620 problems

Let $A$ be a simple abelian variety over $\mathbb{Q}$ of dimension $n$, and let $\mathcal{L}\subset A(\mathbb{Q})$ be a finitely generated subgroup of rank $m\le n$. Let…

Let $A$ be a generic supersingular principally polarized abelian variety of dimension $g\geq 2$, and let $\mathbb Z_m$ denote the cyclic group of order $m$. Oort's conjecture. The…

Let $K$ be a number field, let $g$ be a positive integer, and let $\ell$ be a rational prime. For an abelian variety $A$ over $K$, write $[A]$ for its $K$-isomorphism class, and de…

Let $A$ be an abelian variety of $GL(2,E)$-type over a number field $K$, with conductor $\mathfrak{n}$, and let $\rho_{A,\lambda}$ denote its Galois representation for a prime…

Manin–Mumford conjecture. The subvariety $X$ is a translate of an abelian subvariety.

Let $X$ be a principally polarized complex abelian variety, and let its Kummer variety be the quotient of $X$ by the involution $x\mapsto -x$. A trisecant line is a line meeting th…

Let $(A,\theta)$ be a polarized abelian variety, and let $C\subset A$ be a smooth curve of genus $h$. Write $c$ for the gonality of $C$ and … for its $\theta$-degree. Gonality–genu…

Generalized Sato–Tate conjecture. The moment sequence of the distributions of coefficients of the normalized L-polynomials converges to the moment sequence of the Sato–Tate group.

Manin–Mumford conjecture. A smooth algebraic curve of genus $g\geq 2$ has finitely many torsion points.

Algebraic Sato–Tate conjecture. The groups $\mathcal{G}_\ell$ for varying $\ell$ are all interpolated by a single algebraic group defined over $\mathbb{Q}$, called the algebraic Sa…

Let $A$ be an abelian variety defined over an algebraically closed field ${\mathbb K}$. Given an integer $p \geq 0$, an ample line bundle $L$ on $A$, and an integer $m$, the Green–…

Let $A$ be an abelian variety over an algebraically closed globally valued field $K$. If $K$ is non-archimedean, let $k$ be its constant field and let $(A^{K/k},\operatorname{tr})$…

Let a principally polarized indecomposable abelian variety be given, with its Kummer variety and associated theta functions. A family of flex lines has a fourth-order jet when its…

Geometric Bogomolov conjecture. $X$ contains Zariski dense sets of small points if and only if $X$ is special.

Let $(A,\Theta)\in\mathcal A_g$ be a principally polarized abelian variety, and let a $d$-dimensional subvariety of $A$ have minimal cohomology class, meaning the class is the mini…

Murty–Patankar conjecture. The abelian variety $B$ has absolutely simple reduction for a density one set of primes up to a finite extension if and only if its endomorphism ring is…

Pareschi–Popa conjecture. If $\Theta=X+Y$ decomposes nontrivially as a sum, then either $(A,\Theta)$ is the Jacobian of a smooth projective curve, or $(A,\Theta)$ is the intermedia…

Let $A$) be the Jacobian of a smooth genus-two curve, embedded by $|3\Theta|$ in $\check{\mathbb{P}}^8$, and let $\mathscr{C}_3=Z(F)$ be the unique cubic hypersurface singular alon…

Lang–Silverman conjecture. There exists a positive quantity $c_{23}=c_{23}(K,g)$ such that

David-Hindry's conjecture. For every integer $n\in\mathbb{N}$, there is a constant $c(A/K,\mathcal{L},n)>0$ such that, for every $n$-tuple $(P_1,\ldots,P_n)$ of points of infinite…

Let $A$ be an abelian variety defined over a number field, and let $A_{\mathrm{tors}}$ denote its torsion subgroup. S. David's conjecture. The field $\mathbb{Q}(A_{\mathrm{tors}})$…

Let $v_{l,k}(f)$ be the finite-level local factor and let $v_l(f)$ be its limit. Power-saving local-factor approximation. There is a constant $\delta>0$ such that … This conjecture…

Let $\Pi_{17}$ be the double cover of the Teichmüller curve $W_{17}$, and let its associated supersingular polynomial be the polynomial whose zeros describe the supersingular point…

Let $K$ be a totally real field and $L$ a number field. Let $A/L$ be an abelian variety of $\mathrm{GL}_2$-type such that … For a prime $\mathfrak{p}$ of $K$ above a rational prime…

Let $A$ be an abelian variety over a global field $k$, and let {text{% {fontencoding{OT2}fontfamily{cmr}fontseries{m}fontshape{n}selectfont Sh}}}(A) denote its Tate–Shafarevich g…