13 problems

Kim's conjecture. For $n\gg 0$,

Let $F_\infty$ be the cyclotomic $\mathbf Z_p$-extension of $F$, and let $h^{(1)}$ denote the first derived height pairing. Cyclotomic nondegeneracy conjecture. The pairing…

Localisation injectivity and vanishing conjecture. For all $0\leq i\leq n$, this map is injective, and

The $p$-converse conjecture. If

Let $F$ be a totally real number field of narrow class number $1$, let $\mathfrak p$ be a prime of $F$, and let $K_i/F$ be quadratic extensions in which $\mathfrak p$ is inert, wit…

Let $F$ be a totally real number field of narrow class number $1$, let $\mathfrak p$ be a prime of $F$, and let $K_1,K_2$ be almost totally complex quadratic extensions of $F$ in w…

Let $F$ be a totally real number field of narrow class number $1$, let $\mathfrak p$ be a prime of $F$, and let $K/F$ be an almost totally complex quadratic extension in which…

$p$-saturated strong Chabauty conjecture. There is a finite subscheme $Z\subset A$ and a set of primes $p$ of density $1$ such that

Katz's conjecture. The mod $p$ logarithm $\overline{\log}_p(P)$ is equidistributed in

Let $p$ be inert in $K_D=\mathbb{Q}(\sqrt{D})$, let $2k>2$ be an even weight, and let $\mathfrak Z^\mathrm{How}_{D,r,2k}$ and $\mathfrak Z_{D,r,2k}^\mathrm{Heeg}$ denote respective…

Let $E/K$ be as in the source, let $p$ be a prime, and suppose that the $p$-primary part of the Tate–Shafarevich group of $E/K$ is finite. Let $\mathcal{U}$ be the universal norm m…

Mazur–Rubin's question. As $K$ varies, what are the lines $L_K$, and how are they distributed in $E(\mathbb{Q})\otimes\mathbb{Q}_p$? This asks for a description of the shadow lines…

Arithmetic ring-specific Oort conjecture. There exist $l>0$ and a normal, $C$-Galois, regular branched cover $Y\rightarrow D$ such that $Y_k$ is reduced, $L_m=M$ for $m\gg0$ and…