549 problems

Let $k$ be a number field, let $M\in \mathrm{M}_n(\mathcal{O}_k)$, and suppose that the triangularization variety $X_M$ has a rational point, $X_M(k)\neq\varnothing$. An irreducibl…

For every integer $g\ge 2$ and every prime $p$, let $\mathcal{S}_g$ denote the supersingular locus in the moduli space of principally polarized abelian varieties of dimension $g$ o…

Let $V(k)$) be the set of $k$-rational points of a projective variety $V$ over a number field $k$, and let $A^1_{\infty}$ be the infinite matrix associated with $V$ in the paper. D…

Let $X$ be a proper smooth variety over a local field $k$, and let $V$ be its $i$th $\ell$-adic étale cohomology group. Write $N$ for the monodromy operator, let $W_\bullet$ be the…

Let $M$ be the Rapoport–Zink space of strict $\pi$-divisible $O_F$-modules with quasi-isogeny to $\mathbb{X}$, and let $Z_1,Z_2\to M$ be the closed immersions associated with the m…

Weak Manin's conjecture. There exists a non-empty Zariski open subset $U\subset X$ such that, for every $\epsilon>0$,

Conjectured invariants of $S_q$. If $q$ is even, then

Mazur's uniform Mordell–Lang conjecture. Let $g\ge 2$ be an integer. Then there exists a constant $c(g)\ge 1$ such that, for every smooth curve $C$ of genus $g$ defined over $F$, e…

Homological stability conjecture. The cohomology group

Let $X$ be a smooth, proper, geometrically irreducible variety over a number field $k$. A zero-cycle of degree one on $X$ is a formal integer linear combination of closed points wh…

Let $X$ be a smooth projective variety over $\bC$, let $A=\operatorname{Spec}R$ where $R$ is a finitely generated $\bZ$-algebra containing $\bZ$, and let $X_A\to A$ be a spreading…

Let $K$ be a discretely valued field with separable closure $K^{\operatorname{sep}}$, let $X$ be a variety over $K$ with semistable model, and let $E_r^{p,q}$ be its weight spectra…

Let $r\geq 2$ be an integer. For a nonsingular projective variety $X$ over $\overline{\bold Q}$, let … be the Abel-Jacobi map. If $X_{\bold C}=X\otimes_{\overline{\bold Q}}\bold C$…

The Strong Lang Conjecture. If $X$ is of general type, then there exists a proper subvariety $\Sigma\subsetneqq X$ such that for any finite extension $K'$ of $K$,

The nef anticanonical bundle converse. If the anticanonical bundle $-K_X$ of $X$ is nef, then for some finite extension $K'$ of $K$ the set $X(K')$ of $K'$-rational points of $X$ i…

The negative-canonical-bundle converse. If the canonical bundle $K_X$ of $X$ is negative, then for some finite extension $K'$ of $K$ the set $X(K')$ of $K'$-rational points of $X$…

Let $\alpha^*$ be the inverse-image functor from etale to Zariski motivic complexes, let $j^*{\Bbb Z}_l(n)_S$ be the indicated arithmetic etale Tate-twist complex, and let $n\ge0$.…

Let $S$ be the arithmetic scheme in the source, let $j$ be the relevant open immersion, let $e$ be the indicated cup-product class, and let $A(1)$ be the real Tate-twist object. Co…

Let $S$ be the open arithmetic scheme and let $A(n)$ denote the real Tate-twist object used in the source. Theorem 12.1(a) gives a comparison isomorphism for positive twists betwee…

Let $F$ be a number field with algebraic closure $\overline{F}$, let $X$ be a smooth projective variety over $F$, let $p$ be a prime, and let $S$ contain the places above $\infty$…

Let $Y$ be a scheme over $\mathbb Z$, and let $\{(Y_p,{\mathscr G}_p)\}$ be a family arising from a fixed constructible complex and morphisms as in the setup. Suppose $Y$ is an alg…

Corrected local model flatness conjecture. The scheme $M$ is flat over

Pappas's spin flatness conjecture. The scheme $M_I$ is flat over $\mathcal O_E$; equivalently,

Let $K$ be a global field, and let $V/K$ be a smooth projective variety such that its base change $V_{\overline K}=V\operatorname*{\otimes}\overline K$ is rational and has Picard g…

Let ${\mathscr X}$ be the arithmetic scheme in the paper, with Zariski and small étale sites ${\mathscr X}_{{\mathrm{Zar}}}$ and ${\mathscr X}_{{\mathrm{\acute{e}t}}}$, and let…