61 problems

Let $r(\alpha)$ be the integer that counts the number of rotations of the velocity vector along a curve $\alpha$. Curves with $r(\alpha)=m$ include the curves $\gamma_m$ in the fam…

Mazur's uniform Mordell–Lang conjecture. Let $g\ge 2$ be an integer. Then there exists a constant $c(g)\ge 1$ such that, for every smooth curve $C$ of genus $g$ defined over $F$, e…

Let $k$ be a perfect field of characteristic $2$, let $\Delta$ be a two-dimensional lattice polygon satisfying the conditions from Lemma, and let $f$ be a $\Delta$-non-degenerate (…

SNC index specialization conjecture. Under these assumptions,

IFG field curve-index conjecture. If $K$ is an IFG field and $I\mid 2g-2$, then there exists a curve $C_{/K}$ of genus $g$ with $I(C)=I$.

Ample-cone conjecture. If $C$ is a very general curve of genus $g\ge 4$, then

Let $C$ be a smooth projective geometrically connected curve of genus at least $2$ over a number field $k$, with $C(k)=\emptyset$. Assume that $C$ has a rational divisor class of d…

Let $k$ be a number field, and let $C$ be a smooth projective geometrically connected curve over $k$. In the paper, a curve is very good when its finite descent obstructions accoun…

Let $K$ be any number field and let $E$ be any elliptic curve over $K$. Write $K^{\operatorname{ab}}$ for the maximal abelian extension of $K$. Conjectural strengthenings. The foll…

Uniform boundedness conjecture. For any complex function field $L$ and any integer $g\geq 2$ there exists a number $B_g(L)$ such that for every $C\in{f{\mathcal C}_g(L)}$ one has

Projection-diameter conjecture. There is some orthogonal projection to $\mathbb R^{n-1}$ in which the image of $\gamma$ has diameter at most

Shortest 2-design curve conjecture. The curve $\gamma$ is the shortest $2$-design curve in $S^2$, up to isometry and reparameterization. The conjecture is motivated by the precedin…

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…

Let $k$ be a number field and let $X$ be a smooth projective curve of genus $g\ge 2$ defined over $k$. Mordell's conjecture. The set of rational points $X(k)$ is finite. This is th…

Low-degree curve plane-containment conjecture. Every curve $C$ of degree $\delta \leq d$ must be contained in a $2$-plane. This extends the stated minimality theorem for plane curv…

Let $\mathbb k$ be a field, and let $C$ and $C'$ be smooth projective curves over $\mathbb k$ that are D-equivalent and whose geometric irreducible components have genus $1$. The a…

Let $X$) be a smooth projective curve of genus $g\geq2$ whose Jacobian has Mordell–Weil rank at most $g-2$. Let $U_1^p$ denote the abelianisation of the fundamental group of…

Closed-curve non-spectrality conjecture. The measure $mu$ is not spectral.

González-Alonso–Torelli conjecture. Such a deformation can have rank

Bennett-Siksek conjecture. The only rational points satisfy $y=0$, or are one of the following:

Periodicity conjecture. The minimal periods of the sequences $a^{r_1}(X_n)$ and $a^{r_0}(X_n)$ are equal.

Booher–Cais conjecture. For each $r\geq 1$, there exist an integer $m_r$ and functions $b_r,\nu_r,\lambda_r:\mathbf Z/m_r\mathbf Z\to\mathbf Q$ such that, for $n\gg0$,

Density conjecture. The subset of piecewise $C^{1,\beta}$ 2-paths is dense in $C^{1,\beta}$.

Random-curve taut smoothing conjecture. A random curve has a taut smoothing.

Let $F$ be a field, let $B$ be a quaternion algebra over $F$, and let $g>1$. A curve $C/F$ is $B$-modular if there is a Shimura curve $X_K$ associated with a suitable compact open…