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A result of Bleher, Chinburg, Greenberg, Kakde, Pappas, Sharifi and Taylorhas initiated the topic of higher codimension Iwasawa theory. As ageneralization of the classical Iwasawa main conjecture, they prove arelationship between analytic objects (a pair of Katz's $2$-variable $p$-adic$L$-functions) and algebraic objects (two "everywhere unramified'' Iwasawamodules) involving codimension two cycles in a $2$-variable Iwasawa algebra. Weprove 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 byLoeffler and a fifth $2$-variable $p$-adic $L$-function is due to Hida). On thealgebraic side, we consider modifications of fine Selmer groups over the$\mathbb{Z}_p^2$-extension of $K$. We also provide numerical evidence, usingalgorithms of Pollack, towards a pseudo-nullity conjecture of Coates-Sujatha.

Codimension two cycles in Iwasawa theory and elliptic curves with supersingular reduction