On the images of the Galois representations attached to generic automorphic representations of GSp(4)

Abstract

By making use of Langlands functoriality between GSp(4) and GL(4), we show that the images of the Galois representations attached to "genuine" globally generic automorphic representations of GSp(4) are "large" for a set of primes of density one. Moreover, by using the notion of (n, p)-groups (introduced by Khare, Larsen and Savin) and generic Langlands functoriality from SO(5) to GL(4) we construct automorphic representations of GSp(4) such that the compatible system attached to them has large image for all primes.

On the images of Galois representations attached to low weight Siegel modular forms

“…In this section, we prove Theorem 1.2. Our arguments generalise those of [14][15][16] to the case that 𝜋 is non-cohomological. Moreover, Lemma 7.3 allows us to strengthen the results of [16] even in the cohomological case (see [16,Rmk.…”

“…Indeed, in Lemma 7.3, we use it to prove that 𝜌 𝜆 cannot have an even two-dimensional subrepresentation for infinitely many 𝜆 ∈ . In the cohomological case, this argument allows us to strengthen the result of [16] to apply to all but finitely many primes, rather than just a density 1 set of primes.…”

“…Hence, $$\overline{\rho }_\lambda$$ is irreducible for all but finitely many $$\lambda \in \mathcal {L}$$. The remainder of the proof of Theorem 1.2 is exactly the same as for the cohomological case [16, 3.2–3.5]. By the classification of the maximal subgroups of $$\operatorname{GSp}_4(\mathbf {F}_{\ell ^n})$$ [30], if $$\overline{\rho }_\lambda$$ is irreducible and does not contain $$\operatorname{Sp}_4(\mathbf {F}_\ell )$$, then one of the following cases must hold: (i)the image contains a reducible index two subgroup, that is, $$\overline{\rho }_\lambda$$ is induced from a quadratic extension; (ii)$$\overline{\rho }_\lambda$$ is isomorphic to the symmetric cube of a two‐dimensional representation; (iii)the image is a small exceptional group. By using the description of the image of inertia as in Proposition 7.1, Dieulefait–Zenteno show that each of these cases can only occur for finitely many $$\ell \in \mathcal {L}$$.$$\Box$$…”

“…For high weight Siegel modular forms, irreducibility for all but finitely many primes follows from the work of Ramakrishnan [38], while the analogue of Theorem 1.2 follows from [5, Prop. 5.3.2] and from the work of Dieulefait–Zenteno [16], but only for a set of primes $$\lambda$$ of residual Dirichlet density 1. Conjecturally, $$\rho _\lambda$$ should be irreducible for all primes $$\lambda$$, and $$\overline{\rho }_\lambda$$ should be irreducible for all but finitely many primes $$\lambda$$.…”

On the Lang--Trotter conjecture for Siegel modular forms

“…By work of Ramakrishnan [Ram13], Dieulefait-Zenteno [DZ20] and the third author [Wei18], the image of ρ λ is generically large, in the following sense. Let L ′ be the set of rational primes ℓ such that ℓ ≥ 5, such that…”