17 problems

Let $M$ be a motive over a number field $F$ with Betti realization $M_B$, Mumford–Tate group $G_{MT}$, and adelic Galois representation … Assume that the Betti realization $M_B$ is…

Let $M$ be a motive in the chosen motivic category, let $G_{l,K}^{\operatorname{alg}}$ be the algebraic group attached to its $l$-adic realization, and let…

Let $M$ be a motive in the relevant motivic category, with rational polarized Hodge realization $(V,\psi)$. Let $\operatorname{MMT}_{\sim}(M)$ be its motivic Mumford–Tate group. Se…

Let $A$ be a nonzero abelian variety over a number field $K$. Write, after fixing an embedding $K\subseteq\mathbb C$, $A_\mathbb C$ up to isogeny as $\prod_{i=1}^n A_i^{m_i}$, wher…

Let $E$) be a number field, let $A/E$ be an abelian variety, and let $\mathrm{MT}(A)$ denote its Mumford–Tate group. A point of an algebraic group is unipotent if it is unipotent i…

Let $M=H^w(X)$ arise from a smooth projective variety $X$ over a number field $F$, and fix an embedding $\sigma:F\hookrightarrow\mathbb C$. Let $G_{MT}$ be the Mumford–Tate group o…

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…

Let $(X,x)$, $X_0$, and $f$ be as in the source, let $u\in\operatorname{fpl}(\Q)$, and let $G(\L_{\mathbf{I},u,X_0},x)$ and $\operatorname{Lg}(f)_{\Q_u}$ be the groups defined ther…

Let $p$ be an odd prime, let $X_0$ be a smooth geometrically connected variety or a finitely generated field over F_{\infty}_p, and let f_0:X_0\to\A_{g,F_{\infty}_p} be a morph…

Let $p$ be an odd prime, let $J_{p^3}$ be the Jacobian of $C_{p^3}\colon y^2=x^{p^3}-1$, and write $J_{p^3}\simeq X_2\times X_1\times J_p$, where $X_2$ is the absolutely simple fac…

Triviality conjecture for the defect group. The defect group $P(X)$ should be trivial. Equivalently,

Let $k$ be a finitely generated subfield of $\mathbf C$, let $\ell$ be a prime, and let $M\in\operatorname{AM}(k)$. Let $r_{\mathrm B}(M)$ and $r_\ell(M)$ be its Betti and $\ell$-a…

Generalized Grothendieck period conjecture. One should have $t\geq d$; equivalently, $t\geq 3$ if $\tau$ is CM and $t\geq 7$ otherwise.

Let $A$ be an abelian variety of dimension $g$ over a number field, and fix a prime $\ell$. Let $G_\ell^{\rm zar}\subseteq \operatorname{GSp}_{2g}$ and…

Let $X$ be a smooth projective curve with associated base field $K$, let $X_0$ be the corresponding algebraic curve over $K$, and let $\operatorname{Per}(X_0)$ denote its space of…

Let $(V_{\mathbb Z},\mu\circ s,\mu\circ\rho_{\widetilde{x},l})$ be an $(\infty,l)$-integral structure of Shimura type. Let $H_{\mathbb Q_l}$ be the identity component of the Zarisk…

Let $x=[s,\overline{1}]$ be a point of a Shimura variety as in the source, let $\widetilde{x}$ be a lift defining the $l$-adic representation $\rho_{\widetilde{x},l}$, and let $M$…