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 in arXiv:2106.11452 and arXiv:2210.11635. 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, 2022b]. The next important point is the construction of Mochizuki's $\Theta_{gau}$-links and the set of such links, called Mochizuki's Ansatz in {\S}6. Theorem~6.9.1 establishes valuation scaling property satisfied by points of Mochizuki's Ansatz (i.e. by my version of $\Theta_{gau}$-links). These results lead to the construction of a theta-values set ({\S}8) which is similar to Mochizuki's Theta-values set (differences between the two are in {\S}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), $\mathfrak{log}$-links (Theorem~10.15.1) and Mochizuki's log-Kummer Indeterminacy (Theorem~10.20.1) in my theory.