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Product of Rankin-Selberg convolutions and a new proof of Jacquet's local converse conjecture

This paper constructs a new family of Rankin-Selberg integrals that generalize the classical Jacquet–Piatetski-Shapiro–Shalika integrals to represent products of $L$ -functions, defines associated local gamma factors, and utilizes these tools to provide a new proof of Jacquet's local converse conjecture.

Original authors: Pan Yan, Qing Zhang

Published 2026-08-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Pan Yan, Qing Zhang

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 trying to identify a mysterious stranger in a crowded room. You can't see their face, but you can ask them to sing a song while holding hands with different people. If the stranger sings a unique, recognizable tune every time they hold hands with a specific partner, you can figure out who they are just by listening to those duets. In the world of advanced mathematics, specifically a field called number theory, the "strangers" are complex mathematical objects called representations (which act like hidden symmetries), and the "partners" are other representations. The "song" they sing together is called an L-function. For decades, mathematicians have used these L-functions to solve deep puzzles about how numbers behave, but there was a nagging question: How many different partners do you actually need to hold hands with to be 100% sure you've identified the stranger?

This paper tackles that exact question. It focuses on a famous guess (a conjecture) made by a mathematician named Jacquet, which suggests that you don't need to test the stranger against everyone in the room; you only need to test them against a specific, smaller group of partners to know exactly who they are. To prove this, the authors, Pan Yan and Qing Zhang, invented a new, more powerful "microphone" to listen to these mathematical duets. They built a family of new integrals (which are fancy ways of adding up infinite numbers to find a pattern) that can hear the "song" of two different duets happening at the same time. By using this new tool, they provided a fresh, solid proof that Jacquet's guess is correct. This is a big deal because it simplifies the rules for identifying these hidden mathematical shapes, which helps mathematicians build bridges between different areas of math and understand the fundamental structure of the universe of numbers.

The New Tool: A Double-Acting Microphone

The authors' main achievement is the construction of a new family of integrals. Think of the old method (developed by Jacquet, Piatetski-Shapiro, and Shalika) as a microphone that could only listen to one duet at a time: a main singer (let's call him "GL-l") singing with a backup singer ("GL-m"). The new tool, however, is a super-microphone that can listen to a product of two duets simultaneously. It hears the main singer performing with a backup singer of size "m" and at the same time, performing with a different backup singer of size "n".

The paper shows that this new integral works perfectly. It converges (meaning the math adds up to a sensible number) and, most importantly, it breaks down into a neat product of known L-functions. This means the new tool doesn't just make noise; it produces a clear, structured signal that mathematicians can decode. When the authors tested this tool in the "unramified" case (a simplified, clean version of the problem where the numbers behave nicely), they found that the signal they got was exactly the product of two famous L-functions divided by a third. This confirmed that their new tool was mathematically sound and ready for the real job.

Solving the Identity Puzzle

With this new tool in hand, the authors turned their attention to Jacquet's Local Converse Conjecture. The conjecture asks: If two different "main singers" (representations of GL-l) sound exactly the same when they sing with every possible backup singer of a certain size (up to half the size of the main singer), are they actually the same person?

Before this paper, the answer was "yes," but the proof relied on older, more complicated methods that were hard to generalize. The authors used their new "double-acting" integrals to prove the conjecture again, but this time with a different, more flexible approach. They didn't just prove it for one specific case; they proved it for a whole family of cases where the main singer interacts with two different backup singers of sizes m and n (as long as m + n is less than the size of the main singer).

The proof works like a process of elimination. The authors assume two singers are different. They then use their new integrals to show that if the singers sound the same with all the required partners, the difference between them must be zero. They do this by breaking the problem down into smaller pieces called "partial Bessel functions" (which are like specific notes in the song). They show that if the gamma factors (the mathematical "tuning" of the song) match, then these specific notes must also match. By proving that the notes match for every possible partner, they force the conclusion that the two singers are, in fact, identical.

What This Means and What It Doesn't

The paper proves that Jacquet's conjecture is true for non-archimedean local fields (a specific type of number system used in this branch of math) with a characteristic different from 2. The authors are very confident in this result; they have provided a rigorous, step-by-step proof that leaves no room for doubt within the scope of their assumptions.

