2021/11/08 by Kirti Joshi, Joshi, Kirti · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2111.04890
openalex publication_date 2021/11/08 · openalex created_date 2022/05/05 · openalex updated_date 2026/07/28
This paper deals with three consequences of the existence of Arithmetic\nTeichmuller spaces of arXiv:2106.11452. Let mathscrXF,\ℚp\n(resp. B=B\ℚp) be the complete Fargues-Fontaine curve (resp. the\nring) constructed by Fargues-Fontaine with the datum F=\ℂp^ flat\n(the tilt of \ℂp), E=\ℚp. Fix an odd prime \ℓ, let\n\ℓ^*=\(\ℓ-1)/(2). The construction ( S 7) of an uncountable subset\n\ΣF\⊂ mathscrXF,\ℚp\ℓ^* with a simultaneous\nvaluation scaling property (Theorem 7.8.1), Galois action and other symmetries.\n Now fix a Tate elliptic curve over a finite extension of \ℚp. The\nexistence of \ΣF leads to the construction ( S 9) of a set\n widetilde\Θ\⊂ B\ℓ^* consisting of lifts (to B), of values\n(lying in different untilts provided by \ΣF) of a chosen\ntheta-function evaluated at 2\ℓ-torsion points on the chosen elliptic\ncurve. The construction of widetilde\Θ can be easily adelized.\nMoreover I also prove a lower bound (Theorem 10.1.1) for the size of\n widetilde\Θ (here size is defined in terms of the Fr 'echet structure\nof B).\n I also demonstrate (in S 11) the existence of ``log-links'' in the theory of\n[Joshi 2021].\n