Codimension two cycles in Iwasawa theory and tensor product of Hida families

Abstract

The purpose of this paper is to build on results in higher codimension Iwasawa theory. The setting of our results involves Galois representations arising as cyclotomic twist deformations associated to (1) the tensor product of two cuspidal Hida families F and G, and (2) the tensor product of three cuspidal Hida families F, G and H. On the analytic side, we consider (1) a pair of 3-variable Rankin–Selberg p-adic L-functions constructed by Hida and (2) a balanced 4-variable p-adic L-function (due to Hsieh and Yamana) and an unbalanced 4-variable p-adic L-function (whose existence is currently conjectural). In each of these setups, when the two p-adic L-functions generate a height two ideal in the corresponding deformation ring, we use codimension two cycles of that ring to relate them to a pair of pseudo-null modules.