However, the paper also clarifies what it does not do. It does not claim to have solved the "global" version of this problem (which involves a much larger, more complex system of numbers) yet, though the authors hope their new integrals will help solve that in the future. They also note that while their method works for the "local" case, the specific case where the two backup singers are exactly half the size of the main singer (when the total size is even) requires a slightly different, pre-existing tool, which they incorporate into their proof.

The authors also point out that their method is distinct from other recent proofs of the same conjecture. While other mathematicians used different models (like Kirillov models) to solve the puzzle, this team treated the problem as a "classical group" problem, making their proof independent of those other models. This adds a new, robust perspective to the field.

In short, Yan and Zhang have built a new, versatile mathematical instrument. They used it to listen to the hidden songs of numbers, proving that you don't need to test every possible combination to know who you're dealing with. Their work confirms a long-standing guess and opens the door for even deeper exploration into the symmetries that govern our mathematical universe.

Technical Summary: Product of Rankin-Selberg Convolutions and a New Proof of Jacquet's Local Converse Conjecture

Problem Statement
The paper addresses two interconnected problems in the theory of automorphic representations and $L$ -functions for general linear groups over global and local fields.

Generalization of Integrals: The authors seek to construct a family of Rankin-Selberg type integrals that represent the product of two Rankin-Selberg $L$ -functions, specifically $L(s_1, \pi \times \tau_1)L(s_2, \tilde{\pi} \times \tilde{\tau}_2)$ , where $\pi$ is a cuspidal automorphic representation of $GL_l$ and $\tau_1, \tau_2$ are cuspidal representations of $GL_m$ and $GL_n$ respectively, under the constraint $m+n < l$ . When $n=0$ , these should reduce to the classical Jacquet–Piatetski-Shapiro–Shalika (JPSS) integrals.
Jacquet's Local Converse Conjecture: The paper aims to provide a new proof of Jacquet's local converse conjecture. This conjecture posits that for two irreducible generic representations $\pi_1, \pi_2$ of $GL_l(F)$ (where $F$ is a non-archimedean local field) with the same central character, if their twisted gamma factors $\gamma(s, \pi \times \tau, \psi)$ coincide for all irreducible generic representations $\tau$ of $GL_m(F)$ with $1 \le m \le [l/2]$ , then $\pi_1 \cong \pi_2$ . While previously proved by Chai (2019) and Jacquet–Lapid (2018), the authors aim to prove this using their newly constructed family of integrals, potentially offering a framework applicable to global converse problems.

Methodology
The authors employ a combination of global integral constructions, local zeta integral analysis, and the theory of partial Bessel functions.

Global Construction: They define a global integral $I_j(\phi, f_s)$ involving a cusp form $\phi \in \pi$ on $GL_l(\mathbb{A})$ and an Eisenstein series $E(h, f_s)$ on $GL_{m+n}(\mathbb{A})$ . The integral is taken over $GL_{m+n}(F) \backslash GL_{m+n}(\mathbb{A})$ and involves a specific Fourier coefficient of $\phi$ along a subgroup $Y_j \subset GL_l$ . The embedding $\iota_j: GL_{m+n} \to GL_l$ is carefully chosen to facilitate the unfolding process.
Unfolding and Euler Product: Using the Fourier expansion of cusp forms (generalizing the work of Piatetski-Shapiro and Shalika), they unfold the global integral to obtain an Euler product. This involves computing local zeta integrals $\Psi(W_v, \xi_{f_s, v}; j)$ which represent the product of local $L$ -factors.
Local Theory: In the local setting, they define local zeta integrals and establish the existence of a local gamma factor $\Gamma(s, \pi \times (\tau_1, \tau_2), \psi; j)$ via a functional equation involving an intertwining operator. They compute the unramified case explicitly using the hive model for Littlewood-Richardson coefficients (developed by Knutson and Tao) to show that the local integral equals the ratio of $L$ -functions:
$\Psi(W^0, \xi^0; j) = \frac{L(s_1 + \frac{k-j}{2}, \pi \times \tau_1)L(s_2 - \frac{k-j}{2}, \tilde{\pi} \times \tilde{\tau}_2)}{L(s_1+s_2, \tau_1 \times \tilde{\tau}_2)}$
where $k = l - m - n - 1 - j$ .
Proof of Converse Theorem: To prove the local converse theorem, the authors utilize partial Bessel functions (developed by Cogdell, Shahidi, and Tsai) associated with Howe vectors. They construct a difference of partial Bessel functions $B_i(g, f_1) - B_i(g, f_2)$ $B_{i} (g, f_{1}) - B_{i} (g, f_{2})$ for two representations $\pi_1, \pi_2$ $π_{1}, π_{2}$ . By assuming the equality of gamma factors (the condition $C(m)$ $C (m)$ ), they show inductively that this difference vanishes on specific Bruhat cells.
- They use a double induction on the parameters $m$ and $n$ .
- They leverage the uniqueness of Bessel models to relate the vanishing of integrals to the vanishing of the difference of partial Bessel functions.
- For the case where $l$ is even ( $l=2r$ ) and $m=n=r$ , they incorporate the local gamma factors for unitary groups (as defined in [BAS09] and [Mor25]) to handle the boundary case.

