9 problems

Let $E/\mathbb{Q}$ be an elliptic curve, and let its isogeny class be the set of elliptic curves over $\mathbb{Q}$ that are $\mathbb{Q}$-isogenous to $E$. Mazur's conjecture. The s…

Let $p$ be prime, and let $\ell_1,\ldots,\ell_r$ be primes in an interval with $X\asymp Y\asymp\log^4 p$, as in the source. Let $\varepsilon$ denote the separation parameter for th…

Let $A_\ell$ denote the weighted adjacency matrix of the supersingular $\ell$-isogeny graph $\mathcal{G}(p,\ell)$ for all $\ell\neq p$. Let $v$ be an $L^2$-normalized simultaneous…

Let $p\ge 5$ and let $\ell\ne p$ be prime. Let $\mathcal{G}^{SS}_{2}(\ell,p)$ be the superspecial isogeny graph, with normalized adjacency eigenvalues … where $m=|SS_2(p)|$, and se…

Let $g\ge 2$, and let $\mathcal{G}^{SS}_{g}(\ell,p)$ denote the superspecial isogeny graph associated with the prime $\ell\ne p$. A graph is asymptotically relatively Ramanujan whe…

Uniform spectral bound conjecture. For all $g$ and $\ell$, there exists a fixed $\lambda=\lambda(g,\ell)<1$ such that

Maximal-order containment conjecture. For every $\epsilon>0$, the number of maximal orders containing $\Lambda$ is

Good-extension path conjecture. For every two vertices $A$ and $A'$ in $\mathcal{J}_p\subsetmathcal{G}_p$, such a path exists.

Jacobian-subgraph connectedness conjecture. The subgraph of $\mathcal{G}_p$ supported on $\mathcal{J}_p$ is connected.