Construction of Arithmetic Teichmuller spaces II: Towards Diophantine Estimates

Abstract

This paper deals with three consequences of the existence of Arithmetic Teichmuller spaces of arXiv:2106.11452. Let ${\mathcal{X}}_{F,{\mathbb{Q}}_p}$ (resp. $B=B_{{\mathbb{Q}}_p}$) be the complete Fargues-Fontaine curve (resp. the ring) constructed by Fargues-Fontaine with the datum $F={\mathbb{C}}_p^{♭}$ (the tilt of ${\mathbb{C}}_p$), $E={\mathbb{Q}}_p$. Fix an odd prime $\ell$, let ${\ell }^{\ast }=\frac{\ell -1}{2}$. The construction (§7) of an uncountable subset ${\Sigma }_F\subset {\mathcal{X}}_{F,{\mathbb{Q}}_p}^{{\ell }^{\ast }}$ with a simultaneous valuation scaling property (Theorem 7.8.1), Galois action and other symmetries. Now fix a Tate elliptic curve over a finite extension of ${\mathbb{Q}}_p$. The existence of ${\Sigma }_F$ leads to the construction (§9) of a set $\tilde{\Theta }\subset B^{{\ell }^{\ast }}$ consisting of lifts (to $B$), of values (lying in different untilts provided by ${\Sigma }_F$) of a chosen theta-function evaluated at $2\ell$-torsion points on the chosen elliptic curve. The construction of $\tilde{\Theta }$ can be easily adelized. Moreover I also prove a lower bound (Theorem 10.1.1) for the size of $\tilde{\Theta }$ (here size is defined in terms of the Fréchet structure of $B$). I also demonstrate (in §11) the existence of ``log-links'' in the theory of [Joshi 2021].

This paper is now replaced by arXiv:2303.01662 . This version: 23 pages; This is a preliminary version; comments are welcome

References in corpus (1)

• Probabilistic Szpiro, Baby Szpiro, and Explicit Szpiro from Mochizuki's Corollary 3.12