4 problems

The twist conjecture. Taking $K_{\alpha,\beta}$ as in the displayed formula, the matrix $R_J$ satisfies the QYBE and coincides with $R_{\text{GGS}}$:

Let $\mathcal{E}$ be a twist family, such as a quadratic, cubic, quartic, or sextic twist family. For every $E\in\mathcal{E}$ and prime $p$, let $\overline{a}_p(E)$ be the normaliz…

Let $a,b,c\in\mathbb{Z}\setminus\{0\}$, and let $C_1,C_2,C_3$ be the hyperelliptic curves defined by the source's displayed family ${\rm}$. Three m-twist rank conjecture. The set o…

Simultaneous twist-rank conjecture. The set of $d\in\mathbb{Q}$ such that the Jacobians of the quadratic, $m$-twist and $2m$-twist of $C$ by $d$ all have positive rank is infinite.