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 pseudonullity conjecture of Coates–Sujatha.
Highlights
IntroductionCorresponding to the tensor product of the Galois representation associated to the newform fE and ρK , Hida has constructed a two-variable Rankin–Selberg p-adic L-function, denoted by θ4G,2r , in the fraction field of the Iwasawa algebra Zp[[Γ ]]
Would produce a Zp[[Γ ]]-module that has rank at least two. Another interesting point to note is that when the root number of the elliptic curve E over K equals −1, it is known that ker(πac) belongs to the support of the divisors Div(θ4+,2+) and Div(θ4−,2−)
Because, as we indicate in Remark 8.5, this choice makes it transparent how the cyclotomic specializations of the 2-variable ++ and −− p-adic L-functions are related to certain one-variable p-adic L-functions of Rob Pollack
Summary
Corresponding to the tensor product of the Galois representation associated to the newform fE and ρK , Hida has constructed a two-variable Rankin–Selberg p-adic L-function, denoted by θ4G,2r , in the fraction field of the Iwasawa algebra Zp[[Γ ]]. A similar construction of X(Q, Dρ4,2 ) in these cases would produce a Zp[[Γ ]]-module that has rank at least two (it is exactly two if the Pontryagin dual of any of the corresponding Selmer groups is torsion over Zp[[Γ ]]) Another interesting point to note is that when the root number of the elliptic curve E over K equals −1, it is known that ker(πac) belongs to the support of the divisors Div(θ4+,2+) and Div(θ4−,2−). This choice is opposite to that of Kim and Kobayashi
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