论文标题
Q的四个不可还原兼容系统的单肌
Monodromy of four dimensional irreducible compatible systems of Q
论文作者
论文摘要
让$ f $是一个完全真实的领域，而$ n \ leq 4 $ a自然数字。我们研究任何$ n $二维的严格兼容系统的单型组$ \ {ρ_λ\}_λ$的$λ$ - $ f $ $ f $的代表，具有独特的hodge-tate数字，因此$ρ_{λ_0} $对于某些$λ_0$是不可转让的。当$ f = \ mathbb {q} $，$ n = 4 $，而$ρ_{λ_0} $是完全符号的，则获得以下断言。 （i）表示几乎所有$λ$的表示形式$ρ_λ$都是完全符合的。 （ii）如果此外，如果$ρ_{λ_0} $的similiture $μ_{λ_0} $是奇怪的，则系统$ \ {ρ_λ\}_λ$可能是自动形态的，可能是残留的自动型$ \barρ_λ（\barρ_λ（\barρ_λ） $ \ text {sp} _4（\ mathbb {f} _ \ ell）$几乎所有$λ$。
Let $F$ be a totally real field and $n\leq 4$ a natural number. We study the monodromy groups of any $n$-dimensional strictly compatible system $\{ρ_λ\}_λ$ of $λ$-adic representations of $F$ with distinct Hodge-Tate numbers such that $ρ_{λ_0}$ is irreducible for some $λ_0$. When $F=\mathbb{Q}$, $n=4$, and $ρ_{λ_0}$ is fully symplectic, the following assertions are obtained. (i) The representation $ρ_λ$ is fully symplectic for almost all $λ$. (ii) If in addition the similitude character $μ_{λ_0}$ of $ρ_{λ_0}$ is odd, then the system $\{ρ_λ\}_λ$ is potentially automorphic and the residual image $\barρ_λ(\text{Gal}_\mathbb{Q})$ has a subgroup conjugate to $\text{Sp}_4(\mathbb{F}_\ell)$ for almost all $λ$.
