Diagonal cubic forms and the large sieve

TL;DR

This paper establishes bounds on the number of integral solutions to diagonal cubic forms using a large sieve approach, extending previous results that relied on automorphy and GRH assumptions.

Contribution

It introduces a large sieve inequality method to bound solutions of diagonal cubic forms, bypassing the need for automorphy and GRH assumptions.

Findings

Bound on solutions: $N(X) \\ll X^{3+\\epsilon}$

Extension to forms in ≥4 variables with approximations to Hasse--Weil L-functions

Method provides an alternative to automorphy-based bounds

Abstract

Let $N(X)$ be the number of integral zeros $(x_1,\dots,x_6)\in [-X,X]^6$ of $\sum_{1\le i\le 6} x_i^3$. Works of Hooley and Heath-Brown imply $N(X)\ll_\epsilon X^{3+\epsilon}$, if one assumes automorphy and GRH for certain Hasse--Weil $L$-functions. Assuming instead a natural large sieve inequality, we recover the same bound on $N(X)$. This is part of a more general statement, for diagonal cubic forms in $\geq 4$ variables, where we allow approximations to Hasse--Weil $L$-functions.