The Hasse principle for pairs of diagonal cubic forms

TL;DR

This paper proves that pairs of diagonal cubic forms in thirteen or more variables satisfy the Hasse principle over the rationals, using advanced exponential sum estimates and the Hardy-Littlewood method.

Contribution

It introduces a new mean value theorem for exponential sums and applies it to establish the Hasse principle for these forms.

Findings

Hasse principle holds for pairs of diagonal cubic forms in ≥13 variables

New mean value theorem for exponential sums developed

Application of Hardy-Littlewood method confirmed the result

Abstract

By means of the Hardy-Littlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables.