Kummer-faithfulness for function fields

2023/03/08 by Asayama, Takuya
#11R58 #11S15 (Secondary) #12E30 (Primary) 11G09 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2303.04396

Abstract

A perfect field K is said to be Kummer-faithful if the Mordell-Weil group of every semi-abelian variety over every finite extension of K has no nonzero divisible element. The class of Kummer-faithful fields contains that of sub-p-adic fields and is thought to be suitable for developing anabelian geometry. In this paper, we investigate a function field analogue of the notion of Kummer-faithful fields. We introduce a notion of Drinfeld-Kummer-faithful (DKF) fields using Drinfeld modules. A sufficient condition for a Galois extension of a function field to be DKF is provided in terms of ramification theory. More precisely, a Galois extension with finite maximal ramification break outside the infinite prime (1 / t) over a finite extension of the rational function field \mathbbF_q(t) over the finite field \mathbbF_q of q elements is DKF. Some examples of DKF fields are also given. The construction of these examples is inspired by Ozeki and Taguchi's examples of highly Kummer-faithful fields.

Related

• Kummer-faithfulness over p-adic fields
• A note on highly Kummer-faithful fields
• Kummer theory for function fields
• Mordell--Weil groups over large algebraic extensions of fields of characteristic zero
• Some Kummer extensions over maximal cyclotomic fields, a finiteness theorem of Ribet and TKND-AVKF fields