Symmetric Cube Lifts of Hecke Eigenforms

• Symmetric cubes are lifts of algebraic regular Hecke eigenforms from GL(2) to GL(4) that encode deeper arithmetic data via functoriality.
• Explicit Fourier coefficient bounds and nonvanishing L-values validate the automorphy and effective distinction of these eigenforms.
• Refined period integrals, trace formulas, and Selmer group analyses link these lifts to broader insights in the Langlands program.

Symmetric cubes of algebraic regular Hecke eigenforms occupy a central position in modern automorphic and arithmetic theory. For a cuspidal Hecke eigenform $f$ on $\mathrm{GL}(2)$, its symmetric cube lift—constructed via functoriality—produces an automorphic representation on $\mathrm{GL}(4)$, encoding deeper arithmetic data than the original object. The exploration of these lifts involves fundamental questions of functoriality, the arithmetic of $L$-functions, Galois representations, period integrals, and effective algebraic and analytic distinctions, with implications for broader themes in the Langlands program.

1. Construction and Automorphy of Symmetric Cube Lifts

The symmetric cube construction starts with a classical or Hilbert modular cusp form $f$ (regular, algebraic, often normalized as a Hecke eigenform), or its associated automorphic representation $\pi$ on $\mathrm{GL}_2(\mathbb{A}_F)$. The third symmetric power lift $\operatorname{Sym}^3 \pi$ gives, via Langlands' conjectural parametrization, an automorphic representation of $\mathrm{GL}_4(\mathbb{A}_F)$ whose local factors at unramified places $v$ are determined by the characteristic polynomials of the images of Frobenius under the 2-dimensional Galois representation attached to $\pi$.

The automorphy of these symmetric cubes is now unconditional over $\mathbb{Q}$ in wide generality. The key result establishes that, for any cuspidal Hecke eigenform $f$ of level 1, the symmetric cube lifting $\operatorname{Sym}^3 f$ is automorphic for every $n \geq 1$—as is $\mathrm{Sym}^n f$ in general—by a propagation technique over the eigencurve. Specifically, once automorphy is "seeded" for one member of a connected family of Hecke eigenforms (e.g., of level one), the automorphy "propagates" to all in the family via analytic and geometric properties of eigenvarieties (Newton et al., 2019). This uses "ping-pong" arguments relating twin and companion points and employs the deep geometric understanding of the spectral variety of the Coleman–Mazur eigencurve.

For semistable elliptic curves $E/\mathbb{Q}$, whose modularity gives a weight-2 Hecke eigenform $f_E$, these results ensure that the symmetric cube $\operatorname{Sym}^3 f_E$ is automorphic on GL(4), and the associated $L$-function enjoys analytic continuation and functional equation properties accordingly.

2. Distinction, Classification, and "Icosahedral" Phenomena

A fundamental issue is the extent to which the symmetric cube lifts distinguish the original eigenforms. For eigenforms $f,g$ of weights $k_1 \ne k_2$ on $\Gamma_0(N)$, the comparison of their initial Fourier coefficients suffices to separate them: there exists $n \leq 4(\log N + 1)^2$ with $a_n(f) \neq a_n(g)$ (Ghitza, 2010). This result provides explicit and sharp, even numerically verified, Sturm-type bounds for distinction. When functorial lifts such as symmetric cubes are constructed from these forms, the property that only finitely many (often a single) Fourier coefficient suffices for distinction percolates: the symmetric cube representations constructed from inequivalent eigenforms are arithmetically independent and hence distinguishable by their Fourier expansions or associated Hecke eigenvalues in small degree.

For $\pi, \pi'$ cuspidal automorphic representations of $\mathrm{GL}(2)$, not of solvable polyhedral type, with isomorphic symmetric cubes ($$\operatorname{Sym}^3 \pi \simeq \operatorname{Sym}^3 \pi'$$), the classification is governed by the dichotomy: either $\pi$ and $\pi'$ are twists of one another, or, if not, a certain degree-36 Rankin–Selberg $L$-function $$L(s, \operatorname{Sym}^5 \pi \times (\pi\boxtimes \operatorname{Sym}^2 \pi') \otimes \omega^{-1})$$ has a simple pole at $s=1$ (Ramakrishnan, 2015). When the symmetric fifth power $\operatorname{Sym}^5 \pi$ is automorphic, the occurrence of the pole identifies the "icosahedral" case: the fiber of the symmetric cube map consists of two elements, interchanged by the nontrivial automorphism of $\mathbb{Q}(\sqrt{5})$, corresponding to icosahedral Galois representations (projective image $A_5$). The precise isomorphism

$$\operatorname{Sym}^5 \pi \simeq \pi \boxtimes \operatorname{Sym}^2 \pi' \otimes \omega$$

and its detection through the pole in the $L$-function provide fine-grained analytic separation in such scenarios.

