On the Iwasawa main conjecture for the double product

Let σ \sigma and τ \tau denote a pair of absolutely irreducible p p -ordinary and p p -distinguished Galois representations into GL 2 ⁡ ( F ¯ p ) \operatorname {GL}_2(\overline {\mathbb {F}}_p) . Given two primitive forms ( f , g ) (f,g) such that wt ⁡ ( f ) > wt ⁡ ( g ) > 1 \operatorname {wt}(f)>\operatorname {wt}(g)> 1 and where ρ ¯ f ≅ σ \overline {\rho }_f\cong \sigma and ρ ¯ g ≅ τ \overline {\rho }_g\cong \tau , we show that the Iwasawa Main Conjecture for the double product ρ f ⊗ ρ g \rho _f\otimes \rho _g depends only on the residual Galois representation σ ⊗ τ : G Q → GL 4 ⁡ ( F ¯ p ) \sigma \otimes \tau : G_{\mathbb {Q}}\rightarrow \operatorname {GL}_4(\overline {\mathbb {F}}_p) . More precisely, if IMC( f ⊗ g f\otimes g ) is true for one pair ( f , g ) (f,g) with ρ ¯ f ≅ σ \overline {\rho }_f \cong \sigma and ρ ¯ g ≅ τ \overline {\rho }_g\cong \tau and whose μ \mu -invariant equals zero, then it is true for every congruent pair too.