57 problems

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 (…

Rationality conjecture. The sheaf ${\mathcal F}_0(h)$ gives a rational point on the Jacobian if and only if, for each $k$, there exists a nonzero formal power series $R_k(q)$ such…

Positive-density conjecture. The set of primes $p$ of good reduction for $C$ such that $\bar{\omega}_p$ does not vanish on $C(\mathbb F_p)$ has positive density, unless $J$ splits…

Let $g$ be a sufficiently large genus, and let a smooth projective curve of genus $g$ have a Jacobian of ST type, as in the source. Coleman's conjecture. Only finitely many smooth…

Let $p$ be a prime, and let $J_1(p)$ be the Jacobian of the modular curve $X_1(p)$. A rational cusp difference is a divisor class represented by the difference of two rational cusp…

Let $\mathbb{F}_q$ be a finite field, let $\mathcal{A}=\mathbb{F}_q[T]$, and let $K=\mathbb{F}_q(T)$. For a monic $\mathfrak{n}\in\mathcal{A}$, let $X_0(\mathfrak{n})$ be the Drinf…

Let $N$ be a positive integer. Let $X_0(N)$ be the modular curve of level $N$, let $J_0(N)$ be its Jacobian, and let $C_N(\mathbb{Q})$ be its rational cuspidal subgroup. Let $C(N)$…

Generalised Ogg's conjecture. For every positive integer $N$,

Let $X$, $Y$, $\bar{Y}$, and $W$ be the curves in the diagram … For a suitable choice of a subcover $W$, the algebraic correspondence … induces the isogeny in the paper's hiding el…

Let $\mathfrak{p}$ be a prime of $A$, let $J_0(\mathfrak{p})$ be the Jacobian of the Drinfeld modular curve $X_0(\mathfrak{p})$, let $\mathcal{S}(\mathfrak{p})$ be the Shimura subg…

Let $p$ be prime. Let $\mathcal{S}(p)$ be the Shimura subgroup of $J_0(p)$, and let $\mathcal{M}(p)$ be the largest $\mu$-type subgroup of $J_0(p)$. Ogg's Shimura-subgroup conjectu…

Let $p$ be prime, let $J_0(p)$ be the Jacobian of $X_0(p)$, and let $\mathcal{C}(p)$ be its rational cuspidal divisor class group. Ogg's cuspidal torsion conjecture. The cyclic gro…

Let $C$ be a smooth projective curve over a number field $k$. Let $X_k$ be the set of conjugacy classes of $\operatorname{ST}(\operatorname{Jac}(C)_k)$, and let $\{p_i\}_{i\geq 1}$…

Let $F$ be a number field, let $v$ be a finite place above a prime $p$, and let $J/F$ be a Jacobian with semistable ordinary reduction at $v$. Let $\Theta$ be a theta divisor and s…

Let $E$ be a strong Weil curve over $\mathbb{Q}$ of conductor $M\mid N$. Suppose that $E$ has odd analytic rank, and let $\xi_{E,N}^\vee$ be the dual map associated with the windin…

Let $X$ be a curve defined over a number field $K$, and let $G$ be a finite group of $K$-automorphisms of $X$. Assume that $\Omega^{1}(\operatorname{Jac}_X)$ is self-dual as a $G$-…

Let $p$ be an odd prime, let $J_{p^3}$ be the Jacobian of $C_{p^3}\colon y^2=x^{p^3}-1$, and write $J_{p^3}\simeq X_2\times X_1\times J_p$, where $X_2$ is the absolutely simple fac…

Let $m$ be a positive odd integer, and let $J_m$ denote the Jacobian associated with the curve $C_m\colon y^2=x^m-1$. Write $\phi(d)$ for Euler's totient function. The simple-facto…

Let $B$) be a hyperelliptic curve of genus $\ga$, with a marked Weierstrass point $q$, and let $\Theta_1^0(B)\subset \operatorname{Jac}(B)$ denote the relevant locus of point class…

Let $N$ be a positive integer. Let $C_N(\mathbb{Q})$ be the subgroup of $J_0(N)(\mathbb{Q})$ represented by degree-zero cuspidal divisors, and let $C(N)$ be the subgroup formed by…

Let $C$ be a curve of genus $2$ or $3$ over a number field $K$, with an unramified double cover … and associated Prym variety $\operatorname{Prym}(D/C)$ over $K$. Let…

Let $p$ be a prime, let $A$ be a $4$-dimensional abelian variety over $\overline{\mathbf{F}}_p$, and let a stable curve be allowed to be possibly reducible. Shankar–Tsimerman's con…

Ahmadi–Shparlinski's conjecture. Every ordinary, geometrically simple Jacobian over a finite field has maximal angle rank.

Superelliptic Jacobian group-isomorphism conjecture. There exists an isomorphism of abstract groups

Regular-matroid Jacobian exponent conjecture. For every positive integer $k$, there are only finitely many connected regular matroids $M$ such that the exponent of…