11 problems

Let $p$ be a prime with $p\equiv1,3\pmod8$, and write $p=x^2+2y^2$. Let $A_n$ denote the sequence used in the paper's Beukers-method construction. The first author's conjecture ass…

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good ordinary reduction at $p$. Let the $p$-adic height pairing on $E(\mathbb{Q})$ be the $p$-adic analogue of the usual height…

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…

Chabauty–Kim conjecture. For all sufficiently large $n$, one has

Let $Z$ be an integer scheme and let $p$ be a prime below $Z$, with $Z$ totally split over $p$. For each $n\in\mathbb{N}$, let … be the associated big polylogarithmic Chabauty–Kim…

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…

The authors study discrete Painlevé II equations over finite fields by extending the domain through spaces of initial conditions and by reduction modulo a prime from a local field.…

Let $p$ be prime and $e\geq 1$. Consider solutions $(g,x)$ of … with $g,x\in\{1,\dots,p^e\}$ and $p\nmid g,x$. Fixed-point counting conjecture. The number of these solutions is asy…

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

The constructed-basis conjecture. These elements are always a $p$-integral basis of $K$.