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 $\mathbb{Q}$ , 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.