16 problems

Weak Main Conjecture. There are no rational points at suitably high tower levels.

Uniform non-modularity conjecture. The same conclusion holds for every $p\neq 3$.

Mazur–Merel version of the Strong Main Conjecture. Assuming the hypotheses of the Strong Main Conjecture, for all $p\notin P_{\mathbf C}$, only finitely many $V\in\mathcal V_p(J)$…

Strong Main Conjecture. For all $p\notin P_{\mathbf C}$, only finitely many $V\in\mathcal V_p(J)$ give a space $\mathcal H(V\times^s J,\mathbf C)^{\mathrm{in},\mathrm{rd}}$ having…

High-level genus conjecture. For large $k$, all components of $\overline{\mathcal H}_k$ have genus exceeding $1$.

No-Weigel-branch conjecture. There are no Weigel cusp branches on any infinite $K$ component branch of a modular tower.

g-$p'$ conjecture. Each $K$ component branch on a modular tower is defined by some g-$p'$ cusp branch.

Weak Main Conjecture. There cannot be $K$ points at every level of a modular tower. Consequently, regular realizations of all the $G_k$ over $K$ require an unbounded number of bran…

Let $\mathbb H$ be a modular tower, with levels parametrized by $k$, and let $\mathbb Q$ denote the rational field. A variety is of general type when some multiple of its canonical…

Let $O$ be a modular-tower orbit with geometric monodromy $G_O$, and for each $J'$ let $\mathcal G(\overline{\mathbf S}_{J'})$ denote the isomorphism class of the corresponding dec…

Let $\mathbb H=\{\mathcal H_k\to J_r\}_{k=0}^\infty$ be a projective sequence of absolutely irreducible Hurwitz-space components attached to $(G,\mathbf C,\ell)$, and let the geome…

Let the modular tower be the tower of arithmetic locally symmetric spaces and associated étale fundamental groups arising from the moduli-space constructions discussed in the sourc…

Let $\bar O_k$ be a component at level $k$, and let ${{\mathcal H}}^{{\text{rd}}}_{\bar O_k}$ denote the corresponding reduced Hurwitz-space component. Orbit shortening is the redu…

Assume the Main Conjecture hypotheses, and let $\bar O_k$ be any component at level $k$. Elliptic ramification from $\bar O_k$ to a component at level $k+1$ means the ramification…

Let a Modular Tower be associated to a $p$-perfect centerless group $G_0$ and the relevant collection of conjugacy classes, and let its level-$k$ spaces have nonsingular projective…

Let $p$ be an odd prime, let $G=G_0$ be a $p$-perfect centerless finite group, let ${{\text{\bf C}}}$ be the specified collection of $p'$ conjugacy classes, and let $G_k$ denote th…