The main conjectures of Iwasawa theory provide the only general method known at present for studying the mysterious relationship between purely arithmetic problems and the special values of complex \(L\)-functions, typified by the conjecture of Birch and Swinnerton-Dyer and its generalizations. Over the years, the Iwasawa main conjecture has been formulated in various setups and various guises. The underlying principle in each formulation has been to relate objects on the algebraic side to the objects on the analytic side. On the algebraic side of Iwasawa theory, one studies modules over Iwasawa algebras. An Iwasawa algebra is a completed group ring \(\mathbb{Z}_p[[G]]\), for some \(p\)-adic Lie group \(G\). On the analytic side, one studies \(p\)-adic \(L\)-functions. The \(p\)-adic \(L\)-functions are believed to satisfy certain integrality properties. For example, consider the case when the group \(G\) is isomorphic to \(\mathbb{Z}_p\times \Delta\), for some finite abelian group \(\Delta\). Under suitable conditions, the \(p\)-adic \(L\)-function is known to be a measure. Their results in this paper are motivated by similar integrality properties of \(p\)-adic \(L\)-functions, in the non-commutative setting, as predicted by the non-commutative Iwasawa main conjectures.
Let \(\Lambda\) (isomorphic to \(\mathbb{Z}_p[[T]]\)) denote the usual Iwasawa algebra and \(G\) denote the Galois group of a finite Galois extension \(L/K\) of totally real fields. When the non-primitive Iwasawa module over the cyclotomic \(\mathbb{Z}_p\)-extension has a free resolution of length one over the group ring \(\Lambda[G]\), the authors prove that the validity of the non-commutative Iwasawa main conjecture allows us to find a representative for the non-primitive \(p\)-adic \(L\)-function (which is an element of a \(K_1\)-group) in a maximal \(\Lambda\)-order. This integrality result involves a study of the Dieudonné determinant. Using a cohomological criterion of Greenberg, they also deduce the precise conditions under which the non-primitive Iwasawa module has a free resolution of length one. As one application of the last result, the authors consider an elliptic curve over \(\mathbb{Q}\) with a cyclic isogeny of degree \(p^2\). They relate the characteristic ideal in the ring \(\Lambda\) of the Pontryagin dual of its non-primitive Selmer group to two characteristic ideals, viewed as elements of group rings over \(\Lambda\), associated to two non-primitive classical Iwasawa modules.