The cubic case of the main conjecture in Vinogradov's mean value theorem

We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree 3 for moments of order 2s  , establishing strongly diagonal behaviour for 1⩽s⩽61⩽s⩽6. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding 2.