A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylor has initiated the topic of higher codimension Iwasawa theory. As a generalization of the classical Iwasawa main conjecture, they prove a relationship between analytic objects (a pair of Katz's 2-variable p-adic L-functions) and algebraic objects (two "everywhere unramified" Iwasawa modules) involving codimension two cycles in a 2-variable Iwasawa algebra. We prove a result by considering the restriction to an imaginary quadratic field K (where an odd prime p splits) of an elliptic curve E, defined over Q, with good supersingular reduction at p. On the analytic side, we consider eight pairs of 2-variable p-adic L-functions in this setup (four of the 2-variable p-adic L-functions have been constructed by Loeffler and a fifth 2-variable p-adic L-function is due to Hida). On the algebraic side, we consider modifications of fine Selmer groups over the Z 2 p -extension of K. We also provide numerical evidence, using algorithms of Pollack, towards a pseudo-nullity conjecture of Coates-Sujatha.1 The first subscript d of ρ d,n indicates the dimension of the Galois representation, while the second subscript n denotes a number one less than the Krull dimension of the ring R. In the settings we are interested in, the number n would denote the number of variables in the corresponding p-adic L-functions.2 The Panchishkin condition is a type of "ordinariness" assumption, introduced by Greenberg, while formulating the Iwasawa main conjecture for Galois deformations. See Section 4 in [14] for the precise definition. 2

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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 (i) the tensor product of two cuspidal Hida families F and G, and (ii) the tensor product of three cuspidal Hida families F , G and H. On the analytic side, we consider (i) a pair of 3-variable Rankin-Selberg p-adic L-functions constructed by Hida and (ii) a balanced 4-variable p-adic L-function (whose construction is forthcoming in a work of 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. 4 Description of Z(Q, D d d d,n ) and Z(Q, D ⋆ d d d,n