Iwasawa theory for Rankin-Selberg product 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.