Integrality of $\mathrm{GL}_2\times\mathrm{GL}_2$ Rankin-Selberg integrals for ramified representations
Alexandros Groutides
TL;DR
This work defines a robust notion of integrality for local GL$_2\times$GL$_2$ Rankin–Selberg zeta integrals with ramified representations by introducing \(\Pi\)-integral data and a finite algebra \(A\) capturing Hecke and unitary character data. It proves a local integrality theorem: for tempered GL$_2$ pairs \(\pi_1,\pi_2\) and any \(\Pi=\pi_1\times\pi_2\)-integral datum, the normalized zeta integral decomposes as $$Z(\phi,g_1W_{\pi_1}^{\mathrm{new}},g_2W_{\pi_2}^{\mathrm{new}};s)=L(\Pi,s)\,\Phi(\cdot;q^s)$$ with \(\Phi\) an element of \(A[X,X^{-1}]\). The paper provides a detailed case‑by‑case analysis across the full ramification spectrum (unramified, Steinberg twists, half-/fully-ramified principal series, and supercuspidals) and shows the required integrality holds, balancing potential denominators via volume factors. It then derives a global automorphic consequence for newforms: the global Rankin–Selberg integral factors as the automorphic L-function times a finite polynomial in primes of ramification, with archimedean Gamma factors accounted for. A further application connects the integrality results to trilinear forms in the spirit of Prasad, yielding integral values for the unique trilinear functional arising from the zeta construction. The results sharpen the arithmetic understanding of Rankin–Selberg integrals in ramified settings and provide explicit, computable integral structures for both local and global objects.
Abstract
Let $π_1,π_2$ be irreducible admissible generic tempered representations of $\mathrm{GL}_2(F)$ for some $p$-adic field $F$ of odd residue characteristic. We introduce a natural notion of general $(π_1\timesπ_2)$-integral data $(φ,g_1,g_2)\in \mathcal{S}(F^2)\times\mathrm{GL}_2(F)^2$ at which the Rankin-Selberg integral can be evaluated. This is inspired by work of Loeffler, and previous work of the author, on unramified zeta integrals. We then establish an integral variant of a result of Jacquet-Langlands for the local Rankin-Selberg zeta integral associated to $π_1\timesπ_2$; i.e. we show that for any such integral datum $(φ,g_1,g_2)$, we have $$\frac{Z(φ,g_1W_{π_1}^\mathrm{new},g_2W_{π_2}^\mathrm{new};s)}{L(π_1\timesπ_2,s)}=Φ(φ,g_1W_{π_1}^\mathrm{new},g_2W_{π_2}^\mathrm{new};q^s)\in \mathbf{Z}[q^{-1},Σ^1][q^s,q^{-s}]$$for a finite set $Σ^1\subseteq\mathbf{C}^\times$ of roots of unity and unitary character values, depending only on $π_1,π_2$. This is compatible with the notion of integrality coming from newforms $f_1,f_2$ of even integral weights, satisfying a mild local dihedral condition at $2$. We show that if $π_1,π_2$ are local pieces of $f_1,f_2$ at any prime $p$, the coefficient algebra is $\mathcal{O}_{K}[p^{-1}]$ with $K$ a number field only depending on $f_1,f_2$. Our approach relies on a reinterpretation of the local Rankin-Selberg integral, and works of Assing and Saha on values of $p$-adic Whittaker new-vectors.