On the Iwasawa main conjecture for the double product

Let σ and τ denote a pair of absolutely irreducible p p -ordinary and p p -distinguished Galois representations into GL 2 ⁡ (F ¯ p) GL₂ ({F}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 f and ρ ¯ g ≅ τ g, we show that the Iwasawa Main Conjecture for the double product ρ f ⊗ ρ g f g depends only on the residual Galois representation σ ⊗ τ: G Q → GL 4 ⁡ (F ¯ p): G ₐ GL₄ ({F}ₚ). More precisely, if IMC (f ⊗ g f g) is true for one pair (f, g) (f, g) with ρ ¯ f ≅ σ f and ρ ¯ g ≅ τ g and whose μ -invariant equals zero, then it is true for every congruent pair too.