10 problems

Let $\mathrm{M}_2^6$ and $\mathrm{M}_3^{(2,2)}$ be the motives introduced in the paper, and let $(-3)$ denote the Tate twist. Motive-isomorphism conjecture. The two motives … are i…

Isolated-singularity conjecture. The morphism above is an isomorphism for any $n$ and $\lambda$ when the matrix $N(V_\lambda)+E(V_\lambda)$ is invertible. This would imply unramifi…

Let $n$ be prime and let $\psi:\mathbb F_q\to\overline{\mathbb Q}_\ell^{\times}$ be nontrivial. Let $\Gamma=\varinjlim\widehat{\mathbb F_{q^n}^{\times}}$, and let…

Let $G$ and $G'$ be split almost simple groups over $k$ whose dual groups satisfy $\check{G}\supset\check{G}'$ and appear in the same line of the paper's global-monodromy table. Le…

Strong rigidity characterization conjecture. There exists a generic linear function $\phi:I(1)/I(2)\to{\mathbb G}_a$ such that $L\cong\operatorname{Kl}_{\check{G}}(\phi,\chi)$ up t…

Let $k$ be the finite base field, let $\check{G}$ be the dual group, and let $\operatorname{Kl}_{\check{G}}(\phi,\chi)$ be the Kloosterman sheaf associated with a generic linear fu…

Let $k={\mathbb C}$, let $G$ be almost simple, and let $\mathcal G$ be a quasi-split form of $G$ on ${\mathbb P}^1_{\backslash\{0,\infty\}}$ determined by…

Let $k$ be the finite base field, let $G$ be a semisimple simply connected group over $k$, let $\check{G}$ be its dual group, and let $\phi:I(1)\to{\mathbb Q}_\ell(\mu_p)$ be an af…

Let $b(p)=p(\alpha_p+\beta_p)$, where $\alpha_p$ and $\beta_p$ are the eigenvalues of geometric Frobenius at $p$ under a two-dimensional representation $\rho$ as described above, a…

Automorphy conjecture. The infinite product $\zeta_{k,n}(s)$ is automorphic and thus extends to a meromorphic function in $s\in {\bf C}$. This conjecture concerns the variation, as…