Let $p\ge 5$ be a prime number, $\mathbb{F}$ a finite field of characteristic p and let $\overline{\chi }$ be the mod-p cyclotomic character. Let $\overline{\rho }:{\mathrm{G}}_{\mathbb{Q}}\to {\mathrm{GL}}_2(\mathbb{F})$ be a Galois representation such that the local representation ${\overline{\rho }}_{\mathrm{\upharpoonright }{\text{G}}_{{\mathbb{Q}}_p}}$ is flat and irreducible. Further, assume that $\text{det}\overline{\rho }=\overline{\chi }$. The celebrated theorem of Khare and Wintenberger asserts that if $\overline{\rho }$ satisfies some natural conditions, there exists a normalized Hecke-eigencuspform $f={\sum }_{n\ge 1}{a}_n{q}^n$ and a prime $\mathfrak{p}|p$ in its field of Fourier coefficients such that the associated $\mathfrak{p}$-adic representation ${\rho }_{f,\mathfrak{p}}$ lifts $\overline{\rho }$. In this manuscript we prove a refined version of this theorem, namely, that one may control the valuation of the p-th Fourier coefficient of f. The main result is of interest from the perspective of the p-adic Langlands program.

让 $磷\ge 5$ 成为质数， $\mathbb{F}$特征p的有限域，令$\overline{\chi }$是 mod- p 分圆特征。让$\overline{\rho }：{\mathrm{G}}_{\mathbb{问}}\to {\mathrm{GL}}_2(\mathbb{F})$ 是一个伽罗瓦表示，使得局部表示 ${\overline{\rho }}_{\mathrm{\upharpoonright }{\text{G}}_{{\mathbb{问}}_{磷}}}$是平坦且不可约的。进一步，假设$\text{检测}\overline{\rho }=\overline{\chi }$. 著名的 Khare 和 Wintenberger 定理断言，如果$\overline{\rho }$ 满足一些自然条件，存在归一化的 Hecke-eigencuspform $F={\sum }_{n\ge 1}{\mathrm{一种}}_n{q}^n$ 和一个素数 $\mathfrak{磷}|磷$ 在其傅立叶系数领域，使得相关联 $\mathfrak{磷}$-adic表示 ${\rho }_{F,\mathfrak{磷}}$ 升降机 $\overline{\rho }$. 在这篇手稿中，我们证明了这个定理的一个改进版本，即可以控制f的第 p个傅立叶系数的估值。从p- adic Langlands 程序的角度来看，主要结果很有趣。