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 $\mathscr{X}_{F,\mathbb{Q}_p}$ (resp. $B=B_{\mathbb{Q}_p}$) be the complete Fargues-Fontaine curve (resp. the ring) constructed by Fargues-Fontaine with the datum $F={\mathbb{C}_p^\flat}$ (the tilt of $\mathbb{C}_p$), $E=\mathbb{Q}_p$. Fix an odd prime $\ell$, let $\ell^*=\frac{\ell-1}{2}$. The construction (\S 7) of an uncountable subset $\Sigma_{F}\subset \mathscr{X}_{F,\mathbb{Q}_p}^{\ell^*}$ 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 (\S 9) of a set $\widetilde{\Theta}\subset B^{\ell^*}$ 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 $\widetilde{\Theta}$ can be easily adelized. Moreover I also prove a lower bound (Theorem 10.1.1) for the size of $\widetilde{\Theta}$ (here size is defined in terms of the Fr\'echet structure of $B$). I also demonstrate (in \S 11) the existence of ``log-links'' in the theory of [Joshi 2021].