23 problems

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

A Fano hypersurface is a hypersurface whose anticanonical class is ample; in the statistical model of random Fano hypersurfaces over the rationals, one asks whether such a hypersur…

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…

Let $k$ be a number field, and let $C/k$ be a curve of positive genus with no $k$-rational points. A potential Hasse principle failure (PHPF) is a variety for which there exists an…

A cubic hypersurface in $\mathbb{P}^n$ over $\mathbb{Q}$ is the vanishing locus of a cubic form in $n+1$ variables. The Hasse principle conjecture. When $n\geq 4$, every such cubic…

Let $V$ be a smooth cubic surface over a number field $k$. A point of degree prime to $3$ on $V$ means a closed point whose degree over $k$ is coprime to $3$. The prime-to-3 degree…

Let $C$ be an algebraic curve defined over a number field $K$, of positive genus and without $K$-rational points. Say that $C_{/K}$ is a potential Hasse principle violation if ther…

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…

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

Let $C/\mathbb Q$ be a genus-one hyperelliptic curve, let $\mathcal F_b$ be a twist family, and suppose that for every $d\in\mathcal F_b$ the class $[C]$ lies in…

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…

Poonen–Voloch conjecture. If $n\geq d+1$ and $d\geq 3$, then the proportion of vectors $\boldsymbol a\in\mathfrak{A}(A)$ for which the equation $f_{\boldsymbol a}(\boldsymbol x)=0$…

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 $\mathbb{V}_{d,n}$ be the set of degree $d$ hypersurfaces in $\mathbb{P}^n$ defined over $\mathbb{Q}$, and let $\mathbb{V}_{d,n}(A)$ be the subcollection of height at most $A$.…

Let $n$ be an integer, and consider the Diophantine equation … The sum of three cubes conjecture. If $n\not\equiv \pm 4\pmod 9$, then the equation has an integral solution. The sou…

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…

Folk conjecture. A density of $100\%$ of integral binary $n$-ic forms that locally represent $h$ do not globally represent $h$, meaning that $F(x,y)=h$ has no solution in…

Let plane cubic curves over $\mathbb{Z}$ be ordered by height, and consider those that are everywhere locally soluble. Bhargava's conjecture. The precise proportion of such plane c…

Let $a_1,\dots,a_k$, $b_{11},\dots,b_{1\ell},\dots,b_{k1},\dots,b_{k\ell}$ be non-zero integers, let $c$ be an integer, and consider the equation … with non-negative integer expone…

Local-to-global proportion conjecture. For any $B$, if $V$ is the space of ternary cubic forms, then

Exact-failure-proportion conjecture. For any $B$,

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…