3. Arithmetic Invariants and Periods: L-functions, Bessel Periods, and Decorrelation

Symmetric cube $L$-functions encode rich arithmetic information. For Maass–Hecke forms over fields like $\mathbb{Q}[\sqrt{-3}]$, infinitely many forms exist for which the central value $L(1/2, \operatorname{Sym}^3, \varphi)$ does not vanish—shown via equivalence with the nonvanishing of triple product integrals involving cubic theta functions (Hoffstein et al., 2021). A conjectural period formula of the form

$$|\langle \varphi, |\theta|^2 \rangle|^2 = c_\varphi \frac{L^*(1/2, \varphi, \operatorname{Sym}^3)}{L^*(1, \varphi, \operatorname{Sym}^2)}$$

links the triple period to the ratio of completed symmetric cube and square $L$-functions, generalizing Watson's formula to the cubic/metaplectic context.

On the algebraic side, motivic and Iwasawa-theoretic aspects are tied to the symmetric cube via the Bloch–Kato conjecture: if the critical $L$-value $L(\operatorname{Sym}^3 f, k + j)$ does not vanish, then the Bloch–Kato Selmer group for the critical twist $(\operatorname{Sym}^3 V_{f, \nu})(j)$ vanishes. The divisibility of the characteristic ideal of the appropriate Selmer group by the symmetric cube $p$-adic $L$-function (one inclusion of the Iwasawa main conjecture) reinforces the direct connection between special $L$-values and Galois cohomology (Loeffler et al., 2020). The nontriviality of Selmer classes, constructed via $p$-adically deformed Eisenstein series for $G_2$ and analysis of the preserved alternating trilinear form, is aligned with root number predictions—when the $L$-function vanishes to odd order, there exists a nontrivial Selmer element (Mundy, 2022).

Global Bessel periods arising in the context of $(\mathrm{SO}(5) \times \mathrm{SO}(2))$ correspond to standard quadratic base change components via the refined Gan–Gross–Prasad formula:

$$|B_K(\varphi,1;\psi)|^2 \doteq \frac{L(1/2, \pi)L(1/2, \pi \times \chi_d)}{L(1, \pi, \mathrm{Ad})L(1, \chi_d)}$$

for $K = \mathbb{Q}(\sqrt{d})$. Averaging products of such periods over quadratic fields yields a decorrelation estimate, conditional on GRH, showing that the periods associated to different forms become independent in distribution:

$$\frac{1}{D} \sum_{d \sim D} \prod_{i=1}^m |B_K(\varphi_i,1;\psi)| \ll (\log D)^{-m/8 + \epsilon}$$

(Hua et al., 21 Aug 2025). This quantitative decay signals that the arithmetic periods associated to symmetric cubes lose mutual correlation in mass as quadratic fields vary.

4. Effective Algebraic and Analytic Methods: Bounds, Fourier Coefficients, and Hecke Algebras

The distinction between symmetric cube lifts and their arithmetic independence crucially depends on properties of Fourier coefficients and Hecke eigenvalues. Explicit upper bounds for the minimal number of coefficients needed to distinguish eigenforms have both theoretical and practical consequences in algorithms and computational investigations (Ghitza, 2010). Generalizations under hypotheses such as GRH (or Cramér's conjecture) further sharpen these, reducing the required sample size for a given level N.

Rigidity phenomena for occurrences of polynomial identities between eigenforms (e.g., $h = af^2 + bfg + g^2$ with $f,g,h$ newforms) exhibit that such identities are only possible for finitely many triples (with fixed level N and nonzero $a, b$), and, under Maeda's conjecture, nonvanishing of the relevant Petersson inner products ensures that any identity of the form $f^2 = \sum c_i g_i$ follows from dimension considerations (Bao, 2017).

