49 problems

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…

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…

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…

Cadoret–Tamagawa conjecture. One has

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

Let $G$ be a finite group with a regular realization over $\mathbb Q$, let $P_\ell$ be an $\ell$-Sylow subgroup of $G$, let $m=[G:P_\ell]$, and let $r$ be the branch-point number o…

Let $\ell$ be a prime, $k\geq 0$, and $d$ a fixed dimension. A cyclotomic point means a rational point defined over a cyclotomic extension, and $D_{\ell^{k+1}}$ denotes the dihedra…

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…

Let $d\geq 1$, let $P\in \overline{\mathbb{Q}}[X_1^{\pm 1},\ldots,X_d^{\pm 1}]\setminus\{0\}$, and let $m(P)$ denote the logarithmic Mahler measure of $P$. For…

Primitive torsion-growth classification conjecture. One has

Torsion-growth classification conjecture. The set $\Phi_{\mathbb{Q}}(d)$ consists of the union of all $\Theta(d')$ such that $d'\mid d$:

Let $S=\{(r_i,\theta_i)\}_{i=1}^{4}\in\mathcal{R}_4$ have common order $n$. Let $\Gamma_S=\{\gamma^{-T}\in\operatorname{SL}_2(\mathbb{Z}):\gamma\cdot S=S\}$ be the stabilizer of…

Bogomolov–Tschinkel's uniform common-torsion conjecture. There exists a uniform constant $B$ such that

Let $E$ be an elliptic curve over a number field $K$ of degree $d=[K:\mathbb{Q}]$. Suppose that $E$ has a point of exact order $p^n$ over $K$, where $p$ is prime and $n\geq 1$. Loz…

Strong-torsion bound conjecture. Some expression in $r$ and $m$, independent of the prime of good reduction $v$, bounds $p^u$.

Let $A$ be an abelian variety isogenous to a product … where the $A_i$ are simple abelian varieties that are pairwise non-isogenous. Let $\gamma(A)$ be the optimal exponent governi…

Let $C$ be a curve of genus $g \geq 2$ embedded in its Jacobian $J$, all defined over a number field $K$. Let $J_{\mathrm{tors}}$ be the torsion subgroup of $J$ and let…