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].
Categories: math.AG math.NT