On certain root number $1$ cases of the cube sum problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912822100754432 |
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| author | Das, Shamik Jha, Somnath |
| author_facet | Das, Shamik Jha, Somnath |
| contents | We consider certain families of integers $n$ determined by some congruence condition, such that the global root number of the elliptic curve $E_{-432n^2}: Y^2=X^3-432n^2$ is $1$ for every $n$, however a given $n$ may or may not be a sum of two rational cubes. We give explicit criteria in terms of the $2$-parts and $3$-parts of the ideal class groups of certain cubic number fields to determine whether such an $n$ is a cube sum. In particular, we study integers $n$ divisible by $3$ such that the global root number of $E_{-432n^2}$ is $1$. For example, for a prime $\ell \equiv 7 \pmod{9}$, we show that for $3\ell$ to be a sum of two rational cubes, it is necessary that the ideal class group of $\Q(\sqrt[3]{12\ell})$ contains $\frac{\Z}{6\Z}\oplus \frac{\Z}{3\Z}$ as a subgroup. Moreover, for a positive proportion of primes $\ell \equiv 7 \pmod{9}$, $3\ell$ can not be a sum of two rational cubes. A key ingredient in the proof is to explore the relation between the $2$-Selmer group and the $3$-isogeny Selmer group of $E_{-432n^2}$ with the ideal class groups of appropriate cubic number fields. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_05361 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On certain root number $1$ cases of the cube sum problem Das, Shamik Jha, Somnath Number Theory 11G05, 11R29, 11R34, 11D25, 11A15 We consider certain families of integers $n$ determined by some congruence condition, such that the global root number of the elliptic curve $E_{-432n^2}: Y^2=X^3-432n^2$ is $1$ for every $n$, however a given $n$ may or may not be a sum of two rational cubes. We give explicit criteria in terms of the $2$-parts and $3$-parts of the ideal class groups of certain cubic number fields to determine whether such an $n$ is a cube sum. In particular, we study integers $n$ divisible by $3$ such that the global root number of $E_{-432n^2}$ is $1$. For example, for a prime $\ell \equiv 7 \pmod{9}$, we show that for $3\ell$ to be a sum of two rational cubes, it is necessary that the ideal class group of $\Q(\sqrt[3]{12\ell})$ contains $\frac{\Z}{6\Z}\oplus \frac{\Z}{3\Z}$ as a subgroup. Moreover, for a positive proportion of primes $\ell \equiv 7 \pmod{9}$, $3\ell$ can not be a sum of two rational cubes. A key ingredient in the proof is to explore the relation between the $2$-Selmer group and the $3$-isogeny Selmer group of $E_{-432n^2}$ with the ideal class groups of appropriate cubic number fields. |
| title | On certain root number $1$ cases of the cube sum problem |
| topic | Number Theory 11G05, 11R29, 11R34, 11D25, 11A15 |
| url | https://arxiv.org/abs/2508.05361 |