On the Iwasawa main conjecture for the double product

Let σ σ and τ τ denote a pair of absolutely irreducible p p -ordinary and p p -distinguished Galois representations into GL 2 ⁡ ( F ¯ p ) GL_2(F̄_p) . Given two primitive forms ( f , g ) (f,g) such that wt ⁡ ( f ) > wt ⁡ ( g ) > 1 wt(f)>wt(g)> 1 and where ρ ¯ f ≅ σ ρ ̄_f σ and ρ ¯ g ≅ τ ρ ̄_g τ , we show that the Iwasawa Main Conjecture for the double product ρ f ⊗ ρ g ρ _f⊗ ρ _g depends only on the residual Galois representation σ ⊗ τ : G Q → GL 4 ⁡ ( F ¯ p ) σ ⊗ τ : GQ→ GL_4(F̄_p) . More precisely, if IMC( f ⊗ g f⊗ g ) is true for one pair ( f , g ) (f,g) with ρ ¯ f ≅ σ ρ ̄_f σ and ρ ¯ g ≅ τ ρ ̄_g τ and whose μ μ -invariant equals zero, then it is true for every congruent pair too.