Torsion of rank-two $A$-motives values in odd characteristic cyclotomic towers

TL;DR

This paper extends classical theorems about torsion points of abelian varieties to the setting of rank-two $A$-motives and Drinfeld modules over function fields, showing finiteness results in cyclotomic towers.

Contribution

It provides analogues of Imai's and Ribet's theorems for rank-two $A$-motives and Drinfeld modules in odd characteristic, establishing finiteness of torsion points in cyclotomic extensions.

Findings

Finiteness of torsion points for rank-two $A$-motives over local fields.

Finiteness of torsion points of abelian Anderson $A$-modules.

Analogues of classical theorems in the function field setting.

Abstract

For rank-two $A$-motives defined over local fields with odd characteristic, we give an analogue of a theorem of Imai stating that abelian varieties with good reduction over $p$-adic fields have only finitely many torsion points values in cyclotomic towers. This implies the finiteness of torsion points of abelian Anderson $A$-modules. For rank-two Drinfeld $A$-modules over global function fields, we also give an analogue of a theorem of Ribet on torsion points of abelian varieties values in maximal cyclotomic extensions of number fields.