Key Contributions and Results

New Family of Integrals: The paper constructs a family of integrals $I_j$ that generalize the JPSS integrals. These integrals represent the product of two Rankin-Selberg $L$ -functions.
Unramified Calculation: The authors provide a detailed unramified calculation of these integrals, utilizing the hive model to handle the combinatorics of Schur polynomials and Littlewood-Richardson coefficients. This generalizes previous unramified calculations for $GL_l \times GL_m$ and unitary groups.
Local Gamma Factors: They define local gamma factors $\Gamma(s, \pi \times (\tau_1, \tau_2), \psi; j)$ and prove they are products of standard JPSS gamma factors (up to shifts and twists).
New Proof of Jacquet's Local Converse Conjecture: The primary application is a new proof of Jacquet's local converse conjecture for $GL_l(F)$ $G L_{l} (F)$ (where $F$ $F$ is a non-archimedean local field of characteristic $\neq 2$ $\neq = 2$ ).
- The proof establishes that if $\Gamma(s, \pi_1 \times (\tau_1, \tau_2), \psi; 0) = \Gamma(s, \pi_2 \times (\tau_1, \tau_2), \psi; 0)$ for all $\tau_1, \tau_2$ with $0 \le m, n \le [l/2]$ , then $\pi_1 \cong \pi_2$ .
- The authors note that this condition is equivalent to the standard condition involving single twists $\gamma(s, \pi \times \tau, \psi)$ for $1 \le m \le [l/2]$ .
Technical Distinctions: Unlike previous proofs (e.g., [Cha19] which used partial Bessel functions only on the open Bruhat cell, or [JL18] which relied on Kirillov models), this proof:
- Proves the equality of partial Bessel functions on all Bruhat cells, making it applicable to finite fields (where the topology is discrete).
- Treats $GL_l$ as a classical group, avoiding dependence on the existence of Kirillov models.
- Provides a stronger result (Theorem 5.1) that characterizes representations determined by gamma factors twisted up to $GL_m$ for any $m \le [l/2]$ .

Significance and Claims
The authors claim that their new family of integrals offers a fresh perspective on the local converse problem. While the conjecture was already known to be true, this new proof:

Unifies Approaches: It connects the product of $L$ -functions directly to the converse problem, suggesting that the "product" structure might be more robust for global applications.
Potential for Global Converse: The authors explicitly state that they expect these integrals to be useful for attacking the global converse conjecture (Conjecture 1.3), which seeks to reduce the number of twists required to determine a global automorphic representation. The property (1.1) of their gamma factors provides a new family of equalities that could supplement the existing JPSS integrals in global settings.
Generality: The method is robust enough to cover finite fields and is independent of Kirillov models, potentially making it adaptable to other classical groups and exceptional groups (as hinted by references to work on $G_2$ and classical groups over finite fields).

The paper concludes by noting that the construction is currently limited to the case $m+n < l$ , and generalizing to $m+n \ge l$ is a subject for future work.

The New Tool: A Double-Acting Microphone

Solving the Identity Puzzle

What This Means and What It Doesn't

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