On automorphy of certain Galois representations of GO4-type

Abstract

Let ρ : Gal(Q/Q) → GO 4 (Q p ) be a continuous representation. We prove (potential) automorphy theorems for certain types of ρ. Our results include several cases in which the HodgeTate weights are irregular. Finally, we prove (potential) automorphy for certain compatible systems of representations of GO 4 -type, which includes certain compatible systems constructed from Scholl motives.

Equidistribution theorems for holomorphic Siegel modular forms for $GSp_4$; Hecke fields and $n$-level density

“…The cases (3), ( 4) and ( 5) are excluded since F is genuine with the argument above in the case (3). Then by applying Theorem 1.0.1 of [27] for a sufficiently large ℓ ∈ Σ (note that they used the notation SGO(4) instead of GSO( 4)), we see that F is a Rankin-Selberg convolution which contradicts with the assumption on F .…”

Monodromy of four‐dimensional irreducible compatible systems of Q$\mathbb {Q}$

“…Two impetuses of the problem are the Mumford‐Tate conjecture that concerns the $$\lambda$$‐independence of $$\mathbf {G}_\lambda ^\circ$$ for motivic compatible systems and the irreducibility conjecture that predicts the absolute irreducibility of $$\rho _\lambda$$ for compatible systems attached to algebraic cuspidal automorphic forms of $$\mathrm{GL}_n(\mathbb {A}_K)$$. There have been many studies on the irreducibility and big images of compatible system since 1970s, assuming that $$\lbrace \rho _\lambda \rbrace _\lambda$$ is motivic, see [41, 44, 47] for abelian varieties, [13, 18] for $$n=3$$, and [14, 15] for $$n=4$$, automorphic, see [17, 42, 43, 53] for (Hilbert) modular forms, [2, 13] for $$n=3$$, [11, 12, 16, 40, 57] for $$n=4$$, [4, 25, 58] for $$n\leqslant 6$$, and [1] in general, and more recently, a weakly compatible system (Definition 2.1), see [1, 35, 38, 39], and [18]. …”

“…Denote by $$\mathbf {G}_\lambda ^{\operatorname{der}}$$ the derived group of the identity component $$\mathbf {G}_\lambda ^\circ$$. The representation $$\rho _\lambda$$ is said to be fully orthogonal if $$\mathbf {G}_\lambda ^{\operatorname{der}}=\mathrm{SO}_n$$ with $$\langle \nobreakspace ,\nobreakspace \rangle$$ symmetric; fully symplectic if $$\mathbf {G}_\lambda ^{\operatorname{der}}=\mathrm{Sp}_n$$ with $$\langle \nobreakspace ,\nobreakspace \rangle$$ skew‐symmetric. Below is our main result on four‐dimensional fully symplectic Galois representations of $$\mathbb {Q}$$; see [35] for four‐dimensional fully orthogonal Galois representations. Theorem Suppose $$\lbrace \rho _\lambda :\operatorname{Gal}_\mathbb {Q}\rightarrow \mathrm{GL}_4(\overline{E}_\lambda )\rbrace _\lambda$$ is a strictly compatible system of $$\mathbb {Q}$$ with distinct Hodge–Tate numbers.…”

“…Irreducibility and big images for four‐dimensional compatible system $$\lbrace \rho _\lambda \rbrace _\lambda$$ of $$\mathbb {Q}$$ are studied in the works [11, 15, 16, 35, 40, 57], in which (potential) automorphy techniques such as Serre's conjecture and [1] are used to rule out two‐dimensional factors of $$\rho _\lambda$$. This is also the main idea for Theorem 1.1(iv) and Theorem 1.2.…”

“…The representation 𝜌 𝜆 is said to be • fully orthogonal if 𝐆 der 𝜆 = SO 𝑛 with ⟨ , ⟩ symmetric; • fully symplectic if 𝐆 der 𝜆 = Sp 𝑛 with ⟨ , ⟩ skew-symmetric. Below is our main result on four-dimensional fully symplectic Galois representations of ℚ; see [35] for four-dimensional fully orthogonal Galois representations. Theorem 1.2.…”