Integrality of ${\mathrm{G}\mathrm{L}}_2\times {\mathrm{G}\mathrm{L}}_2$ Rankin-Selberg integrals for ramified representations

Let ${\pi }_1,{\pi }_2$ be irreducible admissible generic tempered representations of ${\mathrm{G}\mathrm{L}}_2(F)$ for some finite extension $F\mathrm{/}{\mathbf{Q}}_p$ 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 $({\pi }_1\times {\pi }_2)$-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 $({\pi }_1\times {\pi }_2)$-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 $p$-adic Whittaker new vectors.