3 problems

For distinct prime numbers $p$ and $q$, let $H_{p,q}$ be the subgroup of $(\mathbb{Z}/p\mathbb{Z})^{\times}$ spanned by the images of $q$ and $-1$ in $\mathbb{F}_{p}^{\times}$, and…

For $N\in\mathbb{Z}_{\geq1}$, let $\kappa(N)$ be the dimension of the cokernel of the weight-two map $D_{2}^{\mathrm{iter}}$. Let $a,b\in\mathbb{Z}_{\geq0}$. Vanishing conjecture f…

Let $p\geq5$ be a prime number. For $N\in\mathbb{Z}_{\geq1}$, let $\kappa(N)$ be the dimension of the cokernel of the weight-two map $D_{2}^{\mathrm{iter}}$. Zhao's conjectures. Th…