4 problems

Conjecture (plane case). For every field $K$ of characteristic $0$ and all $p,q\in K[x,y]$: if $J_F=c$ for some constant $c\in K\setminus\{0\}$, then $F$ has a polynomial inverse.…

Conjecture $T^i(X,\ell)$ (Tate). The map $c^i_\ell$ is surjective. In words, every Galois-invariant class in $H^{2i}_{\mathrm{\acute et}}(X_{\bar k},\mathbb Q_\ell(i))$ is a…

Let $k$ be a field and let $H^*$ be a Weil cohomology theory for smooth projective $k$-varieties with coefficients in a field $F$ of characteristic $0$. Examples include $\ell$-adi…

Definition (Deligne). A Hodge class $\alpha$ is absolute Hodge if for every $\sigma\in\operatorname{Aut}(\mathbb C)$ the pair $\sigma(\alpha_{\mathbb A})$ is the image…