论文标题
Saito-Kurokawa类型的Galois表示限制的不可约性
Irreducibility of limits of Galois representations of Saito-Kurokawa type
论文作者
论文摘要
我们证明了（在某些假设下）一系列不可约的限制$σ_2$的不可约性，本质上是自dual galois表示$σ_K：g _ {\ m athbf {q}}} \ to \ to \ to \ mathrm {gl} $ p $ -adic sense）哪个mod $ p $减少（半简化后）至$ 1 \ oplusρ\ oplusχ$，带有$ρ$不可修复，确定性$χ$的二维，其中$χ$是mod $ p $ p $ cyclotomic cyclotomic carem。更确切地说，我们假设$σ_K$是结晶（特定的权重），而Siegel-disminary则为$ p $。这种表示形式在$ p $ - 亚种模块化表单和限制的属性的研究中出现，因为$ k \ to $ 2 $在偏见的猜想的背景下似乎很重要。结果是从两个Selmer组的有限性中得出的，这些Selmer组由椭圆形模块化表格的$ P $ -ADIC $ L $ - 值（产生$ρ$）来控制，我们认为这是非零的。
We prove (under certain assumptions) the irreducibility of the limit $σ_2$ of a sequence of irreducible essentially self-dual Galois representations $σ_k: G_{\mathbf{Q}} \to \mathrm{GL}_4(\overline{\mathbf{Q}}_p)$ (as $k$ approaches 2 in a $p$-adic sense) which mod $p$ reduce (after semi-simplifying) to $1 \oplus ρ\oplus χ$ with $ρ$ irreducible, two-dimensional of determinant $χ$, where $χ$ is the mod $p$ cyclotomic character. More precisely, we assume that $σ_k$ are crystalline (with a particular choice of weights) and Siegel-ordinary at $p$. Such representations arise in the study of $p$-adic families of Siegel modular forms and properties of their limits as $k\to 2$ appear to be important in the context of the Paramodular Conjecture. The result is deduced from the finiteness of two Selmer groups whose order is controlled by $p$-adic $L$-values of an elliptic modular form (giving rise to $ρ$) which we assume are non-zero.
