8 problems

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…

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…

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…

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…

Open-orbit vanishing conjecture. Suppose that the integral … is zero unless $w=w_0$ and $\nu=\nu_0$. Moreover, $w_0$ can be chosen to be a Weyl element corresponding to the longest…