596 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 prime $p>3$ satisfying $p\equiv 4,7,8\pmod 9$, there exist rational numbers $x,y\in\mathbb{Q}$ such that $p=x^3+y^3$.

For every smooth projective curve $C/\mathbb{Q}$ with genus $g(C)\geq 2$ and every integer $n\geq 1$, let…

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Let $X_\infty=(X_n)_{n\ge 0}$ be a geometric $\mathbf Z_p$-tower of smooth projective curves over a finite field of characteristic $p$, with base $X_0=\mathbf P^1$ and ramification…

For a compact set $K\subset\mathbb{R}$, define the integer Chebyshev constant by…

For every nonsingular Hermitian variety $X$ of dimension $2m$ over a finite field, let $h$ denote the hyperplane class and let $N^m(X)_{\mathbb{R}}$ be the space of codimension-$m$…

Let $f\in\mathbb{Z}[x_1,\ldots,x_n]$ be nonconstant, and define the normalized exponential sum … where $p$ is prime and $m\geq 1$. If $\operatorname{lct}(f)$ denotes the log-canoni…

Let $X$ be a supersingular irreducible symplectic variety over a field of positive characteristic. The conjecture predicts that $X$ satisfies the supersingular Tate conjecture: the…

For every odd prime $p$ and every imaginary quadratic field $K$ whose $p$-class rank is $2$, if the Galois group $G_{K,\varnothing}(p)$ of the maximal unramified pro-$p$ extension…

For every number field $F$, prime $p$, Hilbert modular variety $X/F$, and cohomological degree $i$, the locally analytic $p$-adically completed cohomology $\widetilde{H}^{i}(X)$ sh…

For every integer $g\ge 2$, there are only finitely many cyclic commensurability classes of closed, fibered arithmetic hyperbolic $3$-manifolds having a fiber of genus $g$.

For integers $m\ge 1$ and $n\ge 1$, let $X_m^n\subset \mathbb{P}^{n+1}_{\mathbb{C}}$ be the Fermat hypersurface $x_0^m+x_1^m+\cdots+x_{n+1}^m=0$. The Hodge conjecture for Fermat va…

In a standard uniform formulation: for every choice of complexity bounds for a subvariety $X$ and every integer $r\ge 0$, there is a constant $C$ such that, for every semiabelian v…

The supplied sources identify a Lang–Trotter conjecture concerning the frequency of primes at which reductions of points on an elliptic curve satisfy a primitive-point condition. T…

Let $R$ be an unramified regular local ring, let $K=\operatorname{Frac}(R)$ be its fraction field, and let $G$ be a reductive group scheme over $R$. If a $G$-torsor over $R$ become…

For every global function field $K$ of characteristic $p$, every integer $n\ge 1$, and every effective divisor $D$ on $K$, there are only finitely many isomorphism classes of conti…

For a rigid analytic space $X$ over $\mathbb{C}_p$ and a profinite pro-étale torsor $Y\to X$, if $Y$ is affinoid perfectoid, then the associated geometric Sen morphism is surjectiv…

For every projective klt orbifold pair $(X,\Delta)$ such that $-(K_X+\Delta)$ is ample, the orbifold $(X,\Delta)$ is Campana rationally connected.

Let $(G,X)$ and $(G',X')$ be Shimura data whose underlying groups are pure inner forms of one another, and let $\operatorname{Ig}(G,X)$ and $\operatorname{Ig}(G',X')$ denote the as…

Every equality between convergent periods follows from the standard formal relations for periods: linearity of integration, additivity with respect to domains, and algebraic change…

For every elliptic curve $E/\mathbb{Q}$ with complex multiplication and $\operatorname{rank}E(\mathbb{Q})=2$, and for every good ordinary prime $p$ of $E$, is…

Let $\mathcal{S}$ be a split unitary Shimura variety of signature $(g,g)$ at a split prime $p$, and let $\mathcal{S}_{\mathrm{basic}}$ denote its basic locus. Outside the degenerat…

Let $F$ be a nonarchimedean local field and let $k$ be a field of characteristic $p>0$. Does the tensor-triangulated category $\operatorname{DAM}(F;k)$ of derived Artin motives sat…

Let $S$ be a base scheme and let $G$ be a proper, flat, finitely presented commutative group stack over $S$ with finite flat inertia. Define its dual by…