17 problems

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:

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…

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

Let $F$ be the number field and $\bar F$ its algebraic closure, with absolute Galois group $GammaF=\operatorname{Gal}(bar F/F)$. Let $pi$ be a regular algebraic, conjugate self-dua…

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.

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…