Local solubility of ternary cubic forms

TL;DR

This paper provides an asymptotic count of ternary cubic forms with integer coefficients that are locally soluble everywhere, advancing understanding of their distribution over the integers.

Contribution

It establishes an asymptotic formula for the number of such forms, specifically those of the form  x^3 + b y^3 - z^3 with bounded coefficients, satisfying local solubility conditions.

Findings

Derived an asymptotic formula for locally soluble forms

Quantified the density of such forms among all forms with bounded coefficients

Extended understanding of local-global principles for cubic forms

Abstract

We consider cubic forms $\phi_{a,b}(x,y,z) = ax^3 + by^3 - z^3$ with coefficients $a,b \in \mathbb{Z}$. We give an asymptotic formula for how many of these forms are locally soluble everywhere, i.e. we give an asymptotic formula for the number of pairs of integers $(a, b)$ that satisfy $1 \leq a \leq A$, $1 \leq b \leq B$ and some mild conditions, such that $\phi_{a,b}$ has a non-zero solution in $\mathbb{Q}_p$ for all primes $p$.