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Period relations for Rankin-Selberg convolutions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$

This paper establishes the rationality and period relations for critical values of Rankin-Selberg L-functions for $\mathrm{GL}(n)\times\mathrm{GL}(n)$ over number fields containing a CM field by utilizing a modular symbol approach.

Original authors: Yubo Jin, Jian-Shu Li, Dongwen Liu, Binyong Sun

Published 2026-03-31

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CC BY 4.0

Original authors: Yubo Jin, Jian-Shu Li, Dongwen Liu, Binyong Sun

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a mystery about the hidden "DNA" of numbers. In the world of mathematics, there are special functions called L-functions. Think of these as complex, musical scores that encode deep secrets about prime numbers and the structure of the universe.

The big question mathematicians have been asking for decades is: "When we play a specific note on this score (a 'special value'), is the result a simple, rational number (like 3/4), or is it a messy, irrational number (like $\pi$ or $\sqrt{2}$ )?"

This paper, written by Jin, Li, Liu, and Sun, is a major breakthrough in solving this mystery for a specific, very difficult type of L-function called the Rankin-Selberg convolution for GL(n) × GL(n).

Here is the story of their discovery, broken down into simple concepts:

1. The Problem: The "Messy" Numbers

Imagine you have a machine that takes two complex musical instruments (representing mathematical objects called automorphic representations) and blends them together. The output is a new, incredibly complex sound (the L-function).

Mathematicians suspect that if you stop the machine at a very specific moment (a "critical value"), the volume of the sound isn't random. It should be related to a "period"—a fundamental unit of measurement, like a ruler or a clock tick.

The goal is to prove that if you divide the volume of the sound by this fundamental ruler, the result is a rational number (a clean fraction). If you can prove this, you unlock a deeper understanding of how numbers relate to geometry.

2. The Challenge: The "GL(n) × GL(n)" Wall

For a long time, mathematicians could solve this puzzle for simpler machines (like mixing a big instrument with a small one). But when they tried to mix two identical large instruments (GL(n) × GL(n)), the math got stuck.

It was like trying to bake a cake with two giant, heavy ovens that were fused together. The heat (mathematical complexity) was too intense, and the ingredients (the numbers) wouldn't mix cleanly. Previous attempts to solve this had some cracks in the foundation, leaving the proof incomplete.

3. The New Tool: "Eisenstein Cohomology"

The authors' secret weapon is something called Eisenstein cohomology.

The Analogy: Imagine you are trying to understand the shape of a giant, foggy mountain (the mathematical space). You can't see the whole thing at once. So, you send out a team of explorers (Eisenstein series) who climb the mountain and send back postcards (cohomology classes) describing what they see.
The Innovation: The authors proved that these postcards are "rational." In other words, the explorers are sending back clear, understandable messages that fit perfectly into a rational framework. They fixed the cracks in the previous attempts by carefully checking the "postcards" at the very top of the mountain (the boundary) and proving they align perfectly with the bottom.

4. The "Modular Symbol" Bridge

To connect the messy L-function to the clean rational numbers, they used Modular Symbols.

The Analogy: Think of a Modular Symbol as a universal translator. It takes the complex, foreign language of the L-function and translates it into the simple language of rational numbers.
The authors showed that this translator works perfectly. When you feed the L-function into the translator, it spits out a result that is exactly what the famous mathematician Deligne predicted it should be: a rational number multiplied by a specific "period" (the ruler).

5. The Grand Result

The paper proves a formula that looks like this:

$\frac{\text{The L-Function Value}}{\text{The Period} \times \text{Some Constants}} = \text{A Rational Number}$

This is huge because:

It confirms a 40-year-old guess: It validates a conjecture made by Deligne and Blasius, showing that the universe of numbers is more orderly than we thought.
It works for the hardest case: They solved it for the "GL(n) × GL(n)" case, which was the last major hurdle.
It's a new starting point: Just as a ladder allows you to climb higher, this result gives mathematicians a new, solid rung to climb up to solve even more complex problems in the future.

Summary

In short, Jin, Li, Liu, and Sun built a sturdy bridge (using Eisenstein cohomology and modular symbols) over a deep canyon of complexity. They proved that when you cross this bridge, the destination is always a clean, rational number. This confirms that the hidden patterns of numbers are not random chaos, but a beautifully structured, rational design.

1. Problem Statement

The central problem addressed in this paper is the study of the special values of Rankin-Selberg $L$ -functions associated with the product of two general linear groups, $GL(n) \times GL(n)$ , over a number field $k$ containing a CM field.

Specifically, the authors aim to prove:

Rationality: That certain normalized critical values of these $L$ -functions lie in a specific algebraic number field (the rationality field of the representation and the character).
Period Relations: That these special values satisfy specific equivariance properties under the action of the automorphism group $\text{Aut}(\mathbb{C})$ , relating the $L$ -value of a representation $\Pi$ to that of its conjugate $\sigma\Pi$ .

This work extends previous results on the $GL(n) \times GL(n-1)$ case (studied extensively in [LLS24]) to the more complex and less understood $GL(n) \times GL(n)$ case. The paper focuses on the critical value at $s=0$ for the $L$ -function $L(s, \Pi \times \chi)$ , where $\Pi = \Sigma \boxtimes \Sigma'$ is an irreducible smooth automorphic representation and $\chi$ is an algebraic Hecke character.

2. Methodology

The authors employ a sophisticated combination of modular symbols, Eisenstein cohomology, and Rankin-Selberg integrals. The methodology can be broken down into the following key steps:

A. Modular Symbol Approach

Instead of direct analytic computation, the authors utilize the modular symbol approach. They define global and local modular symbols that relate the $L$ -values to cohomology classes.

