79 problems

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 $E_1$ and $E_2$ be elliptic curves defined over $\mathbb{C}$, given by Weierstrass equations … where $f_1$ and $f_2$ do not have the same roots. Write…

Let $n$ be a positive integer, and consider pairs of elliptic curves whose $n$-torsion modules are isomorphic. The $j$-invariant of an elliptic curve is denoted by $j(E)$. Kani–Sch…

A special subvariety of an abelian variety over an algebraically closed field of characteristic zero is a translate of an abelian subvariety by a torsion point. Let $K$ be a number…

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 an elliptic curve defined over a quadratic field with a point of order $13$, and let $c_E$ denote its Tamagawa number. Krumm's parity conjecture. The valuation…

Let $\pi: \mathfrak{A}_{g,l} \to A_{g,l}$ be the universal family of complex principally polarized abelian varieties of dimension $g$ with symplectic level $l$-structure, and let…

Let $K$ be a finitely generated field over its prime field, let $e$ be a positive integer, and let $\sigma$ range over $\mathrm{Gal}(K^\mathrm{sep}/K)^e$. For almost all such…

Torsion-point conjecture. Each irreducible component of $\Sigma^i_k(X)$ contains a torsion point.

Torsion-order conjecture. There is a constant $C$, independent of $E$ and $K$, such that

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 $k$) 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…

Strong boundedness conjecture. Torsion is strongly bounded for abelian varieties $A/\mathbb{Q}$ in every dimension $d$. This would extend Merel's theorem for elliptic curves over…

Let $p\geq 5$ be a prime number, and suppose that $K/{\mathbb Q}_p$ is an unramified finite extension. Let $X/K$ be a curve of genus $g\geq 2$, embedded in its Jacobian via a $K$-r…

Discriminant-square conjecture. If $D\equiv 0\pmod 8$, then $u+64=DZ^2$ for some $Z\in\mathbb{Q}(j(\alpha))$.

Cadoret–Tamagawa conjecture. One has

Uniform boundedness conjecture. There exists a positive integer $N(g,D)$ such that

Let $E/\mathbb{Q}$ have split multiplicative reduction at a prime $p$, put $G_p=\left(\mathbb{Z}/p\mathbb{Z}\right)^*/\langle-1\rangle$, and let $\lambda(a,p)$ be the even modular…

Revised relative Manin–Mumford conjecture. If $Z(\mathbb{C})\cap\mathcal{G}_{\mathrm{tor}}$ is Zariski dense in $Z$, then either $Z$ is contained in a special subvariety of…

Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{C}$, and for each $j=1,2$ let $\iota$ be the involution of $E_j$ and choose a double cover $\pi_j:E_j\to\mathbb{P}^{1}$ satisfy…

Let $A$ be an abelian variety of dimension $g$ defined over a number field $K$ of degree $d=[K:\mathbb{Q}]$, and let $A(K)_{\operatorname{tor}}$ denote the finite subgroup of torsi…

Let $F$ be a number field of degree $d=[F:\mathbb{Q}]$, and let $E/F$ be an elliptic curve. Write $E(F)[\operatorname{tors}]$ for the subgroup of $F$-rational points of finite orde…

Uniform boundedness conjecture. There exists $C \geq 1$, depending only on $g$ and $d$, such that

Let $K$ be a number field. Say that $K$ has rationally defined complex multiplication (RCM) if there exists a CM elliptic curve defined over $K$ whose endomorphism ring is $K$-rati…