Lifting and automorphy of reducible mod p Galois representations over global fields

作者：N. Fakhruddin, Chandrashekhar B. Khare, Stefan Patrikis · 发表于：Inventiones Mathematicae · 年份：2020 · DOI：10.1007/s00222-021-01085-7 · 被引用次数：12 · 研究领域：Mathematics

We prove the modularity of most reducible, odd representations ρ¯:ΓQ→GL2(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)$$\end{document} with k a finite field of characteristic an odd prime p. This is an analogue of Serre’s celebrated modularity conjecture (which concerned irreducible, odd representations ρ¯:ΓQ→GL2(k)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)$$\end{document}) for reducible, odd representations. Our proof lifts ρ¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{\rho }}$$\end{document} to an irreducible geometric p-adic representation ρ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho $$\end{docum...