Integrality of GL₂×GL₂ Rankin-Selberg integrals for ramified representations

Let π 1 , π 2 \pi _1,\pi _2 be irreducible admissible generic tempered representations of G L 2 ( F ) \mathrm {GL}_2(F) for some finite extension F / Q p F/\mathbf {Q}_p of odd residue characteristic. Inspired by work of Loeffler and previous work of the author on unramified zeta-integrals, we introduce a natural general notion of ( π 1 × π 2 ) (\pi _1\times \pi _2) - integral data at which the Rankin-Selberg zeta-integral can be evaluated. We then establish an integral refinement of Jacquet-Langland’s GCD-result for this zeta-integral, when evaluated at ( π 1 × π 2 ) (\pi _1\times \pi _2) -integral data. This is compatible with the notion of integrality coming from the Fourier coefficients of newforms of even integral weights. Our approach relies on a reinterpretation of the Rankin-Selberg zeta-integral, and works of Assing and Saha on values of p p -adic Whittaker new vectors.