26 problems

Larsen's conjecture. For all sufficiently large primes $\ell$, there exists a semisimple group scheme $\mathcal{G}_\ell$ defined over $\mathbb{Z}_\ell$, with generic fiber…

Automorphic-to-Galois compatibility conjecture. There is an irreducible geometric strongly compatible system $\mathcal{R}$ of l-adic representations such that

Compatible-systems conjecture. The following should hold: (1) every continuous semisimple de Rham representation $R:G_{\mathbb{Q}}\to GL_n(\overline{\mathbb{Q}}_l)$ unramified at a…

Galois-theoretic conjecture. The system satisfies all of the following:

Motivic reciprocity meta-conjecture. Any strictly compatible $L$-integral system as in Definition arises motivically.

Let $X$ be a smooth curve over a finite field of characteristic $p$, let $E$ be a number field, and let ${\mathbf{L}}=\{{\mathcal{L}}_\lambda\}$ be an $E$-compatible system of liss…

Let $X$ be a proper, smooth algebraic variety over a number field ${\mathrm{E}}\subset\mathbb C$, and let ${\mathbf{G}}$ be the Mumford–Tate group of the Hodge structure…

Compatible-systems conjecture.

The motivic Newton–Hodge inequality and ordinariness conjectures. For every integer $k$: (b) the representations $R_{\ell}$ form a strictly compatible system; (c) there exists a fi…

Given a compatible system $\{\rho_\ell\}_\ell$ and its algebraic monodromy groups $\mathbf{G}_\ell\subset\mathrm{GL}_{n,\mathbb{Q}_\ell}$ and images…

Let $E$ be a number field and let $\{\rho_\lambda:\operatorname{Gal}_K\to\mathrm{GL}_n(\overline E_\lambda):\lambda\in\Sigma_E\}$ be a semisimple compatible system of $K$ defined o…

Let $X$ be a scheme and let $\{\rho_\lambda\}_\lambda$ be an $E$-rational compatible system of geometric origin, for example one arising from higher direct image sheaves…

Strict compatibility and automorphy conjecture. Any weakly compatible system of Galois representations is strictly compatible, and is in addition automorphic: there is an algebraic…

Geometric-to-strict compatibility conjecture. Any semisimple geometric representation

Finite-order determinant conjecture. If ${\mathcal L}_\ell$ has determinant of finite order, then for every closed point $x\in X({\mathbb F}_{q^n})$ this characteristic polynomial…

Compatible-system conjecture. La familia $(M_q)_q$ es un sistema compatible de representaciones de Galois.

Let $X/\mathbb{F}_{q}$ be a smooth variety, and let $\mathcal{F}$ be an irreducible coefficient object on $X$ with algebraic determinant. Companions conjecture. There exists a numb…

Let $X/\mathbb{F}_{q}$ be a smooth, geometrically connected, quasi-projective variety, let $l\neq p$ be a prime, and let $L$ be a lisse $\overline{\mathbb{Q}}_{l}$-sheaf of rank…

Finite-monodromy conjecture. The compatible system $\\{\mathscr{L}_{l}\\}_{l\ne p}$ should have finite monodromy.

Compatible-system conjecture. The compatible system comes from algebraic geometry.

Serre's independence-of-ell conjecture. The tautological representation of $\mathbf{G}_\ell^\circ$ is independent of $\ell$. This predicts an $\ell$-independent algebraic descripti…

Let $X$, $E$, $\lambda$, and $\mathcal{E}$ be as in the compatibility conjecture for de Rham lisse sheaves, and let conditions (i) and (ii) denote respectively the coefficient cond…

Let $\ell$ be a rational prime. Let $X$ be an irreducible regular scheme, flat and of finite type over $\mathbb{Z}[\ell^{-1}]$. Let $E$ be a finite extension of $\mathbb{Q}$, let…

Let $p$ and $\ell$ be distinct primes. Let $X$ be a connected normal scheme of finite type over $\mathbb{F}_p$, and let $\mathcal{E}$ be an irreducible lisse…

Serre's conjectural bound. For every $a\in\mathbb{Z}$,