89 problems

Let $X/k$ be a curve with Jacobian $J$, and write $g(X)=\dim J$. A CM Jacobian is a Jacobian with complex multiplication. Degeneracy can only occur when … Coleman's conjecture. Up…

Let $N$ be a prime number. Then $X_0(N)$ has exactly two cusps, $0$ and $\infty$, and let $[0-\infty]$ denote the class in $J_0(N)$ of the divisor $0-\infty$. Ogg's conjecture. The…

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

Let $X$ be a principally polarized complex abelian variety, and let its Kummer variety be the quotient of $X$ by the involution $x\mapsto -x$. A trisecant line is a line meeting th…

Ciliberto–van der Geer's conjecture. If $1\le \ell\le g-3$, then

Schneider's nonvanishing conjecture. The Schneider regulator with respect to the cyclotomic character is nonzero. This conjecture asserts nondegeneracy of the cyclotomic $p$-adic h…

Pareschi–Popa conjecture. If $\Theta=X+Y$ decomposes nontrivially as a sum, then either $(A,\Theta)$ is the Jacobian of a smooth projective curve, or $(A,\Theta)$ is the intermedia…

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 $A$ be a geometrically simple ordinary Jacobian, and let $\delta_A$ denote its angle rank. Ahmadi–Shparlinski's conjecture. Every geometrically simple ordinary Jacobian has max…

Finiteness conjecture. The group

Hindry–Pacheco conjecture.

van Straten–Warmt conjecture. The odd Betti numbers of $J_{\mathcal{R}}$ vanish; equivalently,

Van Geemen–van der Geer's conjecture. If $(A,\Theta)$ is the Jacobian of a smooth curve $C$, then, as a set,

Refined Birch–Swinnerton-Dyer conjecture. The Birch–Swinnerton-Dyer conjecture holds, $\sha(J_X)$ is finite, and

Let $(A,\mathrm{\Theta})$ be a principally polarized abelian variety, let $C\subset A$ be a curve, and let $\mathcal I_C(2\Theta)$ denote the ideal sheaf of $C$ twisted by twice th…

Tropical Torelli conjecture. The map from the moduli space of tropical curves to $\mathcal{J}$ is tropical of degree $1$.

Novikov's conjecture. Just the first equation, with $N=1$, is sufficient for the characterization of the Jacobians.

Let ${\mathcal A}_4$ be the moduli space of 4-dimensional principally polarized abelian varieties, let $\tau\in{\mathcal A}_4$, and suppose that some theta constant at $\tau$ and i…

For a smooth canonical curve, its Jacobian has a theta divisor whose double points have tangent cones, viewed as quadrics in the canonical space, and the canonical curve has an ide…

Let $k$ be an algebraically closed field of characteristic $p$, and let ${\cal M}_g$ be the moduli space of smooth curves of genus $g$. For a curve $X$ of genus $g$, write $J_X$ fo…

Let $f:X\to S$ be the family of reduced curves considered in the paper, and let $J$ denote its relative Jacobian. Suppose that the irreducible components of every fibre of $f$ are…

Let $C$ be a smooth projective curve of genus $g$, let $J_C^d$ be its degree-$d$ Jacobian, and let $x\in J_C^d$ with $1\leq d\leq g$. Let ${\cal L}_x$ be the line bundle on $S^dC$…

Let $C$ be a smooth projective curve of genus $g$, let $J^1$ be its degree-$1$ Jacobian, and let $G_{g-1}$ denote the relevant parameter space of divisors in the linear series cons…

Let $C$ be a very general smooth connected projective curve over $\mathbb{C}$ of genus $g\geq 2s+1$, let $J$ be its Jacobian, and let $e$ be a degree-$1$ divisor on $C$. Write…

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…