111 problems

Hasse–Weil conjecture. The function $L(s,C/K)$ has an analytic continuation, and the completed function satisfies

Watkins's conjecture. The set of quadratic twists $C^{(d)}$ with rank at least $1$ has logarithmic density

A hyperelliptic curve with extra involutions is a smooth hyperelliptic curve possessing an automorphism group larger than the subgroup generated by its hyperelliptic involution. It…

Let $C$ be a hyperelliptic curve over $\mathbb{Q}$ of genus $g \geq 2$, defined by a model … for $f\in\mathbb{Z}[x]$ without repeated roots. For each squarefree integer $d$, let…

Let $\mathfrak X$ be a hyperelliptic curve with extra automorphisms, meaning automorphisms beyond the hyperelliptic involution. Shaska's conjecture. Every such curve can be defined…

Let $g$ be the genus of a hyperelliptic sigma-function, let $\mathscr{P}=\mathbb{Q}[\Lambda]$, and let $\mathscr{Q}=\mathscr{P}[D_1,\dots,D_g]$ be the graded ring of linear differe…

Let $n=14$ or $n=16$, and consider the cyclic Newtonian motions associated with trajectories on the hyperelliptic curve ${\Greekmath 0116}^2={\Greekmath 0117}^n-1$. A trajectory is…

A Somos-$k$ sequence is a sequence defined by a higher-order Somos recurrence. A discrete linear flow on the Jacobian of a hyperelliptic curve is a discrete dynamical evolution giv…

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…

Let $X$ be a hyperelliptic curve over a field of characteristic $0$, and let $K_X$ denote its field of moduli. The automorphism group of $X$ is denoted by $\operatorname{Aut}(X)$.…

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 $X$ be a ruled surface over a smooth projective curve $C$ of genus $g$, and let $L\in \operatorname{Pic}X$ be a line bundle in the numerical class $aC_0+bf$, with $a\geq 1$; le…

Let $X$ be a hyperelliptic curve of genus $g$, let $J(X)$ be its Jacobian variety, and let $\Theta$ be the theta divisor in $J(X)$. For $k\leq g$, write…

Let $X$ be the hyperelliptic curve $w^2-t(z)w+1=0$ of genus $g$, with points $\infty^\pm$ above $z=\infty$, and let $J_{\text{aff}}(t)=X[g]-D$, where $D$ is the divisor locus on wh…

Let $J$ be an Abelian variety with theta divisor $\Theta$, and let $W^g$ be the quotient cohomology space defined in the paper. Hyperelliptic Jacobian characterization conjecture.…

Let $X_{\mathrm{aff}}(g)$ be the $g$th symmetric product of the affine curve, let $D=D_0\cup D_{\infty}$, and define … where $\omega$ is the class defined in the paper. The natural…

Let $\mathbf{C}^{*}_0$ and $\mathbf{C}^{*}_{f^0}$ be the specialized complexes associated with the singular and nonsingular parameter values, respectively. For a nonsingular hypere…

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 supersingular hyperelliptic curve of genus $g\geq 2$. Naive hyperelliptic Oort analogue. The automorphism group of $C$ is … The paper immediately explains that…

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 $\mathcal{V}ec^\mathcal{H}(r,d,g)$ be the universal moduli stack of vector bundles of rank $r$ and degree $d$ over hyperelliptic curves of genus $g$. For a hyperelliptic curve…