Title:  Artin's Conjecture on homogeneous forms over $\mathbb{Q}_p$

Abstract:  A field $k$ is called a $C_i$ field if any homogeneous form of degree $i$ in more than $d^i$ variables has a nontrivial zero in $k$. It is well known that finite fields are $C_1$. What about the p-adics? It was conjectured by Emil Artin that $\mathbb{Q}_p$ is $C_2$. The result turned out to be false. We will investigate some positive results.