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

Technical Summary: Integrality of GL2 × GL2 Rankin-Selberg Integrals for Ramified Representations

Problem Statement
The paper addresses the question of integrality for local Rankin-Selberg zeta-integrals associated with pairs of irreducible admissible generic tempered representations $\pi_1, \pi_2$ of $GL_2(F)$, where $F$ is a finite extension of $\mathbb{Q}_p$ ($p$ odd). While the local Rankin-Selberg integral $Z(\phi, g_1 W^{\text{new}}_{\pi_1}, g_2 W^{\text{new}}_{\pi_2}; s)$ is known to factorize as $L(\pi_1 \times \pi_2, s) \cdot \Phi(\dots; q^s)$, where $\Phi$ is a rational function, the arithmetic nature of the coefficients of $\Phi$ remains subtle.

Specifically, the author notes that the "naive" notion of integrality fails: even if the input data (Schwartz function $\phi$ and group elements $g_1, g_2$) are defined over $\mathbb{Z}$, the resulting polynomial $\Phi$ does not necessarily have coefficients in the ring of integers of a number field (possibly localized at $p$). The central problem is to identify a refined notion of "integral data" such that the resulting zeta-integral, when divided by the local $L$-factor, yields a polynomial with coefficients in a specific arithmetic ring $A$. This is motivated by the need to establish integrality properties compatible with Fourier coefficients of newforms and to support the theory of Rankin-Selberg Euler systems.

Methodology
The approach relies on a technical reinterpretation of the Rankin-Selberg integral and a reduction to the study of specific Whittaker new vectors.

1. Reinterpretation of the Integral: The author constructs a commutative diagram relating the standard Rankin-Selberg zeta-integral $Z_\Pi$ to a linear map $\Lambda_\Pi$ acting on a space of compactly induced functions. This involves:

   • Defining a map $\Xi_c$ that transforms the tensor product of Schwartz functions and characteristic functions into a space of functions on $P(F) \backslash G(F) / K_\Pi$.
   • Utilizing the unfolding of the integral to relate $Z_\Pi$ to an integral involving a "new" Whittaker function $I^{\text{new}}_\Pi$.
   • Establishing that the integrality of the original zeta-integral reduces to the integrality of the term $I^{\text{new}}_\Pi(\gamma; s)$ evaluated on specific coset representatives $\gamma$, scaled by the inverse volume of the stabilizer subgroup.

2. Case-by-Case Analysis: The core of the proof involves a detailed, exhaustive case-by-case analysis of the behavior of Whittaker new vectors $W^{\text{new}}_\pi$ for all classes of irreducible admissible generic representations of $GL_2(F)$. The author leverages explicit formulas for these vectors found in the works of Assing [Ass19b, Ass19a] and Saha [Sah16]. The cases covered include:

   • Unramified and ramified principal series representations (half-ramified, fully-ramified).
   • Steinberg representations (unramified and ramified twists).
   • Supercuspidal representations (associated with quadratic extensions).
   • All possible tensor products $\pi_1 \times \pi_2$ of these types.

3. Gauss Sums and Volume Factors: In each case, the author evaluates the relevant integrals, which often reduce to partial Gauss sums or generalized Kloosterman sums. A critical technical step is demonstrating that the potential denominators (specifically factors of $q-1$ arising from Gauss sums) are always cancelled by the volume factors of the stabilizer subgroups $P(F) \cap \gamma K_\Pi \gamma^{-1}$.

Key Contributions and Results

• Definition of Integral Data: The paper introduces a natural notion of $(\pi_1 \times \pi_2)$-integral data. A datum $(\phi, g_1, g_2)$ is defined as integral if the Schwartz function $\phi$ takes values in a specific principal ideal determined by the volume of the stabilizer of $\phi$ intersected with the conjugated conductor subgroup $K_\Pi$.
• Theorem A (Local Integrality): For any pair of irreducible admissible generic tempered representations, and for any $\Pi$-integral datum, the polynomial $\Phi$ appearing in the factorization of the zeta-integral lies in $A[X, X^{-1}]$, where $A$ is a $\mathbb{Z}[q^{-1}, \mu_{\nu}q^\tau]$-algebra containing the spherical Hecke eigenvalues and specific unramified unitary characters determined by the representations. Here $\tau = \max(c(\pi_1), c(\pi_2))$ and $\nu$ depends on whether the representations are unramified.
  • This result holds for $p=2$ under the additional assumption that supercuspidal representations are dihedral.
• Theorem B (Global Application): The local result is applied to global automorphic forms. For normalized cuspidal new eigenforms $f_1, f_2$ of even weights, the global Rankin-Selberg integral $I(\phi_A, g_1 \phi_{f_1}, g_2 \phi_{f_2}; s)$ is shown to be equal to $L(\pi_{f_1} \times \pi_{f_2}, s)$ multiplied by a polynomial $\Phi$ with coefficients in the ring of integers of a specific number field $E_{f_1, f_2}$ (localized at primes in a finite set $S$).
• Trilinear Forms: The paper establishes an integrality result for the unique (up to scalars) trilinear form in $\text{Hom}_{GL_2(F)}(\pi_1 \otimes \pi_2 \otimes \pi_3, 1)$, realized via the Rankin-Selberg integral, showing it takes values in the ring $A$ on integral new vectors.

Significance and Claims
The author claims that this work provides an "integral refinement" of the Jacquet-Langlands result regarding the greatest common divisor of Rankin-Selberg integrals. The significance is twofold:

1. Representation-Theoretic: It resolves the question of integrality for ramified representations, a setting where naive integrality fails. The result is described as a "natural problem of independent interest through the lenses of integral structures in local representation theory."
2. Arithmetic Geometry: The notion of integral data introduced here has a global algebro-geometric interpretation. The paper notes that locally integral data correspond precisely to classes in the image of the integral étale cohomology of modular curves under the Loeffler-Skinner-Zerbes (LSZ) map. Consequently, Theorem B is presented as the "analogous integrality result on the representation-theoretic side" for the theory of Rankin-Selberg Euler systems and $p$-adic $L$-functions.

The paper remains modest regarding the converse: it acknowledges that while the image of the zeta-integral on integral data is contained in the principal fractional ideal generated by the $L$-factor, it is not yet known if this image is the entire ideal or if it is principal. This is identified as an open problem with potential implications for optimizing local factors in Euler system theory.