Iwasawa theory for Rankin-Selberg convolution at an Eisenstein prime

Let $p$ be an odd prime, $f$ be a $p$-ordinary newform of weight $k$ and $h$ be a normalized cuspidal $p$-ordinary Hecke eigenform of weight $l < k$. In this article, we study the $p$-adic $L$-function and $p^{\infty}$-Selmer group of the Rankin-Selberg product of $f$ and $h$ under the assumption that $p$ is an Eisenstein prime for $h$ i.e. the residual Galois representation of $h$ at $p$ is reducible. We show that the $p$-adic $L$-function and the characteristic ideal of the $p^\infty$-Selmer group of the Rankin-Selberg product of $f, h$ generate the same ideal modulo $p$ in the Iwasawa algebra i.e. the Rankin-Selberg Iwasawa main conjecture for $f \otimes h$ holds mod $p$. As an application to our results, we explicitly describe a few examples where the above congruence holds.