Triple Product L-Functions

• Triple product L-functions are degree-8 automorphic L-functions attached to triples of modular forms that integrate analytic, arithmetic, and geometric perspectives.
• They are constructed via period integrals, Euler products, and integral representations, providing functional equations and explicit central value formulas.
• Their p-adic analogues and subconvexity bounds reveal deep connections to Iwasawa theory, non-vanishing results, and the arithmetic of modular forms.

Triple product L-functions are automorphic $L$-functions of degree 8 (or higher, in generalizations) attached to triples of automorphic forms, most notably modular or cuspidal representations of $\mathrm{GL}_2$. These objects integrate automorphic representations or modular forms over products of groups or via period integrals, capturing deep arithmetic and analytic data—including central values critical for applications in number theory, arithmetic geometry, and quantum chaos. The development of triple product $L$-functions involves analytic, algebraic, and $p$-adic perspectives, and connects with period formulas, non-vanishing results, mean values, quantum unique ergodicity, and explicit arithmetic constructions.

1. Analytic Theory and Integral Representations

The analytic foundation of triple product $L$-functions is established via Euler products attached to triples of modular forms or automorphic representations. For example, for normalized newforms $f, g, h \in S_2(\Gamma_0(N))$, the triple product $L$-function is

$$L(s, f \otimes g \otimes h) = \prod_p L_p(s, f \otimes g \otimes h)$$

where each local factor is constructed from the local Langlands parameters of the forms at $p$ (Feigon et al., 2010). These $L$-functions admit analytic continuation and functional equations due to work by Garrett, Piatetski–Shapiro–Rallis, et al. Integral representations, notably those obtained by unfolding global period integrals on groups such as $\mathrm{GSp}_6$, are fundamental: for automorphic forms $\phi_1, \phi_2, \phi_3$, period integrals of the form $\int_{[G]} \phi_1(g)\phi_2(g)\phi_3(g)E(g,s)dg$ (where $E(g,s)$ is an Eisenstein series) are connected, via unfolding, to $L(s, \pi_1 \otimes \pi_2 \otimes \pi_3)$ (Getz, 2019, Getz et al., 27 Mar 2025). The Euler product converges absolutely for $\mathfrak{Re}(s)$ sufficiently large, and standard arguments extend $L(s, f \otimes g \otimes h)$ to a meromorphic function with a functional equation.

A cornerstone for the evaluation and interpretation of central values is the period formula of Gross and Kudla, which expresses the central $L$-value as a finite sum over quaternionic ideal classes: $$\frac{4\pi N}{(f,f)(g,g)(h,h)}L(2, f \otimes g \otimes h) = \sum_{i=1}^{n} w_i^2 f'(I_i) g'(I_i) h'(I_i)$$ where the $I_i$ are representatives of left ideals in a maximal order of a definite quaternion algebra ramified at $N$ and $\infty$, $w_i$ are normalization factors, and $f',g',h'$ are images under the Jacquet–Langlands correspondence (Feigon et al., 2010).

2. Exact Formulas, Mean Values, and Non-vanishing

Exact formulas for averaged central values are available when averaging over modular forms of fixed weight and level, especially prime level. For $N$ prime and $h\in S_2(\Gamma_0(N))$, the weighted sum

$$(4\pi N)\sum_{f,g\in F_2(N)}\frac{L(2, f \otimes g \otimes h)}{(f,f)(g,g)}$$

admits a closed expression involving either explicit rational numbers or products of lower central $L$-values $L(1,h)$ and its quadratic twists, depending on $N \bmod 12$ (Feigon et al., 2010). For instance,

$$= \begin{cases} \frac{1}{12}(N-1)\cdot (h,h), & N \equiv 1 \pmod{12} \ \frac{1}{6}\sqrt{3} L(1,h)L(1,h \otimes \chi_{-3}), & N \equiv 5 \pmod{12} \ 4L(1,h)L(1,h \otimes \chi_{-4}), & N \equiv 7 \pmod{12} \ 6\sqrt{3} L(1,h)L(1,h \otimes \chi_{-3}) + 4L(1,h)L(1,h \otimes \chi_{-4}), & N \equiv 11 \pmod{12} \end{cases}$$

These formulas allow deduction of effective non-vanishing results and quantitative lower bounds: for large $N$, for every $h\in F_2(N)$ there exist $f,g$ with $L(2, f \otimes g \otimes h) \ne 0$ and the number of non-vanishing pairs grows like $N^{3/4-\epsilon}$ (Feigon et al., 2010).

