Local solubility of ternary cubic forms

Abstract

We consider cubic forms $\ddot{I}{•}_{a,b}(x,y,z)=a{x}^3+b{y}^3-z^3$ 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 $1\le a\le A$, $1\le b\le B$ and some mild conditions, such that $\ddot{I}{•}_{a,b}$ has a non-zero solution in ${\mathbb{Q}}_p$ for all primes $p$.

21 pages