Controlling λ-invariants for the double and triple product p-adic L-functions

In the late 1990s, Vatsal showed that a congruence modulo p between two modular forms implied a congruence between their respective p-adic L-functions. We prove an analogous statement for both the double product and triple product p-adic L-functions, Lp(f ⊗ g) and Lp(f ⊗ g ⊗ h): the former is cyclotomic in its nature, while the latter is over the weight-space. As a corollary, we derive transition formulae relating analytic λ-invariants of congruent Galois representations for Vf⊗Vg, and for Vf⊗Vg⊗Vh, respectively.