8 problems

Deligne's weight-monodromy conjecture. The action of Frobenius on $\operatorname{gr}^M_j H^i(Y_{\overline{K}},\mathbb{Q}_\ell)$ should be pure of weight $i+j$.

Let $A$ be a nonsingular projective $g$-dimensional variety over $\mathbb C$, and suppose its Sato–Tate group $\operatorname{ST}(A)$ is defined as a maximal compact Lie subgroup of…

Let $\mathbb{G}$ be a one-dimensional formal group of finite height $n$ over a perfect field $k$ of characteristic $p$, and let $\ell$ and $p$ be distinct primes. Write $H^{n-1}$ f…

Effective Galois descent conjecture. If $H$ satisfies $K/k$-Galois-$\ell$-descent, then there exists a motive $H_0$ in $\mathrm{Mot}_{k^s,\ell}$ and an isomorphism

Let $X$ be a smooth projective variety of dimension $n$ over an algebraically closed field $k$, and let $\ell$ be a prime distinct from the characteristic of $k$. Numerical equival…

Let $\ell$ be a prime, let $n,d>0$, let ${\mathcal G}^{\mathrm{cris},\ell}_1$ be the crystalline $\ell$-adic realization group, let ${\mathcal G}^{\mathsf{MEM}}_1$ be the motivic g…

Let $F$ be a local field of residue characteristic $p>0$, let $X$ be a variety over $F$, and let $\ell\neq p$. By the $\ell$-adic monodromy theorem, one can attach a Weil–Deligne r…

Let $M$ be a motive over $\mathbb Q$. Its $\ell$-independence conjecture predicts that the $L$-series of $M$ is independent of the choice of the prime $\ell$ and the choice of…