Arithmetic constraints on admissible Fourier coefficients—such as those arising from Lehmer's conjecture or Swinnerton-Dyer/Modular-congruence methods—limit the possible values these coefficients can achieve. For the Ramanujan $\tau$-function, explicit sets of forbidden values are determined, both unconditionally and under GRH, as in the result:

$$\tau(n) \notin \{-9, \pm 15, \pm 21, -25, -27, -33, \pm35, \pm45, \pm49, -55, \pm63, \pm77, -81, \pm91\}$$

for $n > 1$ (Hanada et al., 2020). The nonvanishing of such coefficients has direct bearing on the non-degeneracy and automorphy of symmetric cubes and their associated $L$-functions.

The effective generation of Hecke algebras under precise Sato–Tate-type bounds (conditional on GRH) carries over to symmetric cubes. For a Hecke eigenform $f$, the set of symmetric cube Hecke operators $\{T_p : p \leq X\}$ (with $$a_p(\operatorname{Sym}^3 f) = \alpha_p^3 + \alpha_p + \alpha_p^{-1} + \alpha_p^{-3}$$) generates the relevant Hecke algebra when $X$ is chosen in accordance with explicit analytic bounds on Chebyshev theta sums for $L(\operatorname{Sym}^3 f, s)$ (Moore, 2023).

5. Trace Formulas, Fundamental Lemmas, and the Characterization of Lifts

The analysis of symmetric cube $L$-functions and their periods often proceeds through comparisons of trace formulas for different groups, necessitating generalized fundamental lemmas. A key advance is the establishment of a fundamental lemma for Hecke correspondences between the spherical Hecke algebra of $PGL_3(F)$ and the spherical Hecke algebra of anti-genuine functions on the cubic cover $G'$ of $SL_3(F)$ (Friedberg et al., 9 Jul 2025). The correspondence

$\mathrm{Sh} : H \longrightarrow H'$

is defined so that relative orbital integrals match up to explicit transfer factors for all basis elements, satisfying

$$O(\xi, \omega(f)\phi_0) = t(\xi, \gamma) \cdot O'(\gamma, f')$$

for corresponding orbits $\xi, \gamma$. This isomorphism and orbital matching provide the key local ingredients for comparing relative trace formulas, eventually enabling the construction and characterization of global cubic Shimura lifts and explicating their images via period correspondences.

The comparison hinges on controlling and relating combinatorial and arithmetic entities—cubic exponential sums and Kloosterman sums—appearing on both sides, requiring precise recursive formulas and number-theoretic identities.

6. Analytic Behavior, Decorrelation, and Distributional Properties

The analytic properties of symmetric cube $L$-functions and related periods have implications for distributional results and quantum unique ergodicity (QUE) phenomena, as suggested by decorrelation estimates for Hecke eigenforms. For distinct forms $f, g$, quantitative decorrelation of values (as $\operatorname{Re}(s) \to \infty$ or weight $k \to \infty$) and the effective equidistribution of mass and zeros for their linear combinations follow from precise Watson-type formulas linking mixed moments to special values of high-rank $L$-functions and subconvexity estimates (Huang, 2022). Extension of these methods to higher symmetric power lifts, such as symmetric cubes, is a plausible direction, as the same analytic machinery can undergird similar results for equidistribution and mass phenomena in higher rank settings.

7. Open Problems and Further Developments

Outstanding directions include:

• Complete description and detection of "icosahedral" forms via symmetric cube fibers and associated degree-36 poles, and explicit criteria for non-twist equivalence.
• Full realization of Iwasawa main conjecture equalities (rather than one-sided inclusions) for symmetric cube Selmer groups, especially over higher level and Hilbert modular settings.
• Extension of explicit Hecke algebra generation for symmetric cube and higher symmetric power lifts in the absence of GRH.
• Further analytic progress in the unconditional nonvanishing of symmetric cube $L$-functions at the center for Maass and holomorphic forms, especially in settings beyond $\mathbb{Q}$.
• Generalizations of fundamental lemmas and trace formula comparisons to broader classes of automorphic coverings and lifts.

Symmetric cubes of algebraic regular Hecke eigenforms thus stand at the meeting point of several active and interrelated fields: functoriality and automorphy, explicit arithmetic invariants, Galois deformation theory, effective algebraic and analytic estimates, and the analytic theory of automorphic periods and $L$-functions. Their paper informs and motivates progress across the algebraic, analytic, and arithmetic frontiers of automorphic forms and the Langlands program.