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On the exterior square $\varepsilon$ -factors of $GL_n$

This paper establishes, using global methods, that the $\varepsilon$ -factors associated with the exterior square of a generic representation of $GL_r(F)$ over a $p$ -adic field coincide when defined via Jacquet-Shalika integrals and Langlands-Shahidi methods, while also providing a new proof of the local functional equation.

Original authors: Ravi Raghunathan

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ravi Raghunathan

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 involving a very complex, invisible object called a "representation" (let's call it $\pi$ ). This object lives in a mathematical world called a " $p$ -adic field," which is a bit like a universe made of numbers that behave differently than the ones we use in everyday life.

The goal of this paper is to prove that three different detectives, using three completely different sets of tools, all arrive at the exact same conclusion about a specific property of this object called the $\varepsilon$ -factor (think of this as a unique "fingerprint" or "signature" of the object).

Here is the breakdown of the three detectives and their tools:

The Galois Detective (The Arithmetic Way): This detective uses a "Rosetta Stone" called the Local Langlands Correspondence. It translates the object $\pi$ into a different language (Galois theory) and reads the fingerprint directly from there.
The Shahidi Detective (The Structural Way): This detective uses a method called the "Langlands-Shahidi method." It looks at the internal structure and symmetries of the object to calculate the fingerprint.
The Jacquet-Shalika Detective (The Integral Way): This detective uses a very specific, complicated recipe involving "integrals" (a type of mathematical summing process). They pour the object into a machine, run it through a filter, and see what comes out to find the fingerprint.

The Big Problem

For a long time, mathematicians knew that the Galois and Shahidi detectives agreed on the fingerprint. However, no one could prove that the Jacquet-Shalika detective (who uses the messy integral recipe) was getting the same result.

Why was this hard?

The integral recipe is tricky. In the "real number" world (archimedean fields), the recipe was already proven to work. But in the " $p$ -adic" world (the focus of this paper), the recipe was known to work for very simple, "super-crisp" objects (supercuspidal representations), but no one knew if it worked for the more complex, "nearly tempered" objects that make up the bulk of the mathematical universe.
It's like knowing a specific cooking recipe works perfectly for a simple boiled egg, but you aren't sure if it works for a complex, multi-layered soufflé.

The Author's Solution: A Global Detour

The author, Ravi Raghunathan, solves this by taking a "global" detour. Instead of trying to prove the recipe works for every single object directly in the local $p$ -adic world, he does the following:

The "Globalization" Trick: He imagines taking a local object from the $p$ -adic world and embedding it into a much larger, global world (a "number field," which is like a vast network of number systems).
The "Dense Subset" Strategy: He uses a powerful theorem (by Shin) to show that any complex object in the local world is surrounded by a "cloud" of simpler objects that can be embedded into this global world.
The Comparison: In this global world, he compares the results of the Jacquet-Shalika recipe against the Shahidi method. Because the global world has a "functional equation" (a rule that links the left side of a mirror to the right side), he can prove that the two methods must agree for these simpler, embeddable objects.
The "Smoothie" Argument (Continuity): Since the simpler objects are "dense" (meaning they are everywhere, like grains of sand in a beach), and the mathematical functions behave smoothly (like a liquid), if the two methods agree on the grains of sand, they must agree on the whole beach.

The Key Steps in Plain English

Step 1: The Easy Cases. First, the author proves the two methods agree for the simplest objects (supercuspidal representations) using a known trick from the 1980s.
Step 2: The "Nearly Tempered" Middle Ground. He then extends this to "nearly tempered" objects. He does this by showing that the mathematical functions involved are "analytic" (smooth and predictable). If two smooth curves touch at many points, they are likely the same curve.
Step 3: The Final Stretch. Using a sophisticated tool from a paper by Beuzart-Plessis, he shows that because the functions are smooth and the agreement holds for the "dense" set of objects, the agreement must hold for all generic representations, no matter how complex.

The Conclusion

The paper concludes with a definitive "Yes."

Theorem: For any generic representation $\pi$ in the $p$ -adic world, the fingerprint calculated by the messy integral recipe (Jacquet-Shalika) is exactly the same as the fingerprint calculated by the structural method (Shahidi) and the arithmetic method (Galois).

Why does this matter?
In the world of automorphic forms (a branch of math connecting number theory and symmetry), it is crucial that different ways of defining these "L-functions" and " $\varepsilon$ -factors" all yield the same result. If they didn't, the entire theoretical framework would be inconsistent. This paper removes a major doubt, confirming that the "Integral Way" is just as reliable as the other established ways, even in the tricky $p$ -adic setting.

In short: The author built a bridge from the known (simple cases) to the unknown (complex cases) using a global perspective and the smoothness of mathematical functions, proving that three different paths lead to the exact same destination.

