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
$\mathbb{Q}$, 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
$\mathbb{Z}_{p}^{2}$-extension of
$K$. We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.