Codimension Two Cycles in Iwasawa Theory and Elliptic Curves With Supersingular Reduction

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 Zₚ² -extension of K . We also provide numerical evidence, using algorithms of Pollack, towards a pseudonullity conjecture of Coates–Sujatha.