Hasse-Minkowski for cubic forms

Question

We know that an analogue of the Hasse-Minkowski theorem does not hold for all cubic forms, e.g. because Selmer's cubic: $$ 3x^3 + 4y^3 + 5z^3 = 0 $$ has solutions over $\mathbb{R}$ and $\mathbb{Q}_p$ for all $p$, but no solutions over $\mathbb{Q}$.

My questions are:

1. Can we find a (non-trivial) class of cubic forms where an analogue of the Hasse-Minkowski theorem does hold?
2. Is there any intuition for why the local-global principle holds for quadratic but fails for cubic forms?
3. Are there higher degree forms where the local-global principle holds again?
4. Are questions like these addressed anywhere in the literature?

Many thanks.

Answer 1

1. Let $L/\mathbb{Q}$ a Galois cyclic extension of degree $3$. Then the norm form $N:x\in L\mapsto N_{L/\mathbb{Q}}(x))$ is a cubic form in the coordinates of $x$ in a fixed basis of $L$. Now fix $a\in\mathbb{Q}$

By a theorem of Hilbert, Hasse principle holds for the affine cubic form $x\mapsto N(x)-a$ ( a rational number is globally a norm of a cyclic extension if and only if it is locally)

Note this is true if you replace $3$ by any integer, provided you restrict yourself to cyclic extensions. You can also replace $\mathbb{Q}$ ny a global field.

2. I don't know if the following will give you "intuition", but certainly it will explain a lot (but not all) counter examples to Hasse principle. The keywords are: Brauer-Manin obstruction, and you can find a lot of things about it in the literature. This is an obstruction coming from Brauer-Hasse-Noether theorem for central simple algebras.

You can find a nice introduction to the definition here: http://homepages.warwick.ac.uk/staff/S.Siksek/arith/notes/brauermanin.pdf

3. I don't think so. Using Brauer-Manin obstruction, you can construct counterexamples of any prescribed degree.

For example, Bayer, Lee and Parimala computed the Brauer Manin obstruction for multinorm equations, and I'm pretty sure that we can use their work to construct counterexamples of arbitrary large degree. The full paper is available on Parimala's web page.