6 problems

Let $\mathcal{C}$ be a non-hyperelliptic curve, let $\operatorname{Jac}(\mathcal{C})$ be its Jacobian, let $\Sigma$ denote the relevant theta divisor, and let the basis $\wp$-funct…

Let $g$ be a positive integer, let $u,\alpha\in\operatorname{\mathbb{C}}^g$, and let $u_{g+1}\in\operatorname{\mathbb{C}}$. Let $\sigma$ be the genus-$g$ hyperelliptic sigma-functi…

Basis-construction conjecture. In general, a basis for $\Gamma(m)$ can always be constructed using the preceding basis, $m$th-order $\mathcal{R}$-functions, and the first derivativ…

Let $\Gamma(m)$ denote the space of Abelian functions with poles of order at most $m$ along the relevant theta divisor, and consider a basis of $\Gamma(m)$ whose entries have weigh…

Let $J$ be the Jacobian of an $(n,s)$-curve, and let $\Gamma\bigl(J,\mathcal{O}(k\Theta^{[g-1]})\bigr)$ be the corresponding space with a basis of Abelian functions. Polynomial eva…

Surjectivity conjecture. The map $\text{ev}^{gr}$ is surjective. Equivalently, $\text{gr}^{KP}\,A$ is generated as a ${\cal D}$-module by $1\in \text{gr}^{KP}_0A$ and the elements…