Triple Product L-functions Overview

• Triple product L-functions are automorphic L-functions formed from the tensor product of three representations and characterized by analytic continuation, functional equations, and boundedness.
• They serve as a testing ground for functorial transfers and have been instrumental in advancing converse theorems and Rankin–Selberg methods.
• Their analytic control underpins proofs of the Ramanujan conjecture for GLₙ and facilitates the reduction of complex functorial transfers to explicit tensor products.

Triple product $L$-functions are a class of automorphic $L$-functions associated to the tensor product of three automorphic representations of general linear groups. They play a fundamental role in the analytic theory of automorphic forms, exhibit deep connections to functoriality in the Langlands program, and serve as testing grounds for conjectures about special values, arithmetic geometry, and $p$-adic variation. Their explicit construction, analytic properties, and arithmetic ramifications have been at the center of intensive research.

1. Fundamental Definition and Analytic Properties

Let $F$ be a number field with adèle ring $\mathbb{A}_F$. For $n_1, n_2, n_3 \geq 1$, and cuspidal automorphic representations $\pi_i$ of $GL_{n_i}(\mathbb{A}_F)$, the triple product $L$-function is defined via the $L$-group formalism: $$L(s, \pi_1 \times \pi_2 \times \pi_3) := L(s, \pi_1 \otimes \pi_2 \otimes \pi_3, \otimes^3),$$ where

$$\otimes^3 : {}^L(GL_{n_1} \times GL_{n_2} \times GL_{n_3}) \longrightarrow GL_{n_1 n_2 n_3}(\mathbb{C})$$

is the external triple tensor product. At each place $v$, the local factor $L(s, \pi_{1,v} \times \pi_{2,v} \times \pi_{3,v})$ is defined via the local Langlands correspondence.

The "expected analytic properties" of such $L$-functions—termed "niceness"—are: analytic continuation to $\mathbb{C}$ bounded in vertical strips, a standard functional equation, and holomorphicity in a region containing the critical points. In practice, this requires that, for $1 \leq n_3 < n_1 n_2$ and all unitary cuspidal automorphic $\pi_3$ unramified outside a finite set $S$, the function $L(s, \pi_1 \times \pi_2 \times \pi_3)$ is entire, bounded in vertical strips, and satisfies the usual global functional equation: $$L(s, \pi \times \tau) = \varepsilon(s, \pi \times \tau) L(1-s, \pi^\vee \times \tau^\vee),$$ where $\varepsilon(s, \pi \times \tau)$ is the global root number.

2. Converse Theorems and Consequences for Automorphy

A key analytic input is the converse theorem for $GL(n)$, particularly due to Cogdell and Piatetski-Shapiro: If for all $1 \leq m < n$ and for all cuspidal automorphic representations $\tau$ of $GL_m(\mathbb{A}_F)$ ramified only at $S$, the functions $L(s, \pi \times \tau)$ are nice, then there exists an isobaric automorphic representation $\pi'$ of $GL_n(\mathbb{A}_F)$ matching the local components of $\pi$ outside $S$.

The Rankin–Selberg transfer (functorial tensor product transfer)

$\pi_1 \boxtimes \pi_2$

is an automorphic representation of $GL_{n_1 n_2}(\mathbb{A}_F)$ characterized locally and by the global $L$-functions. Existence of such a transfer for all (unitary cuspidal) pairs $(\pi_1, \pi_2)$ is the central mechanism by which triple product $L$-function analytics feed into broader automorphy results and the structure theory of automorphic representations.

3. Implications for the Ramanujan Conjecture

Assuming these analytic properties and the corresponding Rankin–Selberg transfers, the paper establishes that the Ramanujan conjecture for $GL_n$ holds: for any unitary cuspidal automorphic representation $\pi$ of $GL_n(\mathbb{A}_F)$ unramified at $v$, the local component $\pi_v$ is tempered (all Satake parameters of absolute value $1$).

