4 problems

Let $K$ be a number field. A proper variety $X$ over $K$ violates the local-global principle if $X(K_v)\neq\emptyset$ for every nontrivial place $v$ of $K$, while $X(K)=\emptyset$.…

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…

Let $C$ be a smooth curve of genus one embedded in ${\mathbb P}^{n-1}$ by the complete linear system of a $K$-rational divisor of degree $n\geq 3$. Let $A$ be the associative $K$-a…

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