[1] Roger Heath-Brown showed[2] that every cubic form over the integers in at least 14 variables represents 0, improving on earlier results of Davenport.
It is known that Heath-Brown's result is best possible in the sense that there exist non-singular cubic forms over the rationals in 9 variables that do not represent zero.
[5] However, Hooley showed that the Hasse principle holds for the representation of 0 by non-singular cubic forms over the rational numbers in at least nine variables.
[7] Counterexamples by Fujiwara and Sudo show that the Hasse–Minkowski theorem is not extensible to forms of degree 10n + 5, where n is a non-negative integer.
The Hasse principle for algebraic groups was used in the proofs of the Weil conjecture for Tamagawa numbers and the strong approximation theorem.