Construction of Arithmetic Teichmuller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12
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arXiv
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2023
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| _version_ | 1866915168291651584 |
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| author | Joshi, Kirti |
| author_facet | Joshi, Kirti |
| contents | 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 $Θ_{gau}$-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 theta-values 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), $\mathfrak{log}$-links (Theorem~10.15.1) and Mochizuki's log-Kummer Indeterminacy (Theorem~10.20.1) in my theory. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_01662 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Construction of Arithmetic Teichmuller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12 Joshi, Kirti Algebraic Geometry Number Theory 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 $Θ_{gau}$-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 theta-values 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), $\mathfrak{log}$-links (Theorem~10.15.1) and Mochizuki's log-Kummer Indeterminacy (Theorem~10.20.1) in my theory. |
| title | Construction of Arithmetic Teichmuller spaces II: Proof of a local prototype of Mochizuki's Corollary 3.12 |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2303.01662 |