26 problems

Let $G$ be a complex simple group of type $E_8$, let $\widehat{G}^d$ denote its Dirac series, and let $K$ be a maximal compact subgroup of $G$. A representation…

Let $A$ be an arbitrary Albert algebra, and let ${\rm Str}(A)$ be its structure group. The inner structure group is generated by the so-called $U$-operators, and central homothetie…

Let the magic square be indexed by composition algebras $\mathbb A$ and $\mathbb B$, with varieties in its chart associated to the tensor product $\mathbb A\otimes\mathbb B$. Weins…

Let $\operatorname{\bf{G}}_0$ be a simple group of type $F_4$, and let $F_0$ be the generalized Frobenius map defining a Ree group of type $F_4$. Write…

Nonexistence conjecture. No such maximal subgroup $H$ exists.

Let $n$ be such that there is a cuspidal Hecke eigenform of weight $2n+12$ for $\operatorname{SL}_{2}(\mathbb{Z})$, and let $\pi$ be the associated level-one algebraic automorphic…

The F4-type Galois representation counting conjecture. The number of equivalence classes of such representations with $\mathrm{HT}(\rho)=\mathrm{HT}(a,b,c,d)$ and $a,b,c,d\geq 1$ i…

Kim's conjecture. The parameter $\psi$ is associated with a distinguished orbit $\mathcal{O}$ of $G^{\vee}$, and

Pole classification conjecture. For $\operatorname{Re}(s)>0$, if $\chi$ is trivial, then $\mathcal{E}_{P_2}^{E_8}(s,\chi)$ can have a double pole at $s=\frac32,\frac52,\frac72$ and…

Quaternionic exceptional groups possess a distinguished class of automorphic functions called quaternionic modular forms, with Fourier expansions and Fourier coefficients. Let $G$…

Let $J=\{\alpha_1,\alpha_2,\ldots,\alpha_6\}$. Let $\mathfrak{j}$ be either of the two elements of $\mathcal{M}_{\mathbf G}$ of the form $(\mathbf L_J,\mathcal{O}_0,\zeta_3)$, wher…

Let $G$ be an exceptional group, let $k$ have good characteristic for $G$, and let the two unipotent conjugacy classes under consideration consist of elements whose orders need not…

Let $g_1=s_1u_1$ be one of the two possibilities described in the source, with $u_1$ the unipotent component and $x_{14}$ the specified unipotent element. Unipotent-component conje…

Let $\mathbf G$ be an adjoint group of type $E_6$ over a field of good characteristic, and let $\mathbf G(p)$ denote its group of points over the prime field. Consider the characte…

Pole–reducibility conjecture. In the right complex half-plane, a degenerate Eisenstein series has a pole if and only if the corresponding local degenerate principal series is reduc…

Let $k$ be a local field. Let $J={\mathrm{SL}}_2(k)\ltimes V$ be the subgroup of the split group ${\mathrm{G}}_2(k)$ described above, let $\widetilde{\mathrm{SL}}_2(k)$ be the meta…

Let $q$ be a power of the prime $p$, let $\tilde G$ be an exceptional Chevalley group over $\boldsymbol{F}_q$, and let $V$, $W$, and $\tilde G$, $hat G$ be the modules and groups d…

Exceptional U-duality conjecture. The U-dualities of M-theory compactified on an $n$-torus are given by a sequence of discrete exceptional groups

Let $G$ be a group of type ${\rm E}_8$ over a field of characteristic $p=5$, let $\mathfrak{g}=\operatorname{Lie}(G)$, and let $\mathfrak{w}$ be the maximal subalgebra referred to…

Consider the commuting pair $(SO_3,G_2)$ and a cuspidal irreducible representation $\pi$ of $SO_3({\bf A})$. In the analogous classical setting, if the lift to…

Let $F$ be a field of cohomological dimension $2$, and let $H_{{\text{et}}}^1(F,\mathrm{E}_8)$ denote the pointed set of isomorphism classes of groups of type $\mathrm{E}_8$ over…

Smoothness conjecture. The scheme-theoretic stabilizer of $kf$ in $\operatorname{PGL}(V)$ is smooth for all fields $k$, and the scheme-theoretic stabilizer $S$ of $f$ is smooth whe…

The Mermin pentagrams are configurations of commuting four-qubit observables, and their total number is $12096$, equal to the order of $G_2(2)$. The split Cayley hexagon conjecture…

Let $E_7$ and $E_8$ denote the indicated exceptional finite reductive groups, and consider respectively their principal $\Phi_{14}$- and $\Phi_{24}$-blocks. Conjecture on the E7 an…

Local monodromy conjecture at zero. Under $\rho_k$, a topological generator of the tame inertia group $I_0^{\operatorname{tame}}$ gets mapped to an element conjugate to…