Computing the local $2$-component of a non-selfdual automorphic representation of $\mathrm{GL}_3$

Abstract: In this paper, we explicitly determine the local $2$-adic component of a non-selfdual automorphic representation $Π$ of $\mathrm{GL}<em>3$ constructed by van Geemen and Top. We prove that $Π_2$ is a parabolically induced representation of $\mathrm{GL}_3(\mathbb{Q}_2)$ given by $$Π_2 = \mathrm{Ind}_P^{{\mathrm{GL}_3(\mathbb{Q}_2)}(π\boxtimes} χ)$$, where $P$ is the standard parabolic subgroup of $\mathrm{GL}_3$ with Levi subgroup $\mathrm{GL}_2 \times \mathrm{GL}_1$, $χ$ is an unramified character of $\mathbb{Q}_2<sup>\times$ satisfying $χ(2) = -2\sqrt{-1}$, and $π$ is a supercuspidal representation of $\mathrm{GL}_2(\mathbb{Q}_2)$. Furthermore, we describe $π$ explicitly as a compactly induced representation $$π= \mathrm{c-Ind}</em>{J_α}^{{\mathrm{GL}_2(\mathbb{Q}_2)}} Λ$$ and determine the representation $Λ$ explicitly. The proof relies on explicit computations of Hecke eigenvalues using computer calculations. The automorphic representation $Π$ is realized in the cuspidal cohomology of the congruence subgroup $Γ<em>0(128) \subset \mathrm{SL}_3(\mathbb{Z})$. By computing the Hecke eigenvalues of an associated Hecke eigenvector, we are able to uniquely identify the local structure of $Π_2$. As an application, we obtain an explicit description of the $2$-adic local component of the Galois representation $ρ</em>{\mathrm{vGT},\ell}$ associated with $Π$.

• The paper computes the local 2-adic component by explicitly determining its inducing data from a supercuspidal GL2 representation and an unramified character.
• It employs rigorous Hecke eigenvalue computations and Bushnell-Kutzko theory to rule out non-inducible cases and confirm the parabolic induction structure.
• The study demonstrates vital connections to Galois representations and local-global compatibility within the Langlands program.

Computation of the Local 2-Adic Component of a Non-Selfdual Automorphic Representation of $\mathrm{GL}_3$

Overview and Context

This work provides a comprehensive computation of the local $2$-adic component $\Pi_2$ of a non-selfdual cuspidal automorphic representation $\Pi$ of $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$ originally constructed by van Geemen and Top. The significance of $\Pi$ lies in its non-selfduality and its connection to a Galois representation arising in the transcendental motive of a surface. The author establishes, via explicit Hecke eigenvalue computations and representation-theoretic analysis, an explicit description of the local constituent $\Pi_2$ as a parabolically induced representation from a supercuspidal representation $\pi$ of $\mathrm{GL}_2(\mathbb{Q}_2)$ and an unramified character $\chi$.

The methodology leverages both classical techniques and computational tools, with careful attention to the structure of modular symbols, Hecke operators, and the Bushnell-Kutzko theory of types, ultimately producing the explicit type data needed to define the arising supercuspidal representation.

Main Results

The primary result asserts that the local $2$0-component $2$1 has the following explicit form: $2$2 where $2$3 is the standard parabolic subgroup with Levi $2$4, $2$5 is an unramified character of $2$6 with $2$7, and $2$8 is the supercuspidal representation of $2$9 compactly induced from a character $\Pi_2$0 of a compact-mod-center subgroup $\Pi_2$1 associated to a ramified simple stratum.

The character $\Pi_2$2 and the component $\Pi_2$3 are determined as follows:

• $\Pi_2$4 yields a ramified quadratic extension $\Pi_2$5.
• $\Pi_2$6, with $\Pi_2$7 an open compact-mod-center subgroup of $\Pi_2$8.
• $\Pi_2$9 is determined on generators by:
  • $\Pi$0,
  • $\Pi$1,
  • $\Pi$2 for a suitable additive character $\Pi$3.

