On Selmer groups and factoring p-adic L-functions

Abstract

Samit Dasgupta has proved a formula factoring a certain restriction of a 3-variable Rankin-Selberg p-adic L-function as a product of a 2-variable p-adic L-function related to the adjoint representation of a Hida family and a Kubota-Leopoldt p-adic L-function. We prove a result involving Selmer groups that along with Dasgupta's result is consistent with the main conjectures associated to the 4-dimensional representation (to which the 3-variable p-adic L-function is associated), the 3-dimensional representation (to which the 2-variable p-adic L-function is associated) and the 1-dimensional representation (to which the Kubota-Leopoldt p-adic L-function is associated). Under certain additional hypotheses, we indicate how one can use work of Urban to deduce main conjectures for the 3-dimensional representation and the 4-dimensional representation. Dasgupta's method of proof is based on an earlier work of Gross in 1980. Gross's work involved factoring a certain restriction of a 2-variable p-adic L-function associated to an imaginary quadratic field (constructed by Katz) into a product of two Kubota-Leopoldt p-adic L-functions. In 1982, Greenberg proved the corresponding result on the algebraic side involving classical Iwasawa modules, as predicted by the main conjectures for imaginary quadratic fields and Q. Our methods are inspired by this work of Greenberg. One key technical input to our methods is studying the behavior of Selmer groups under specialization.