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

Alexandros Groutides

arxiv: 2501.16972 · v2 · submitted 2025-01-28 · 🧮 math.NT · math.RT

Integrality of GL₂timesGL₂ Rankin-Selberg integrals for ramified representations

Alexandros Groutides This is my paper

Pith reviewed 2026-05-23 04:50 UTC · model grok-4.3

classification 🧮 math.NT math.RT

keywords Rankin-Selberg zeta-integralGL(2) representationsintegralityJacquet-Langlandsp-adic Whittaker new vectorsramified representationsnewformstempered representations

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The pith

The Rankin-Selberg zeta-integral for GL2 × GL2 evaluates to integers at (π1 × π2)-integral data for tempered representations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper defines a general notion of (π1 × π2)-integral data for the Rankin-Selberg zeta-integral attached to a pair of irreducible admissible generic tempered representations of GL2 over a finite extension of Qp. It proves that the integral takes values in the ring of integers at such data, refining the GCD result of Jacquet and Langlands to an integrality statement. The construction is shown to be compatible with the integrality of Fourier coefficients of newforms of even integral weights. The argument proceeds by reinterpreting the zeta-integral and applying results on p-adic Whittaker new vectors.

Core claim

For irreducible admissible generic tempered representations π1 and π2 of GL2(F) where F is a finite extension of Qp of odd residue characteristic, the Rankin-Selberg zeta-integral evaluated at (π1 × π2)-integral data lies in the ring of integers, giving an integral refinement of Jacquet-Langlands' GCD-result.

What carries the argument

The notion of (π1 × π2)-integral data, which permits evaluation of the zeta-integral while preserving integrality and extends the unramified case.

If this is right

The integrality holds for both ramified and unramified representations.
The result is compatible with integrality coming from Fourier coefficients of newforms of even integral weights.
The reinterpretation of the zeta-integral combined with p-adic Whittaker new vector results yields the refinement.
The construction generalizes earlier work on unramified zeta-integrals to the ramified setting.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This integrality may support the definition of integral structures or measures attached to automorphic L-functions.
The refinement could connect to special-value problems in the arithmetic of motives or elliptic curves over number fields.
Direct computation of the integral for small primes and low conductor representations would provide concrete checks.

Load-bearing premise

The representations π1 and π2 are irreducible admissible generic tempered representations of GL2(F) for F a finite extension of Qp with odd residue characteristic.

What would settle it

An explicit pair of such representations together with (π1 × π2)-integral data where the zeta-integral lies outside the ring of integers would falsify the claim.

read the original abstract

Let $\pi_1,\pi_2$ be irreducible admissible generic tempered representations of $\mathrm{GL}_2(F)$ for some finite extension $F/\mathbf{Q}_p$ 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 $(\pi_1\times\pi_2)$-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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This extends the author's unramified integrality work to ramified GL2 x GL2 zeta integrals via a new notion of integral data and Assing-Saha's Whittaker vectors.

read the letter

The paper defines a class of (π1 × π2)-integral data for tempered generic irreducible representations of GL2 over p-adic fields with odd residue characteristic, then proves that the Rankin-Selberg zeta integral at those points satisfies an integrality statement refining the Jacquet-Langlands GCD result. It reinterprets the integral and invokes Assing-Saha on p-adic new vectors to get the result, with explicit compatibility to the integrality coming from newform Fourier coefficients of even weight. That compatibility is the part that could matter for people doing p-adic interpolation of L-functions. The hypotheses are stated up front and match the tools used, so there is no obvious mismatch or post-hoc fitting visible in the setup. The argument stays within the existing program rather than introducing new machinery from scratch. The main limitation is the narrow scope: it is a local, technical refinement that assumes the external results apply cleanly, and the strength of the conclusion depends on how tightly those results fit the new data class. No circularity appears in the outline. This is for specialists already working on local zeta integrals, Whittaker models, or Iwasawa-theoretic applications of automorphic forms. A reader who knows the unramified case and Assing-Saha will see the incremental value immediately. It is a solid, self-contained technical step that deserves a serious referee who can check the reinterpretation and the invocation of the cited results.

Referee Report

0 major / 3 minor

Summary. The paper introduces a general notion of (π1 × π2)-integral data for the local Rankin-Selberg zeta integral attached to a pair of irreducible admissible generic tempered representations π1, π2 of GL2(F), F/Qp finite with odd residue characteristic. It proves that the zeta integral evaluated at such data satisfies an integrality statement that refines the Jacquet-Langlands GCD result, with compatibility to the integrality of newform Fourier coefficients of even weight. The argument proceeds by reinterpreting the zeta integral and applying results of Assing-Saha on p-adic Whittaker new vectors.

Significance. If the central claim holds, the work supplies a natural p-adic integrality refinement for ramified local zeta integrals, extending the author's prior unramified results and Loeffler's framework. The explicit reliance on Assing-Saha Whittaker new-vector theorems provides a concrete, checkable foundation rather than an ad-hoc construction. This strengthens the local input for p-adic L-functions and integral structures on automorphic forms.

minor comments (3)

The definition of (π1 × π2)-integral data (presumably in §2 or §3) should include an explicit comparison with the unramified case treated in the author's earlier work, to clarify the precise extension.
The compatibility statement with newform Fourier coefficients (mentioned in the abstract) would benefit from a short dedicated paragraph or remark citing the precise normalization of newforms used.
A brief remark on the necessity of the odd-residue-characteristic hypothesis (beyond the applicability of Assing-Saha) would help readers assess the scope.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We are grateful to the referee for their positive assessment of the manuscript and for recommending it for minor revision. The report does not raise any specific major comments or concerns. Accordingly, we do not propose any changes to the manuscript at this stage.

Circularity Check

1 steps flagged

Minor self-citation to author's prior unramified work; central derivation relies on external Assing-Saha results

specific steps

self citation load bearing [Abstract]
"Inspired by work of Loeffler and previous work of the author on unramified zeta-integrals, we introduce a natural general notion of (π1×π2)-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 (π1×π2)-integral data."

The notion of integral data draws inspiration from the author's prior work, but the load-bearing steps for the refinement itself are the reinterpretation plus Assing-Saha citations (external to the present paper). This qualifies only as a minor, non-load-bearing self-citation under the scoring guidelines.

full rationale

The paper defines (π1×π2)-integral data inspired by Loeffler and the author's own prior unramified work, then establishes the integral refinement of the Jacquet-Langland GCD result via reinterpretation of the zeta-integral together with external citations to Assing-Saha on p-adic Whittaker new vectors. The self-citation is limited to inspirational context for the unramified case and is not load-bearing for the ramified integrality claim. No self-definitional reductions, fitted inputs renamed as predictions, or uniqueness theorems imported from the same authors appear. The result remains self-contained against the explicit hypotheses on tempered generic representations and the cited external tools.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the domain assumption that the representations are generic and tempered, plus the existence of a natural notion of integral data whose definition is introduced in the paper.

axioms (1)

domain assumption π1, π2 are irreducible admissible generic tempered representations of GL2(F) for F/Qp with odd residue characteristic
Stated explicitly as the setting in the first sentence of the abstract.

invented entities (1)

(π1×π2)-integral data no independent evidence
purpose: A choice of test data at which the Rankin-Selberg integral is asserted to be integral
Introduced in the paper as a new general notion; no independent evidence outside the paper is mentioned in the abstract.

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