论文标题
$ gsp_4 $的塞雷重量完全真实的领域
Serre weights for $GSp_4$ over totally real fields
论文作者
论文摘要
我们证明了给定自重的mod $ p $ p $ galois代表$ \overlineρ：{\ rm gal}（\ overline {f}/f）\ longrightArrow {\ rm gsp} _4（\ rm gsp） Prime $ p> 2 $在充足条件下使用Barnet-Lamb，Gee，Geraghty和Taylor开发的自动式提升技术。作为一个应用程序，当$ p $完全分为$ f $时，我们证明了Serre的权重猜想的变体，以$ \overlineρ$。通过遵循托比·吉（Toby Gee）的哲学来完成我们的塞雷猜想的表述。将这些结果应用于$ f = \ mathbb {q} $的情况下，并详细研究具有某些规定的特性，我们还定义了经典的（幼稚）塞雷的重量。从某种意义上说，这种重量将是可能的经典重量之间的最小重量，这在霍洛型Siegel Hecke eigen cusp的候选中发生了，其水平为PRIME至$ p $。主要的任务是通过仅假设足够的条件来构建一个潜在的普通汽车升降机，以$ \Overlineρ$构建。本文的主要定理还扩展了Barnet-Lamb，Gee和Geraghty获得的许多结果。
We prove the existence of a potentially diagonalizable lift of a given automorphic mod $p$ Galois representation $\overlineρ:{\rm Gal}(\overline{F}/F)\longrightarrow {\rm GSp}_4(\overline{\mathbb{F}}_p)$ for any totally real field $F$ and any rational prime $p>2$ under the adequacy condition by using automorphic lifting techniques developed by Barnet-Lamb, Gee, Geraghty, and Taylor. As an application, when $p$ is split completely in $F$, we prove a variant of Serre's weight conjecture for $\overlineρ$. The formulation of our Serre conjecture is done by following Toby Gee's philosophy. Applying these results to the case when $F=\mathbb{Q}$ with a detailed study of potentially diagonalizable, crystalline lifts with some prescribed properties, we also define classical (naive) Serre's weights. This weight would be the minimal weight among possible classical weights in some sense which occur in candidates of holomorphic Siegel Hecke eigen cusp forms of degree 2 with levels prime to $p$. The main task is to construct a potentially ordinary automorphic lift for $\overlineρ$ by assuming only the adequacy condition. The main theorems in this paper also extend many results obtained by Barnet-Lamb, Gee and Geraghty for potentially ordinary lifts and Gee and Geraghty for companion forms.
