We classify to some extent the pairs of group morphisms

which are element-conjugate but not globally conjugate. As an application, we study the case where

\mathrm{\Gamma }

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document}

is the Weil group of a p-adic local field, which is relevant to the recent approach to the local Langlands correspondence for

{\text{G}}_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{G}_2$$\end{document}

and

{\text{PGSp}}_6

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{PGSp}_6$$\end{document}

in Gan and Savin (Forum Math Pi 11:e28, 2023). As a second application, we improve some result in Kret and Shin (J Eur Math Soc 25(1):75–152, 2023) about

{\text{GSpin}}_7

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{GSpin}_7$$\end{document}

-valued Galois representations.

We consider

and

G_{48}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$G_{48}$$\end{document}

as finite subgroups of the Morava stabilizer group which acts on the height 2 Morava E-theory

{\mathbf{E}}_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textbf{E}}_2$$\end{document}

at the prime 2. We completely compute the G-homotopy fixed point spectral sequences of

{\mathbf{E}}_2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\textbf{E}}_2$$\end{document}

. Our computation uses recently developed equivariant techniques since Hill, Hopkins, and Ravenel. We also compute the

(\ast -{\sigma }_i)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(*-\sigma _i)$$\end{document}

-graded

Q_8

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Q_8$$\end{document}

- and

{\text{SD}}_{16}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\textrm{SD}_{16}$$\end{document}

-homotopy fixed point spectral sequences, where

{\sigma }_i

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sigma _i$$\end{document}

is a non-trivial one-dimensional representation of

Q_8

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$Q_8$$\end{document}

.

Sarnak’s Möbius disjointness conjecture states that Möbius function is disjoint to any zero entropy dynamics. We prove that Möbius disjointness conjecture holds for one-frequency analytic quasi-periodic cocycles which are almost reducible, which extends (Liu and Sarnak in Duke Math J 164(7):1353–1399, 2015; Wang in Invent Math 209:175–196, 2017) to the noncommutative case. The proof relies on quantitative version of almost reducibility.

We study abelian subcategories and torsion pairs in Abramovich–Polishchuk’s heart. And we apply the construction from Liu (J Reine Angew Math 770:135–157, 2021) on a full triangulated subcategory

in

D(X\times S)

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$D(X\times S)$$\end{document}

, for an arbitrary smooth projective variety S. We also define a notion of l-th level stability, which is a generalization of the slope stability and the Gieseker stability. We show that for any object E in Abramovich–Polishchuk’s heart, there is a unique filtration whose factors are l-th level semistable, and the phase vectors are decreasing in a lexicographic order.

We prove that the weight polytope of the Hurwitz form of a polarized smooth toric variety coincides with the convex hull of the characteristic vectors introduced in Ogusu and Sano (Characteristic vectors for the Hurwitz polytopes of toric varieties, preprint 2023) with respect to all regular triangulations of the momentum polytope. Our proof relies on the combination of the two slope formulas of K-energy (Boucksom et al. in J Eur Math Soc 21(9):2905–2944, 2019; Paul in Ann Math 175(1):255–296, 2012) in the toric setting.