This explicit determination follows from stepwise elimination of other types by detailing the paramodular invariance properties, the structure of Hecke eigenvalues, and intricate calculations with coset representatives under the action of congruence subgroups.

Technical Analysis

Cohomological and Hecke Theoretic Framework

The automorphic representation $\Pi$4 is realized within cuspidal cohomology $\Pi$5 of the congruence subgroup $\Pi$6. The associated Hecke eigenvector $\Pi$7 is uniquely determined through computational calculation of Hecke eigenvalues for various double cosets and the use of modular symbol techniques.

These calculations, based on algorithms from [ASH1984412] and [MR1474576], provide strong numerical evidence for the invariance and uniqueness of the modular form and play a crucial role in identifying the type of the local representation.

Elimination of Supercuspidal and Non-inducible Cases

The paper gives a careful argument showing that $\Pi$8 is not itself a supercuspidal representation of $\Pi$9, based on the nonvanishing of eigenvalues for certain Hecke operators. By exploiting explicit double coset decompositions and modular symbol computations, all alternative cases (including discrete series and other parabolic types) are systematically excluded using invariance properties and eigenvalue identities.

Identification of Supercuspidal Data

The representation $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$0 is shown to be compactly induced from an explicit extension $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$1 and a character $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$2 of the corresponding type lattice subgroup. This is achieved by reconciling the calculation of Hecke eigenvalues with the local Langlands and Bushnell-Kutzko theories, as well as through explicit manipulations of generators and relations for the relevant compact open subgroups.

The computation of Hecke operators indexed by various nontrivial elements allows the extraction of the values of $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$3 on specific group elements, and thus a full determination of the type.

Application to Galois Representations

Given the compatibility between the automorphic and Galois representations established in previous work (Ito et al., 2018), this computation of the local component $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$4 gives an explicit description of the inertia and Frobenius action on the associated $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$5-adic Galois representation $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$6 (for $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$7), when restricted to the decomposition group at $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$8: $\mathrm{GL}_3(\mathbb{A}_\mathbb{Q})$9 where $\Pi$0 denotes the two-dimensional irreducible representation attached to $\Pi$1 via the local Langlands correspondence.

Numerical and Structural Highlights

• Dimension and uniqueness of Hecke eigenspaces: The paper verifies the dimension of the relevant cohomological space as $\Pi$2, and the eigenspace for the Hecke operator at $\Pi$3 as $\Pi$4. This ensures isolation of the automorphic representation in question.
• Explicit Hecke eigenvalue data: The computed eigenvalues for the Hecke operators at $\Pi$5 and further indexes crucially support the explicit identification and rule out all incorrect conjectural types for $\Pi$6.
• Determination of the conductor: The supercuspidal constituent $\Pi$7 has $\Pi$8, a substantial conductor, indicating deep ramification properties.

Theoretical and Practical Implications

The explicit determination of a local component at a ramified prime for a non-selfdual representation with Galois-theoretic significance is especially valuable in the study of the local-global compatibility in the Langlands program. The approach demonstrates the necessity of integrating computational tools with theoretical representation-theoretic methods for such explicit calculations.

The results also provide concrete data for the test and refinement of conjectures within the broader $\Pi$9-adic and mod-$\Pi_2$0 local Langlands correspondence, particularly in the study of types and explicit local transfers. The explicit local factors determined here serve as essential input for computations of $\Pi_2$1-functions, local constants, and further arithmetic invariants of associated motives.

Future directions include the application of similar methods to higher rank settings and to the explicit computation of local automorphic and Galois types in more general situations, especially those involving non-generic or non-tempered constituents.

Conclusion

This paper establishes a precise and explicit description of the local $\Pi_2$2-adic component $\Pi_2$3 of a distinguished non-selfdual automorphic representation of $\Pi_2$4, fully characterizing its inducing data, supercuspidal support, and compatibility with the associated Galois representation. The work exhibits the deep interplay between cohomological modular forms, Hecke theory, the theory of types, and explicit computation, providing a template for analogous local analyses in more intricate automorphic and arithmetic settings.

Reference: "Computing the local $\Pi_2$5-component of a non-selfdual automorphic representation of $\Pi_2$6" (2603.25367)