Technical Summary: On the Exterior Square $\varepsilon$ -Factors of $GL_n$

Problem Statement
The paper addresses the consistency of three distinct definitions of exterior square $L$ -factors and $\varepsilon$ -factors for generic representations $\pi$ of $GL_r(F)$ , where $F$ is a local field of characteristic 0. These definitions arise from:

Galois/Arithmetic factors ( $L_G, \varepsilon_G$ ): Defined via the Local Langlands Correspondence (LLC).
Langlands-Shahidi factors ( $L_{Sh}, \varepsilon_{Sh}$ ): Defined via the Langlands-Shahidi method.
Jacquet-Shalika factors ( $L_{JS}, \varepsilon_{JS}$ ): Defined via integral representations (Jacquet-Shalika integrals) and a local functional equation.

While the equality of the $L$ -factors ( $L_{JS} = L_{Sh} = L_G$ ) was largely established, and the equality of $\varepsilon$ -factors was known for the Langlands-Shahidi and Galois definitions (and recently for the archimedean Jacquet-Shalika case), the equality $\varepsilon_{JS}(s, \pi, \wedge^2, \psi) = \varepsilon_{Sh}(s, \pi, \wedge^2, \psi)$ remained an open problem in the $p$ -adic case. The author notes that proving this equality is more subtle than for Rankin-Selberg or Asai factors due to the complexity of the archimedean theory of Jacquet-Shalika integrals and the lack of a prior formula expressing $\varepsilon_{JS}$ in terms of inducing data.

Methodology
The paper employs global methods to resolve the local equality, following a strategy similar to that used for Asai $L$ -functions in [BP21]. The core logical flow is as follows:

Globalisation of Supercuspidal Representations: The author utilizes a theorem by Henniart (Theorem 6.1) which allows a supercuspidal representation $\pi$ of $GL_r(F)$ to be embedded as the local component at a place $v_0$ of a cuspidal automorphic representation $\Pi$ of $GL_r(\mathbb{A}_k)$ , where $\Pi$ is unramified at all other finite places.
Comparison of Global Functional Equations: By comparing the global functional equations for the Jacquet-Shalika integrals and the Langlands-Shahidi method, the author deduces that the local $\varepsilon$ -factors must coincide for supercuspidal representations (Proposition 6.2).
Density and Continuity Arguments: To extend the result from supercuspidal representations to all generic representations, the paper uses a consequence of Shin's theorem (Theorem 7.1) to show that any open neighborhood of a tempered representation in the Fell topology contains a representation that can be globalized to a cuspidal automorphic form with specific local properties (supercuspidal or unramified).
Analytic Continuation: The proof relies heavily on the analytic properties of the Jacquet-Shalika integrals and the $\varepsilon$ $ε$ -factors with respect to the parameters of induced representations. Specifically, it utilizes:
- The existence of analytic sections of Whittaker functions (Proposition 2.1, following [BP21]).
- The holomorphy and finite order of the integrals and $\gamma$ -factors in vertical strips.
- A technical lemma (Proposition 2.8.1 of [BP21]) allowing the extension of analytic functions from products of half-planes to the entire complex plane.

Key Contributions and Results

Main Theorem (Theorem 1.1): For a $p$ -adic field $F$ and any generic representation $\pi \in \text{Irr}_{gen}(G_r)$ , the Jacquet-Shalika $\varepsilon$ -factor coincides with the Langlands-Shahidi $\varepsilon$ -factor:
$\varepsilon_{JS}(s, \pi, \wedge^2, \psi) = \varepsilon_{Sh}(s, \pi, \wedge^2, \psi)$
Combined with existing results ( $\varepsilon_{Sh} = \varepsilon_G$ ), this establishes the full equality of all three definitions (Theorem 1.2).
New Proof of the Local Functional Equation: The paper provides a new proof of the local functional equation for Jacquet-Shalika integrals in the $p$ -adic case (Theorem 4.7). Unlike previous proofs by Matringe and Cogdell-Matringe which relied on discrete series or globalizable representations, this proof extends the functional equation to all generic representations using globalisation and continuity arguments.
Analytic Properties: The paper establishes that the normalized Jacquet-Shalika integrals ( $\Xi$ ) are entire functions of finite order in vertical strips for nearly tempered representations (Proposition 7.6), a necessary step for the analytic continuation arguments used to prove the main theorem.

Significance
The paper claims that the equality of these $\varepsilon$ -factors is intrinsic to the development of the theory of automorphic $L$ -functions and is a required component of the Local Langlands Conjectures for $GL_r$ . The author emphasizes that the proof for exterior square factors is more difficult than for Rankin-Selberg or Asai factors because the archimedean theory of Jacquet-Shalika integrals was previously less developed, and the necessary formulas expressing $\varepsilon_{JS}$ in terms of inducing data were unavailable. By establishing this equality, the paper fills a gap in the theory, ensuring that the $\varepsilon$ -factors defined via integral representations are consistent with those defined via the Langlands-Shahidi method and the local Langlands correspondence for all generic $p$ -adic representations.

The Big Problem

The Author's Solution: A Global Detour

The Key Steps in Plain English

The Conclusion

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