50 problems

Mazur's growth number conjecture. The Mordell–Weil rank of $E$ stays bounded along any ${\mathbb Z}_p$-extension of $K$, unless the extension is anticyclotomic and the root number…

Let $A$ be an abelian variety over an algebraic number field $k$, and let $k^{\mathrm{ab}}$ be the maximal abelian extension of $k$. The Mordell–Weil group is $A(k^{\mathrm{ab}})$.…

Honda's conjecture. There exists a constant $c_A>0$, depending on $A$, such that

Let $E$ range over elliptic curves over $\mathbb{Q}$, ordered in a suitable standard height ordering. Folklore rank conjecture. Almost all elliptic curves over $\mathbb{Q}$ have Mo…

Let an elliptic fibration geometrically engineer a gauge group, and let its Mordell–Weil group be the group of rational sections of the fibration. The center of the gauge group is…

Silverman's conjecture. The Mordell–Weil rank should satisfy

Let $k$ be a finitely generated infinite field, let $\overline{k}$ be a separable closure of $k$, and write $G_k:=\operatorname{Gal}(\overline{k}/k)$. For $\sigma\in G_k$, define t…

Let $p$ be a prime, let $\mathbb{Q}^{p\text{-cyc}}$ denote the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, and let $E/\mathbb{Q}$ be an elliptic curve with good ordinary r…

Silverman's analytic form of Nagao's conjecture.

Let $(a,b,m,n)\in\mathcal{M}$ be a Master-Hit, and let $f_1$ and $\mathcal{P}(a,b,m,n)$ denote the associated integer and set of primes. For a prime $\ell$, write $v_\ell(f_1)$ for…

Let $X\to B$ be as in Theorem MMP, with $X$ a Calabi–Yau variety. Suppose that $B$ is birational to a Mori fiber space $B'\to U$ with fiber $F$, and let…

Non-rationality conjecture. The specializations are not rational in $t$ with coefficients in $K_0(\operatorname{Stck}_{\mathbb{C}})[\mathbb{L}^{-1}][v]$ and…

Let a Weierstrass model be defined over $\mathbb{P}^2$ with an $SU(2)$ gauge symmetry tuned over a degree-nine curve. The associated threefold has a Mordell–Weil group whose additi…

Let $p\geq7$ and let $(p,p-2)$ be a twin-prime pair. Define the twin-prime elliptic curve by … Write $r(E_p)$ for its Mordell-Weil rank. Corrected twin-prime rank conjecture. … The…

Let $p\geq7$ be prime with $p-2$ also prime, and let $E_p$ be the twin prime elliptic curve … Its Mordell-Weil rank is the rank of $E_p(\mathbb Q)$. Beers' rank-two conjecture. If…

Let $p\geq 7$ be prime with $p-2$ also prime, and let $E_p$ be the elliptic curve over $\mathbb Q$ given by … Its Mordell-Weil rank means the rank of the group $E_p(\mathbb Q)$ of…

Let $C$ be a smooth, projective curve over a field $k$, and let $p:A\to C$ be a smooth group scheme with irreducible fibers whose generic fiber $A_K$ is an Abelian variety. Let…

Let $k$ be a number field, let $g\geq 1$ be an integer, and let $\epsilon>0$. For an abelian variety $A$ over $k$, let $N_A$ be the norm of its conductor ideal. Mordell–Weil rank c…

The Diophantine exponent conjecture.

Let $\mathcal{A}$ be a simple abelian variety of dimension $g$ over a number field $K$ embedded in $\mathbb{R}$. Let $\mathcal{A}(\mathbb{R})^0$ be the identity component, let…

Weaker Watkins conjecture. If $m_{\mathsf{E}}$ is odd, then

Rank-zero fibre conjecture. There are infinitely many $b\in\mathbb{P}^1(\mathbb{Q})$ such that

Non-isotrivial elliptic-surface rank conjecture. If $\pi$ is non-isotrivial, then both $\mathscr{J}(X,\pi)$ and $\mathscr{N}(X,\pi)$ are infinite.

Eight-line non-torsion conjecture. Points in the intersection of 8 lines are not always torsion; in particular, there should exist an example of a non-torsion point contained in th…

Let $E$ be an elliptic curve over $\mathbb{Q}$, and let $\mathbf L\subset\mathbb{Q}^{\operatorname{ab}}$ be a real abelian field containing only finitely many extensions of…