Global Modular Symbol ( $P$ ): Defined via the integration of automorphic forms against Eisenstein series over the adelic quotient space $X_n = GL_n(k) \backslash GL_n(\mathbb{A}) / K_\infty$ .
Local Modular Symbols: Defined via local Rankin-Selberg integrals at archimedean and non-archimedean places.
Factorization: The global $L$ -value is expressed as a product of local periods and a global modular symbol, allowing the rationality of the $L$ -value to be deduced from the rationality of the cohomological maps.

B. Eisenstein Cohomology and Rationality

A major technical hurdle in the $GL(n) \times GL(n)$ case is the rationality of the Eisenstein cohomology.

The authors construct an Eisenstein map ( $Eis_\eta$ ) from the cohomology of a degenerate principal series representation $J_\eta$ to the cohomology of the locally symmetric space $X_n$ .
Theorem 1.3 (Main Technical Result): They prove that this Eisenstein map is $\text{Aut}(\mathbb{C})$ -equivariant. This means the map commutes with the action of field automorphisms on the coefficients.
Key Innovation: Unlike the $GL(n) \times GL(n-1)$ case, proving the rationality of Eisenstein cohomology for $GL(n) \times GL(n)$ requires handling the boundary cohomology and intertwining operators more delicately. The authors explicitly construct generators for the cohomology spaces and analyze the behavior of normalized intertwining operators under $\text{Aut}(\mathbb{C})$ .

C. Regular Algebraic Tamely Isobaric Representations

The paper introduces a broader class of representations called tamely isobaric representations (extending the definition in [LLS24]).

These are induced representations from cuspidal representations on smaller $GL$ groups.
The authors prove that if $\Sigma$ is tamely isobaric and $\Sigma'$ is cuspidal, the rationality results hold. This generalization allows the results to serve as a base case for inductive arguments on general Rankin-Selberg convolutions.

D. Period Relations via Diagrams

The proof of the main theorem relies on constructing a large commutative diagram involving:

Local and global cohomology spaces.
The $\text{Aut}(\mathbb{C})$ -action on these spaces.
The modular symbols (archimedean and non-archimedean).
The Deligne periods and Whittaker periods.

By showing that all parts of the diagram commute (specifically the "cubic diagram" in Section 7.5.3), they establish the equivariance of the $L$ -value quotient.

3. Key Contributions and Results

Main Theorem (Theorem 1.2)

Let $\Pi = \Sigma \boxtimes \Sigma'$ be a regular algebraic automorphic representation of $GL(n) \times GL(n)$ , with $\Sigma$ tamely isobaric and $\Sigma'$ cuspidal. Let $\chi$ be a $(\mu, \nu)$ -balanced critical Hecke character. Then the following quantity is $\text{Aut}(\mathbb{C})$ -equivariant:

$\frac{L(0, \Pi \times \chi)}{c(\omega_\Pi \chi^n) \cdot G(\chi)^{\frac{n(n-1)}{2}} \cdot \Omega_\infty(\Pi, \chi) \cdot \Omega(\Pi)} \in \mathbb{Q}(\Pi, \chi)$

Where:

$c(\omega_\Pi \chi^n)$ is the Deligne period associated with the central character.
$G(\chi)$ is the Gauss sum of $\chi$ .
$\Omega_\infty(\Pi, \chi)$ is an explicit archimedean period depending on the weights and the character at infinity.
$\Omega(\Pi)$ is the Whittaker period of $\Pi$ .
$\mathbb{Q}(\Pi, \chi)$ is the compositum of the rationality fields of $\Pi$ and $\chi$ .

The equivariance means that for any $\sigma \in \text{Aut}(\mathbb{C})$ , applying $\sigma$ to the numerator and denominator yields the corresponding expression for the twisted representation $\sigma\Pi$ and character $\sigma\chi$ .

Theorem 1.3 (Eisenstein Cohomology Rationality)

The Eisenstein map $Eis_\eta$ is $\text{Aut}(\mathbb{C})$ -equivariant. This is a standalone result of significant importance for the arithmetic of automorphic forms, providing a new proof of Deligne's conjecture for Hecke characters (independent of previous work by Kurihara) and establishing the rationality of Eisenstein cohomology for $GL(n)$.

4. Significance and Impact

Resolution of a Long-Standing Case: The $GL(n) \times GL(n)$ case is fundamental in the Rankin-Selberg theory. While the $GL(n) \times GL(n-1)$ case was well-understood, the $GL(n) \times GL(n)$ case posed significant difficulties regarding the rationality of Eisenstein cohomology. This paper resolves these difficulties.
Generalization of Tamely Isobaric Representations: By relaxing the conditions on the representations (allowing $\Sigma$ to be tamely isobaric rather than just cuspidal), the authors provide a more flexible framework that facilitates inductive proofs for general Rankin-Selberg $L$ -functions.
Consistency with Deligne's Conjecture: The results are fully compatible with Deligne's conjecture on the special values of motives and Blasius's period relations. The paper explicitly connects the automorphic $L$ -values to the periods of pure motives.
Correction of Previous Work: The authors address issues in previous literature (specifically [Gr03] and [Hi94]), providing rigorous proofs for the cohomological generators and the behavior of intertwining operators that were previously flawed or incomplete.
Foundation for Future Research: The techniques developed, particularly the explicit construction of generators for Eisenstein cohomology and the handling of the boundary cohomology, provide a toolkit for studying special values of $L$ -functions for other groups and higher-rank convolutions.

In summary, this paper establishes a rigorous arithmetic framework for the special values of $GL(n) \times GL(n)$ Rankin-Selberg $L$ -functions, proving their rationality and period relations via a deep synthesis of modular symbols, cohomology, and automorphic forms.