In the analytic setting, mean value estimates and asymptotics for triple product $L$-functions appear in the paper of $L^4$-norms of automorphic forms (e.g., Maass forms). These mean values can be expanded, with power-saving error, using approximate functional equations, trace formulas, and analysis of Kloosterman sums and Bessel functions. An illustrative result is

$$\frac{1}{T^2}\sum_j e^{-t_j^2/T^2} \frac{L(1, u_j) L(1, u_j \times \chi_d) L(1, u_j \times f)}{L(1, \mathrm{sym}^2 u_j)} = \frac{2L(1,\chi_d) L(1,f)L(1,f \times \chi_d)}{\pi^2 L(2,\chi_d)} \log T + C + O_d(T^{-\delta})$$

(Buttcane et al., 2014).

3. Period Relations, Ichino’s Formula, and Local Integrals

The evaluation of central values is connected to period integrals and Ichino's formula, which generalizes the Gross–Kudla period formula to express the square of a global period as the central value of the $L$-function times a product of local factors: $$\frac{|I(\phi \otimes \bar{\phi})|^2}{\langle \phi, \phi \rangle} = C \frac{L(1/2, \Pi, r)}{\prod_{i=1}^3 L(1, \operatorname{sym}^2 \pi_i)} \prod_v \frac{I_v(\phi_v \otimes \bar{\phi}_v)}{\langle \phi_v, \phi_v \rangle_v}$$ where $I_v$ are local period integrals at each place and $C$ is a collection of explicit constants (Cheng, 2018). For local computations, a key result (e.g., Proposition 6.8 in (Cheng, 2018)) asserts that archimedean and non-archimedean trilinear period integrals can be written as products of Rankin–Selberg type integrals, greatly simplifying explicit calculation.

These period/lattice sum formulas play a crucial role in spectral reciprocity, in bounding moments, and in facilitating arithmetic applications via quaternionic or automorphic correspondence.

4. $p$-adic Triple Product $L$-functions and Interpolation

$p$-adic analogues of the complex triple product $L$-functions have been constructed via Hida theory and variations adopting Coleman or finite-slope/overconvergent methods (Hsieh, 2017, Greenberg et al., 2015, Fukunaga, 2019, Hsieh et al., 2019, Blanco, 2020). Given families of ordinary (or more general) $p$-adic modular forms, one builds a multivariable $p$-adic $L$-function by interpolating squares of normalized trilinear period integrals at critical points; for instance, for Hida families $(f, g, h)$ and weights $(k_1, k_2, k_3)$,

$$\mathcal{L}(Q)^2 = \Gamma_{V^\dagger}(0) \frac{L(V^\dagger, 0)}{(\sqrt{-1})^{2k_1}\Omega_f^2(Q)} \mathcal{E}_p(\mathrm{Fil}^+ V^\dagger) \prod_{\ell \in \Sigma_{\mathrm{exc}}}(1+\ell^{-1})^2$$

where $\mathcal{E}_p$ is a carefully engineered Euler factor at $p$, and the construction ensures interpolation at all critical specializations (Hsieh, 2017, Hsieh et al., 2019). Key technical inputs include:

• Explicit computation of local zeta integrals (including at $p$ and the archimedean place),
• Use of regularized diagonal cycles and theta elements in the quaternionic setting,
• Matching with period formulas and carefully chosen canonical periods.

In the balanced critical region (where the weights satisfy triangle inequalities), multiple distinct $p$-adic triple product $L$-functions with the same interpolation region can arise due to the structure of Euler-like factors at $p$; their vanishing is governed by the dimensions of certain Nekovář period spaces (Greenberg et al., 2015). The interpolation formula universally links the $p$-adic $L$-function's values at arithmetic points to the (square roots of) algebraic parts of central $L$-values up to explicit period and Euler-factor corrections.

