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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2412.14980 |
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| _version_ | 1866917874862391296 |
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| author | Wolff, Golo |
| author_facet | Wolff, Golo |
| contents | We consider cubic forms $ϕ_{a,b}(x,y,z) = ax^3 + by^3 - z^3$ 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 $ϕ_{a,b}$ has a non-zero solution in $\mathbb{Q}_p$ for all primes $p$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_14980 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Local solubility of ternary cubic forms Wolff, Golo Number Theory 11N45 (Primary), 11R45 (Secondary) We consider cubic forms $ϕ_{a,b}(x,y,z) = ax^3 + by^3 - z^3$ 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 $ϕ_{a,b}$ has a non-zero solution in $\mathbb{Q}_p$ for all primes $p$. |
| title | Local solubility of ternary cubic forms |
| topic | Number Theory 11N45 (Primary), 11R45 (Secondary) |
| url | https://arxiv.org/abs/2412.14980 |