5 problems

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $K$ be a number field. For each prime $p$, let $R_p(E/K)$ denote the $p$-adic regulator. Density-zero regulator conjecture. T…

The $t$-adic refinement. There is a topological $\mathbb{Q}$-algebra isomorphism from $\mathcal{Z}_{\widehat{\mathcal{A}}}$ onto $\overline{\mathcal{Z}}\llbracket t\rrbracket$ whic…

Let $Z$ be a totally real integer scheme, and let $\mathfrak{p} in Z$ be a totally split prime. For each $n$, let $X(\mathcal{O}_\mathfrak{p})_n$ denote the associated polylogarith…

For $k,m,t\in\mathbb{N}$, let $\mathrm{SPS}(k,m,t)$ denote the class of polynomials represented as sums of $k$ products of $m$ integer polynomials, each having at most $t$ nonzero…

Higher Newton polygon basis conjecture. The following family is a $p$-integral basis of $K$: