Elliptic curves and their principal homogeneous spaces: splitting Severi--Brauer varieties
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| Format: | Preprint |
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2024
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| author | Mackall, Eoin Rekuski, Nick |
| author_facet | Mackall, Eoin Rekuski, Nick |
| contents | We consider the question: which elliptic curves appear as the Jacobian of a smooth curve of genus one splitting a Severi--Brauer variety? We provide three new examples.
First, we show that if $E$ is any elliptic curve over an algebraically closed field $k$ and if $F/k$ is a perfect field extension, then there exists a principal homogeneous space for $E_F$ splitting a Severi--Brauer variety $X=\mathrm{SB}(A)$ over $F$ if and only if $A$ is Brauer equivalent to a cyclic algebra. Along the way, we also give a uniform proof of a generalization of results due to O'Neil, Clark and Sharif, and Antieau and Auel.
Second, we give an example of an elliptic curve $E$ over a field $k$ together with a central simple algebra $A/k$ of degree $4$ such that $E$ is the Jacobian of a smooth genus one curve $C$ embedded in the Severi--Brauer variety $X=\mathrm{SB}(A)$ as a degree 8 curve and such that $E$ is not the Jacobian of any genus one curve of smaller degree contained in $X$. Our example is, in some sense, as small as possible in both dimension and arithmetic complexity.
Third, we show that for any odd integer $n\geq 3$ there is a central simple algebra $A$ of degree $n$ over a local field $k$ which is split by the principal homogeneous space of an elliptic curve $E/k$ but not by any principal homogeneous space for any quadratic twist of $E$. This generalizes a recent result of Saltman in the case of surfaces to arbitrary even dimension. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_10493 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Elliptic curves and their principal homogeneous spaces: splitting Severi--Brauer varieties Mackall, Eoin Rekuski, Nick Algebraic Geometry Number Theory Rings and Algebras 16K50, 14H52 We consider the question: which elliptic curves appear as the Jacobian of a smooth curve of genus one splitting a Severi--Brauer variety? We provide three new examples. First, we show that if $E$ is any elliptic curve over an algebraically closed field $k$ and if $F/k$ is a perfect field extension, then there exists a principal homogeneous space for $E_F$ splitting a Severi--Brauer variety $X=\mathrm{SB}(A)$ over $F$ if and only if $A$ is Brauer equivalent to a cyclic algebra. Along the way, we also give a uniform proof of a generalization of results due to O'Neil, Clark and Sharif, and Antieau and Auel. Second, we give an example of an elliptic curve $E$ over a field $k$ together with a central simple algebra $A/k$ of degree $4$ such that $E$ is the Jacobian of a smooth genus one curve $C$ embedded in the Severi--Brauer variety $X=\mathrm{SB}(A)$ as a degree 8 curve and such that $E$ is not the Jacobian of any genus one curve of smaller degree contained in $X$. Our example is, in some sense, as small as possible in both dimension and arithmetic complexity. Third, we show that for any odd integer $n\geq 3$ there is a central simple algebra $A$ of degree $n$ over a local field $k$ which is split by the principal homogeneous space of an elliptic curve $E/k$ but not by any principal homogeneous space for any quadratic twist of $E$. This generalizes a recent result of Saltman in the case of surfaces to arbitrary even dimension. |
| title | Elliptic curves and their principal homogeneous spaces: splitting Severi--Brauer varieties |
| topic | Algebraic Geometry Number Theory Rings and Algebras 16K50, 14H52 |
| url | https://arxiv.org/abs/2401.10493 |