56 problems

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…

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…

For an odd prime $p$ and each non-empty subset $S\subset GF(p)$, let $X_S$ be the hyperelliptic curve defined by … Write $B(c,p)$ for the assertion that $|X_S(GF(p))|\leq c\,p$ for…

Let $0\leq f<g$, let ${\cal M}_g$ be the moduli space of smooth curves of genus $g$, and let ${\cal H}_g$ be the hyperelliptic locus when $p\geq3$. Write $V_{g,f}$ for the locus of…

Let $X$ be a hyperelliptic Riemann surface of genus $g\geq 2$, let $\tau$ be a period matrix associated to $X$, and let … be a fundamental system, where $\mathcal{F}$ is the set of…

Let $q$ be a prime power with $q\geq 3$, and let $\mathcal{X}$ be the maximal hyperelliptic curve over $\mathbb{F}_{q^2}$ given by … Let $u_B\in\mathbb{F}_{q^2}[x]$ be an irreducib…

Let $(A,\theta)$ be an indecomposable principally polarized abelian variety of dimension $g\geq 4$. The hyperelliptic characterization conjecture. If $\varepsilon(A,\theta)<2$, the…

Let $C$ be a generic supersingular hyperelliptic curve of genus $g$ over a field of characteristic $2$. Refined hyperelliptic Oort conjecture. If $g$ is not a power of $2$, then ……

Let $C$ be a generic hyperelliptic curve of genus $g$ and $2$-rank $0$ over a field of characteristic $2$. Generic automorphism conjecture for 2-rank-zero hyperelliptic curves. If…

Let $f$ be a genus-$g$ Lefschetz fibration with slope $\lambda_f$, and call $f$ hyperelliptic when its monodromy lies in the hyperelliptic mapping class group. Xiao's hyperelliptic…

Let $g$ be the genus of a hyperelliptic curve. Consider locally soluble hyperelliptic curves that have points of odd degree. Bhargava's conjecture. As the family is ordered in the…

Density-degree conjecture. Most pairs of hyperelliptic curves $(C,D)$ of sufficiently large genera satisfy

Let $\Pi_3(q)$ be the proportion of isogeny classes of $3$-dimensional abelian varieties over $\mathbb{F}_q$ that do not contain the Jacobian of a genus-$3$ hyperelliptic curve. Le…

Let $A$ be a $3$-dimensional abelian variety over a finite field of characteristic $2$, with Weil polynomial having Newton polygon whose first slope is $\frac{1}{3}$; equivalently,…

Branch-point characterization. All expressions given by the corresponding finite-gap formula comply with the first reality condition on…

Assume $p\equiv 1\pmod 4$. Let $C$ be a hyperelliptic curve of genus at least $3$ whose branch points are all defined over $\mathbb{F}_p$. Rational-branch-point p-rank conjecture.…

Let $\mathcal{H}_3^0$ denote the moduli space of smooth hyperelliptic curves of genus $3$ and $p$-rank $0$ in characteristic $p\geq 3$. Let $c_{3,0,p}$ be the number of geometric i…

Let $k$ be a field of characteristic $p\geq 3$, and let $\mathcal{H}_g$ be the moduli space of smooth hyperelliptic curves of genus $g$ over $k$. Denote by $\mathcal{H}_g^{g-2}$ th…

Let $k$ be a field of characteristic $p\geq 3$, and let $\mathcal{H}_g$ be the moduli space of smooth hyperelliptic curves of genus $g$ over $k$. Write $\mathcal{H}_g^{g-1}$ for it…

Let $H_a$ be a genus-4 hyperelliptic curve over an algebraic closure of $\mathbb{F}_p$, with parameter $a$. Field-of-definition conjecture. If $H_a$ is superspecial, then…

Let $H_a$ be a genus-4 hyperelliptic curve, let $M_{H_a}$ be its Cartier–Manin matrix, and let $a$ denote the parameter defining the curve. Cartier–Manin matrix conjecture. The gre…

Continue with $p=3$, let $F={{\mathbb{F}_q}}(X)$, and for $\epsilon\in\{1,2\}$ let $\mu'_{\epsilon,g}(a)$ denote the proportion of genus-$g$ hyperelliptic curves in the correspondi…

Superspecial genus-4 dihedral automorphism conjecture. The following statements hold:

Nontrivial-unit conjecture. If $\deg g>\deg h$ and $h\ne 0$, then

Erdős–Graham–Granville–Selfridge conjecture. One has