These $p$-adic objects facilitate Iwasawa-theoretic analysis, Euler system constructions, and proofs of refined main conjectures, including the exceptional zero conjecture for elliptic curves via anticyclotomic specializations (Hsieh, 2017).

5. Subconvexity, Level-Aspect Asymptotics, and Siegel Zeros

Subconvex and Weyl-type bounds for triple product $L$-functions are established in certain level and spectral aspects using trace formulas, Voronoi summation, and analytic large sieve techniques, often achieving exponents better than the convexity barrier (Miao et al., 19 Aug 2025, Miao, 27 Dec 2024, 2207.14449). For instance, when $f, g$ are fixed level 1 cusp forms of weight $k$ and $h$ varies through newforms of weight $2k$ and level $p^3$, a Weyl bound is proved: $$L(1/2, f \times g \times h) \ll_{k,\varepsilon} p^{2 + 2\theta + \varepsilon}$$ with $\theta$ the best-known exponent toward the Ramanujan–Petersson conjecture (Miao et al., 19 Aug 2025). The proof integrates several sophisticated tools: refined Petersson and Kuznetsov trace formulas, Jutila's circle method, spectral large sieve inequalities, and explicit control of Kloosterman sum/Bessel function sums.

Spectral reciprocity and first moment analysis for triple product $L$-functions yield subconvex bounds in the level aspect by linking twisted moments and spectral expansions of triple product periods (Miao, 27 Dec 2024). These results are critical for nonvanishing results and for understanding the distribution of central $L$-values.

Recent work establishes that generic triple product $L$-functions for $$\mathrm{GL}(2)\times\mathrm{GL}(2)\times\mathrm{GL}(2)$$ or $$\mathrm{GL}(2)\times\mathrm{GL}(2)\times\mathrm{GL}(3)$$ over a number field have no Siegel zeros; the only possible exceptions occur in the degenerate case where the representations are dihedral or twist-equivalent, reducing to classical Dirichlet $L$-function issues or, at worst, symmetric cube $L$-functions (Zhao, 8 Aug 2025).

6. Arithmetic and Geometric Applications

The arithmetic impact of triple product $L$-functions is wide-ranging. Exact period and $L$-value formulas yield explicit nonvanishing results and lower bounds for families of modular forms (Feigon et al., 2010, Buttcane et al., 2014), provide insight into the arithmetic of rational points on elliptic curves via $p$-adic heights and logarithms (Lauder, 2013), and feed into deep connections with the theory of Euler systems, Iwasawa main conjectures, and the algebraicity of $L$-values (Greenberg et al., 2015, Hsieh, 2017, Hsieh et al., 2019, Chen, 2021). For instance, special value formulas directly relate $p$-adic $L$-values to $p$-adic logarithms of global points on elliptic curves: $$\log_E(P_g) = 2 d_g \frac{C_0(g)C_1(g)}{(g, f, g)} L_p(g, f, g)(2, 2, 2)$$ with each factor explicitly computable from Fourier coefficients and $p$-depletion data (Lauder, 2013).

The method of period integrals yields algebraicity results for critical $L$-values in the balanced case, extending results by Garrett and Harris and verifying new cases of Deligne's conjecture for symmetric cube $L$-functions of Hilbert modular forms (Chen, 2021). In the $p$-adic context, construction and factorization of $p$-adic $L$-functions confirm conjectural relations between $p$-adic periods, Galois cohomology, and special values.

Applications extend to quantum chaos, spectral theory, and subconvexity problems, where the analysis of triple products is critical to proving quantum unique ergodicity, mean value results, and sup-norm bounds for automorphic forms (Marshall, 2010, Buttcane et al., 2014, Miao, 27 Dec 2024).

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In conclusion, triple product $L$-functions are central objects lying at the crossroads of analytic number theory, arithmetic geometry, automorphic representation theory, and $p$-adic arithmetic. Modern developments—spanning exact average formulas, period integral methods, $p$-adic and Iwasawa-theoretic construction, and the strongest analytic bounds—demonstrate the deep and multifaceted roles of triple product $L$-functions in the arithmetic of automorphic forms.