On Selmer groups and factoring $p$-adic $L$-functions

TL;DR

This paper explores the relationship between Selmer groups and factorizations of $p$-adic $L$-functions, providing evidence for main conjectures related to various Galois representations and utilizing specialization techniques.

Contribution

It establishes Selmer group results consistent with main conjectures for 3- and 4-dimensional Galois representations, building on Dasgupta's factorization of $p$-adic $L$-functions.

Findings

Selmer groups behave predictably under specialization.

Results support main conjectures for certain Galois representations.

Connections between $p$-adic $L$-functions and Selmer groups are clarified.

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. One key technical input to our methods is studying the behavior of Selmer groups under specialization.