Squares in arithmetic progression over quadratic extensions of number fields
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866911651288055808 |
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| author | González-Jiménez, Enrique |
| author_facet | González-Jiménez, Enrique |
| contents | We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we determine the set of $K$-quadratic points on this curve under certain conditions on the base field $K$. Our main results rely on the algebraic properties of specific elliptic curves after performing a base change to suitable number fields. As a consequence, we establish that, under appropriate assumptions, any non-elementary arithmetic progression of five or six squares properly defined over a quadratic extension of $K$ must be of a specific form. Moreover, we prove the non-existence of such progressions of length greater than six under these assumptions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03251 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Squares in arithmetic progression over quadratic extensions of number fields González-Jiménez, Enrique Number Theory Algebraic Geometry Primary: 11B25, 11G05, Secondary: 14G05, 14H45 We study arithmetic progressions of squares over quadratic extensions of number fields. Using a method inspired by an approach of Mordell, we characterize such progressions as quadratic points on a genus $5$ curve. Specifically, we determine the set of $K$-quadratic points on this curve under certain conditions on the base field $K$. Our main results rely on the algebraic properties of specific elliptic curves after performing a base change to suitable number fields. As a consequence, we establish that, under appropriate assumptions, any non-elementary arithmetic progression of five or six squares properly defined over a quadratic extension of $K$ must be of a specific form. Moreover, we prove the non-existence of such progressions of length greater than six under these assumptions. |
| title | Squares in arithmetic progression over quadratic extensions of number fields |
| topic | Number Theory Algebraic Geometry Primary: 11B25, 11G05, Secondary: 14G05, 14H45 |
| url | https://arxiv.org/abs/2602.03251 |