Construction of Arithmetic Teichmuller spaces II: Towards Diophantine Estimates

TL;DR

This paper explores the construction of arithmetic Teichmuller spaces and their implications for Diophantine estimates, including the creation of special sets with valuation properties and bounds on their size, advancing number theory and p-adic geometry.

Contribution

It introduces a new uncountable subset with valuation scaling, constructs lifts of theta-values on elliptic curves, and establishes size bounds, extending the framework of arithmetic Teichmuller spaces.

Findings

Construction of an uncountable subset with valuation scaling property

Development of a set of lifts of theta-values on elliptic curves

Proven lower bounds for the size of the constructed set

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].