This is achieved using a "Langlands trick": The tensor product transfers—constructed via triple product $L$-function analytics and the converse theorem—impose, through the multiplicative behavior of Satake parameters, constraints that force temperedness for all unramified places. The analytic control achievable through the triple product $L$-function is thus transmitted, via functoriality, to the full spectrum of the general linear group.

4. Reduction of General Functorial Transfers to Tensor Products

An important structural outcome is the reduction of functorial transfers for a general reductive group $G$ to $GL_n$. For any algebraic representation $r: \widehat{G} \to GL_n$, functorial transfer at the level of stably automorphic representations decomposes into finitely many tensor product transfers associated to the fundamental representations of $\widehat{G}$ (via Chevalley tensor construction). All irreducible representations of $GL_n$ are recovered as exterior tensor combinations of fundamental ones, and thus any transfer, for arbitrary highest weight, is a finite algebraic combination of basic tensor product transfers.

This connects the analytic theory of triple product $L$-functions to the higher-level representation-theoretic structure of the Langlands program: mastering the analyticity of triple product $L$-functions suffices to construct all functorial transfers up to isobaric twists (outside a finite ambiguous set determined by $G$).

5. Central Role in the Langlands Program and Functoriality

Analytic properties of triple product $L$-functions are now seen as a universal analytic test for functoriality from a group $G$ to $GL_n$. The existence and control of Rankin–Selberg tensor product transfers are building blocks for all more general functorial lifts. The converse theorems for $GL(n)$ tie the global analytic structure of these $L$-functions to the algebraicity and automorphy of all more complicated functorial images.

This circles back repeatedly to the central conjectures of the Langlands program, where $L$-functions and their analytic continuation and functional equations are expected to encode, and indeed determine, the functorial transfer between automorphic spectra and $L$-groups.

6. Summary of Main Logical Relationships

Concept	Main Assumption/Result	Role/Connection
Triple product $L$-functions	Analytic properties: functional equation, continuation	Feed into converse theorems and automorphy
Converse theorem (Cogdell–Piatetski-Shapiro)	Niceness of $L(s, \pi \times \tau)$ for all $\tau$	Implies automorphy of tensor product representations
Rankin–Selberg (tensor) transfer	Constructed via triple product $L$-function theory	Provides explicit functorial transfer
Ramanujan conjecture	Consequence of automorphy of all tensor powers	Follows from functorial transfers and $L$-function control
Functorial transfers in Langlands	Reduction to fundamental representations	All functoriality built from tensor products

7. Conclusion

Triple product $L$-functions, via their expected analytic properties (holomorphicity, boundedness, continuation, and standard functional equation), form the cornerstone for establishing functoriality between groups in the Langlands paradigm and for proving deep conjectures such as the Ramanujan conjecture for $GL_n$. Mastery of their analytic structure allows the reduction of all possible transfers to finite explicit algebraic cases depending only on the group $G$. This positions triple product $L$-functions as the primary testing ground for the analytic side of automorphic functoriality and unlocks strong spectral consequences for automorphic representations.

Key Formulas

• Triple product $L$-function:

$$L(s, \pi_1 \times \pi_2 \times \pi_3) = L(s, \pi_1 \otimes \pi_2 \otimes \pi_3, \otimes^3)$$

• "Niceness" property:

$$L(s, \pi \times \tau) = \varepsilon(s, \pi \times \tau) L(1-s, \pi^\vee \times \tau^\vee)$$

• Functorial transfer reduction:

$$r(\pi^S) = \rho_1^{\otimes m_1}(\pi^S) \boxtimes \cdots \boxtimes \rho_n^{\otimes m_n}(\pi^S)$$

for $\rho_i$ fundamental representations and $S$ the ramified set.

• Ramanujan via tensor powers (for each unramified Satake parameter $\alpha$)

$|\alpha|^k \leq q^{1/2} \implies |\alpha| = 1$

for all $k$.

These results set a benchmark for the role of analytic behavior in the broader Langlands program and have enduring implications for the theory of automorphic forms and beyond (Getz et al., 17 Sep 2025).