论文标题
潜在的对角线大量的模块化升降机
Potentially diagonalizable modular lifts of large weight
论文作者
论文摘要
We prove that for a Hecke cuspform $f\in S_k(Γ_0(N),χ)$ and a prime $l>\max\{k,6\}$ such that $l\nmid N$, there exists an infinite family $\{k_r\}_{r\geq 1}\subseteq\mathbb{Z}$ such that对于每个$ k_r $,在s_ {k_r}中都有一个cusp表格$ f_ {k_r} \ in s_ {k_r}(γ_0(n),χ)$,使得deligne表示$ρ_{f_ {f_ {k_r,l}} $是crystaline且潜在的diagonalizable of $ diagonalizable of $的$ plip__} l} l} f} f} f}。 When $f$ is $l$-ordinary, we base our proof on the theory of Hida families, while in the non-ordinary case, we adapt a local-to-global argument due to Khare and Wintenberger in the setting of their proof of Serre's modularity conjecture, together with a result on existence of lifts with prescribed local conditions over CM fields, a flatness result due to Böckle and a local dimension result by Kisin.我们讨论了我们在$ \ mathrm {gl} _ {2n} $ - 在更高级别案例中的表示的持续研究中的结果和暂定的未来应用。
We prove that for a Hecke cuspform $f\in S_k(Γ_0(N),χ)$ and a prime $l>\max\{k,6\}$ such that $l\nmid N$, there exists an infinite family $\{k_r\}_{r\geq 1}\subseteq\mathbb{Z}$ such that for each $k_r$, there is a cusp form $f_{k_r}\in S_{k_r}(Γ_0(N),χ)$ such that the Deligne representation $ρ_{f_{k_r,l}}$ is a crystaline and potentially diagonalizable lift of $\overlineρ_{f,l}$. When $f$ is $l$-ordinary, we base our proof on the theory of Hida families, while in the non-ordinary case, we adapt a local-to-global argument due to Khare and Wintenberger in the setting of their proof of Serre's modularity conjecture, together with a result on existence of lifts with prescribed local conditions over CM fields, a flatness result due to Böckle and a local dimension result by Kisin. We discuss the motivation and tentative future applications of our result in ongoing research on the automorphy of $\mathrm{GL}_{2n}$-representations in the higher level case.