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arXiv · 2412.14980

Local solubility of ternary cubic forms

Golo Wolff
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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 $p$.

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Golo Wolff. 2024-12-19. Local solubility of ternary cubic forms. https://arxiv.org/abs/2412.14980

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