17 problems

Let $G$ and $H$ be the groups occurring in the Local Langlands correspondence, let $\boldsymbol{Z}[[\sigma]$ be the group ring of…

Dwyer–Mahowald–Kuhn conjecture. The displayed isomorphism on homotopy groups is induced by an equivalence of spectra

Let $X$ be a finite spectrum of type $n-1$, let $L_n$ denote localization at the $n$th monochromatic level, let $X_p^{\wedge}$ denote the $p$-completion of $X$, and let $t$ denote…

Let $L/k$ be a Galois extension of number fields with Galois group $G$, and let $\mathcal{O}_L$ be the ring of integers of $L$. For each $i\in\mathbb{Z}$, write…

Let $G$ be a reductive algebraic group over a non-Archimedean local field $F$ of residue characteristic $p$, let $\sigma$ be an $F$-defined automorphism of prime order $\ell\neq p$…

Let $G$ be the reductive group and $\rho$ the order-$\ell$ automorphism described in the paper, with $\Gamma=\langle\rho\rangle$; let $\Pi$ be an irreducible $\ell$-modular represe…

In the setting of mod-$l$ representations of a reductive algebraic group $\textbf{G}$, let $\gamma$ be an automorphism of $\textbf{G}$ of prime order $l$, and let…

Let $F$ be a non-Archimedean local field, let $G$ be the $F$-points of a connected reductive group defined over $F$, and let $\sigma$ be an automorphism of $G$ of prime order $l$.…

The local Langlands correspondence is expected to relate mod $\ell$ representations through Tate cohomology. Treumann–Venkatesh's conjecture. Tate cohomology should realize functor…

Let $l$ be a prime number, let $F$ be a number field, let $\operatorname{{\bf G}}$ be a reductive algebraic group over $F$, and let $\sigma$ be an automorphism of order $l$ of…

Treumann–Venkatesh's linkage conjecture. Linkage is compatible with functorial transfer: if $\pi$ is linked to $\Pi$, then $\pi$ transfers to $\Pi$ under the Local Langlands corres…

Let $V$ be a self-dual integral form of the monster vertex algebra, let $g$ be a Fricke element, and suppose that $g^j$ is a Fricke element for every integer $j>0$. Weaker Borcherd…

Composite-order graded Brauer character conjecture. Each of these series is a linear combination of Hauptmoduls. This predicts a uniform Hauptmodul description for the graded Braue…

Let $V^\natural_{\mathbb{Z}}$ be a self-dual integral form of $V^\natural$ with Monster symmetry, and let $W_\infty M$ denote the big Witt vectors of a graded module $M$.…

Let $\mathbb{M}$ be the Monster group, let $V^\natural_{\mathbb{Z}}$ be a self-dual integral form of the Moonshine vertex operator algebra, and let $g\in\mathbb{M}$ be a Fricke ele…

Let $F\supset E$ be a degree $l$ extension of characteristic-zero non-archimedean local fields with residue characteristic $p$, where $p$ and $l$ are distinct primes. Let $\pi$ be…

Let $G$ be a finite group and let $k$ be a field of characteristic $p$. Let $M$ be an indecomposable finitely generated $kG$-module such that $\operatorname{H}^*(G,M)\neq\{0\}$. Th…