9 problems

Infinitude conjecture. There are infinitely many values of $c'\not\equiv 3\pmod 7$ such that every prime other than $2$ and $7$ occurring in the factorization of $3^{3+6c'}+1$ has…

A variational problem may be formulated on chain spaces with coefficients in $\mathbb{R}$, $\mathbb{Z}$, and $\mathbb{Z}/n\mathbb{Z}$ for $n\geq 2$. Obstructions to the Hasse princ…

Let $C/\mathbb Q$ be a genus-one hyperelliptic curve, let $E$ be its Jacobian, let $\Sigma$ be the specified finite set of places, and let $\mathcal F_b$ be the corresponding twist…

Skorobogatov's conjecture. The Brauer–Manin obstruction is the only obstruction to the Hasse principle for algebraic K3 surfaces over number fields.

Let $k$ be a $p$-adic field, and let $F$ be the function field of a smooth projective geometrically integral curve defined over $k$. Let $\Omega$ denote the set of discrete (rank 1…

For each integer $m$, let $\mathcal{U}_m$ denote the corresponding integral Markoff surface, and let $\mathbb{A}_{\mathbb{Z}}$ denote the adelic space used to formulate the integra…

Let $d_i,n_i$ be positive integers for $i\in\{1,\ldots,k\}$, and consider multidegree $(d_1,\ldots,d_k)$ hypersurfaces in $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_k}$ over…

Persistent genus-one Hasse principle violation conjecture. For every global field $k$, there is a genus one curve $C$ defined over $k_0$ such that $C$ violates the Hasse Principle…

Genus-one Hasse principle violation conjecture. For every global field $k$, there exists a genus one curve $C_{/k}$ which violates the Hasse Principle. Equivalently, there exists a…