On Rankin-Selberg integral structures and Euler systems for GL₂× GL₂

We study how Rankin-Selberg periods interact with integral structures in spherical Whittaker type representations. Using this representation-theoretic framework, we show that the local Euler factors appearing in the construction of the motivic Rankin-Selberg Euler system for a product of modular forms are integrally optimal; i.e. any construction of this type with any choice of integral input data would give local factors appearing in tame norm relations at p which are integrally divisible by the Euler factor Pₚ'(Frobₚ⁻¹) modulo $p-1$.