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

Abstract

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 pseudo-nullity conjecture of Coates-Sujatha.

arXiv comment61 Pages, comments welcome. Incorporated comments and feedback from referees, correcting many inaccuracies and which led to many improvements in the paper

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Cited by in corpus (2)

• On the growth of Mordell-Weil ranks in $p$-adic Lie extensions
• Akashi series and Euler characteristics of signed Selmer groups of elliptic curves with semistable reduction at primes above p