328 problems

Let $X_0$ be a geometric connected smooth variety over $\mathbb{F}_q$, let $f_0:X_0\rightarrow \mathscr{A}_{g,\mathbb{F}_q}^{{\mathrm{ord}}}$ be a morphism as above, and let $A$ be…

Let $q$ range over powers of $2$, and consider isogeny classes of $3$-dimensional abelian varieties over the finite field of characteristic $2$ with $q$ elements. The classes under…

Let $p$ be an odd prime, and let $X$ be a smooth curve of genus $3$ over $\overline{\mathbf F}_p$. An unramified double cover of $X$ is a degree-two étale morphism $\pi:Y\to X$; by…

Let $\mathcal M_g^{\leq 1}$ be the moduli space of stable curves of genus $g$ with degenerations of torus rank at most $1$, and let…

Let $\ell$ be a rational prime, let $K$ be a number field, and let $K_\infty=K\cdot\mathbb{Q}_{\infty,\ell}$ be the cyclotomic $\mathbb{Z}_\ell$-extension of $K$. Let $A/K_\infty$…

Generalized relative Manin–Mumford conjecture. Under these hypotheses, $Y$ is contained in a proper torsion coset.

Let $A$ be a complex abelian variety, let $\Gamma$ be a subgroup of finite rank in $A(\mathbb{C})$, and let $X$ be a subvariety of $A$. Here $\overline{X(\mathbb{C}) \cap \Gamma}$…

Correlation conjecture. The intersection ${\mathcal P}\cap H$ is correlated with respect to the morphism $H\to B$.

Let $K$ be a number field, let $A/K$ be an abelian variety, and for each prime $p$ let $\selrk{p}(A/K)$ denote the $p$-Selmer rank. p-independence conjecture. The parity of…

Let $(B,\Xi)$ be a simple principally polarized abelian variety of dimension $g$, and suppose that $(B,\Xi)\notin N_{g,i}$ for every $i\geq k\geq1$. Here $N_k(B,\Xi)\subset B$ is t…

Let $g$ and $k$ be integers with $1\leq k\leq g-3$, and let $N$ be an irreducible component of the Andreotti–Mayer locus $N_{g,k}$ whose general point corresponds to a principally…

Hyperelliptic Jacobian torsion conjecture. For sufficiently large $k$, there is no such $C$ for which $G_K$ acts on $C$ through its cyclotomic action on…

Strong Torsion Conjecture. If $n\geq n(g,d)$, then there are no dimension-$g$ abelian varieties $A$ defined over $K$, with $[K:\mathbb Q]\leq d$, that have a $K$-rational torsion p…

Let $X$ be an abelian variety, let $A$ be an ample line bundle on $X$, and let $m(A)$ denote its index of $M$-regularity. For non-negative integers $p\ge m$, Pareschi–Popa's syzygy…

Let $k$) be a number field, let $S$ be a finite set of places of $k$ containing the archimedean places, and let ${\mathcal O}_{k,S}$ be the ring of $S$-integers. Let $A/k$ be an ab…

Let $k$ be a number field, let $A$ be an abelian variety over $k$, and let $S_A$ be the set of places of good reduction for $A$. For $u\in(0,1)$, call an integer $B^u$-smooth if al…

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…

Abelian Lehmer problem. There is a constant $c(A/K,\mathcal{L})>0$ such that, for every point $P\in A(\overline{K})$ of infinite order modulo every proper abelian subvariety of $A$…

Let ${\mathbb C}_p$ be the completion of the algebraic closure of ${\mathbb Q}_p$, with $v$ the extension of the $p$-adic valuation, and let … be its ring of integers. Let…

Let ${\mathcal X}_{\mathbb C}$ be the complex universal abelian variety considered in the paper, let $Y\subseteq {\mathcal X}_{\mathbb C}$ be an irreducible subvariety, and let…

Let $(A,L)$ be a polarized abelian variety of dimension $g$. A polarization is of type $(d_1,\ldots,d_g)$ when its elementary divisors are $d_1\mid\cdots\mid d_g$. Normal generatio…

Let $(A,L)$ be a polarized abelian variety of dimension $g$. A polarization is of type $(d_1,\ldots,d_g)$ when its elementary divisors are $d_1\mid\cdots\mid d_g$. Normal generatio…

Abelian-variety entire-curve conjecture. Every non-constant holomorphic map

David's conjecture. For every $\varepsilon>0$, there exists a constant $c(A/K,\mathcal{L})>0$ such that

Generalized Lehmer conjecture. There exists a positive constant $c=c(A,K,\mathcal L)$ such that