Suppose that $C$ and $C'$ are cubic forms in at least 19 variables over a
$p$-adic field $k$. A special case of a conjecture of Artin is that the
forms $C$ and $C'$ have a common zero over $k$. While the conjecture of
Artin is false in general, we try to argue that, in this case, it is
(almost) correct! This is still work in progress (joint with
Heath-Brown), so do not expect a full answer.
As a historical note, some cases of Artin's conjecture for certain
hypersurfaces are known. Moreover, Jahan analyzed the case of the
simultaneous vanishing of a cubic and a quadratic form. The approach
we follow is closely based on Jahan's approach, thus there might be
some overlap between his talk and this one. My talk will anyway be
self-contained, so I will repeat everything that I need that might
have already been said in Jahan's talk.