Truncated Euler systems over imaginary quadratic fields

Abstract

Let $K$ be an imaginary quadratic field and let $F$ be an abelian extension of $K$. It is known that the order of the class group $\textup{Cl}_{F}$ of $F$ is equal to the order of the quotient $U_{F}/El_{F}$ of the group of global units $U_{F}$ by the group of elliptic units $El_{F}$ of $F$. We introduce a filtration on $U_{F}/El_{F}$ made from the so-called truncated Euler systems and conjecture that the associated graded module is isomorphic, as a Galois module, to the class group. We provide evidence for the conjecture using Iwasawa theory.