Height one specializations of Selmer groups

TL;DR

This paper studies the behavior of Selmer groups associated with tensor products of Hida families and classical forms under specialization, establishing control theorems and verifying consistency with main conjectures involving multivariable p-adic L-functions.

Contribution

It provides new control theorems relating Selmer groups of tensor products and their deformations, and confirms the compatibility of these results with Hida's multivariable p-adic L-functions and main conjectures.

Findings

Control theorems linking Selmer groups of tensor products and deformations.

Verification of the compatibility of p-adic L-functions with Selmer group behavior.

Consistency of specialization results with main conjectures in Iwasawa theory.

Abstract

We provide applications to studying the behavior of Selmer groups under specialization. We consider Selmer groups associated to four dimensional Galois representations coming from (i) the tensor product of two cuspidal Hida families $F$ and $G$, (ii) its cyclotomic deformation, (iii) the tensor product of a cusp form $f$ and the Hida family $G$, where $f$ is a classical specialization of $F$ with weight $k \geq 2$. We prove control theorems to relate (a) the Selmer group associated to the tensor product of Hida families $F$ and $G$ to the Selmer group associated to its cyclotomic deformation and (b) the Selmer group associated to the tensor product of $f$ and $G$ to the Selmer group associated to the tensor product of $F$ and $G$. On the analytic side of the main conjectures, Hida has constructed one variable, two variable and three variable Rankin-Selberg $p$-adic $L$-functions. Our specialization results enable us to verify that Hida's results relating (a) the two variable $p$-adic $L$-function to the three variable $p$-adic $L$-function and (b) the one variable $p$-adic $L$-function to the two variable $p$-adic $L$-function and our control theorems for Selmer groups are completely consistent with the main conjectures.