Classifying lambda-modules up to isomorphism and applications to Iwasawa theory

In Iwasawa theory, one studies how an arithmetic or geometric object grows as its field of definition varies over certain sequences of number fields. For example, let F/ Q be a finite extension of fields, and let E : y2 = x3 + Ax + B with A, B ∈ F be an elliptic curve. If F = F0 ⊆ F1 ⊆ F2 ⊆ ··· Finfinity = ∪infinityi=0 Fi, one may be interested in properties like the ranks and torsion subgroups of the increasing family of curves E(F0) ⊆ E( F1) ⊆ ··· ⊆ E( Finfinity). The main technique for studying this sequence of curves when Gal(Finfinity/F) has a p-adic analytic structure is to use the action of Gal( Fn/F) on E(Fn) and the Galois cohomology groups attached to E, i.e. the Selmer and Tate-Shafarevich groups. As n varies, these Galois actions fit into a coherent family, and taking a direct limit one obtains a short exact sequence of modules 0→EFinfinity ⊗Qp/Z p→ SelEFinfinity p→IIIE Finfinity p→0 over the profinite group algebra Zp [[Gal(Finfinity/F)]]. When Gal(Finfinity/F) ≅ Zp , this ring is isomorphic to Λ = Zp [[T]], and the Λ-module structure of Sel E(Finfinity)p and IIIE(Finfinity) p encode all the information about the curves E( Fn) as n varies.;In this dissertation, it will be shown how one can classify certain finitely generated Λ-modules with fixed characteristic polynomial f (T) ∈ Zp [T] up to isomorphism. The results yield explicit generators for each module up to isomorphism. As an application, it is shown how to identify the isomorphism class of SelE( Qinfinity )p in this explicit form, where Qinfinity is the cyclotomic Zp -extension of Q , and E is an elliptic curve over Q with good ordinary reduction at p, and possessing the property that E( Q ) has no p-torsion.