansatz (i.e. by my version of Θ gau -links). These results lead to the construction of a thetavalues set ( § 8) which is similar to Mochizuki's Theta-values set (differences between the two are in § 8.7.1). Finally Theorem 9.2.1 is established. For completeness, I provide an intrinsic proof of the existence of Mochizuki's log-links (Theorem 10.9.1), log-links (Theorem 10.15.1) and Mochizuki's log-Kummer Indeterminacy (Theorem 10.20.1) in my theory.