Chain modular units and explicit unit groups in Ray class fields over imaginary quadratic fields

Abstract

We introduce chain modular units: products of Siegel functions arranged along cyclic chains of ℓ-isogenies between CM elliptic curves of type Of. Their CM values take place in the ray class field Kfℓn, and we construct a finite, explicit family that generates, modulo roots of unity, a subgroup of finite index in OKfℓn×. We describe the Galois action via the ideal class action on isogeny chains, give effective criteria for multiplicative independence using Baker-Wustholz bounds, and determine the eventual index in the ℓ-power ray class tower. Computations for D = -7 and D = -11 confirm the theoretical predictions. Our approach exploits the combinatorics of isogeny chains to control conductor growth and produce many independent units.