7 problems

Knapp's conjecture.

Exact value conjecture. If $3\mid d$, then

Artin's additive quartic conjecture. Artin's conjecture holds for all additive quartic forms over ramified quadratic extensions of $\mathbb{Q}_2$. In particular,

Artin's additive form conjecture.

Artin's conjecture. Every homogeneous form of degree $d$ in $d^2+1$ variables over $K$ has a nontrivial zero, regardless of its coefficients.

Let $k\geq 1$ be a natural number, let $a_i$ and $b_i$ be integers for $1\leq i\leq s$, and let $p$ be a prime. A solution $(x_1,\ldots,x_s)\in\mathbb{Q}_p^s$ is non-trivial if it…

Artin's Additive Form Conjecture.