Abstract:

We study short crystalline, minimal, essentially self-dual deformations of a mod $p$ non-semisimple Galois representation $\overline{\sigma}$ with $\overline{\sigma}^{{\rm ss}}=\chi^{k-2}\oplus\rho\oplus\chi^{k-1}$, where $\chi$ is the mod $p$ cyclotomic character and $\rho$ is an absolutely irreducible reduction of the Galois representation $\rho_f$ attached to a cusp form $f$ of weight $2k-2$. We show that if the Bloch-Kato Selmer groups $H^1_f({\bf Q},\rho_f(1-k)\otimes{\bf Q}_p/{\bf Z}_p)$ and $H^1_f({\bf Q},\rho(2-k))$ have order $p$, and there exists a characteristic zero absolutely irreducible deformation of $\overline{\sigma}$ then the universal deformation ring is a dvr. When $k=2$ this allows us to establish the modularity part of the Paramodular Conjecture in cases when one can find a suitable congruence of Siegel modular forms. As an example we prove the modularity of an abelian surface of conductor 731. When $k>2$, we obtain an $R^{{\rm red}}=T$ theorem showing modularity of all such deformations of $\overline{\sigma}$.

摘要：

我们研究mod $ p $非半简单Galois表示$ \ overline {\ sigma} $与$ \ overline {\ sigma} ^ {{\ rm ss}} = \\ chi的短晶体，最小，基本自对偶变形^ {k-2} \ oplus \ rho \ oplus \ chi ^ {k-1} $，其中$ \ chi $是mod $ p $的环线特征，而$ \ rho $是绝对不可减少的伽罗瓦表示$ \ rho_f $附加到权重为$ 2k-2 $的尖端形式$ f $中。我们表明如果Bloch-Kato Selmer将$ H ^ 1_f（{\ bf Q}，\ rho_f（1-k）\ otimes {\ bf Q} _p / {\ bf Z} _p）$和$ H ^ 1_f组（{\ bf Q}，\ rho（2-k））$的阶次为$ p $，并且存在$ \ overline {\ sigma} $零绝对不可约的特征变形，则通用变形环为dvr。当$ k = 2 $时，如果可以找到合适的Siegel模块化形式的全等，则这使我们能够建立Paramodular Conjecture的模块化部分。