Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field
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| Format: | Preprint |
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2021
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| _version_ | 1866910063554199552 |
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| author | Goodman, Pip |
| author_facet | Goodman, Pip |
| contents | In this article we study the endomorphism algebras of abelian varieties $A$ defined over a given number field $K$ with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of $A$ to be defined over $K(A[2])$, the field generated by its 2-torsion.
When $K= \mathbb{Q}$ and $\mathrm{Gal}(\mathbb{Q}(A[2])/\mathbb{Q})$ is cyclic of prime order $p = 2 \dim(A) +1$, we prove that there are only finitely many possibilities for the geometric endomorphism algebra $\mathrm{End}(A) \otimes \mathbb{Q}$.In fact, when $\dim (A) \not \in \{3,5,9,21,33,81\}$, we show $\mathrm{End}(A) \otimes \mathbb{Q}$ is a proper subfield of the $p$-th cyclotomic field. In particular, when $g=2$, $\mathrm{End}(A) \otimes \mathbb{Q}$ is isomorphic to either $\mathbb{Q}$ or $\mathbb{Q}(\sqrt{5})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_01565 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field Goodman, Pip Number Theory 11G10, 14H40, 14K15 In this article we study the endomorphism algebras of abelian varieties $A$ defined over a given number field $K$ with large cyclic 2-torsion fields. A key step in doing so is to provide criteria for all the endomorphisms of $A$ to be defined over $K(A[2])$, the field generated by its 2-torsion. When $K= \mathbb{Q}$ and $\mathrm{Gal}(\mathbb{Q}(A[2])/\mathbb{Q})$ is cyclic of prime order $p = 2 \dim(A) +1$, we prove that there are only finitely many possibilities for the geometric endomorphism algebra $\mathrm{End}(A) \otimes \mathbb{Q}$.In fact, when $\dim (A) \not \in \{3,5,9,21,33,81\}$, we show $\mathrm{End}(A) \otimes \mathbb{Q}$ is a proper subfield of the $p$-th cyclotomic field. In particular, when $g=2$, $\mathrm{End}(A) \otimes \mathbb{Q}$ is isomorphic to either $\mathbb{Q}$ or $\mathbb{Q}(\sqrt{5})$. |
| title | Endomorphism algebras of abelian varieties with large cyclic 2-torsion field over a given field |
| topic | Number Theory 11G10, 14H40, 14K15 |
| url | https://arxiv.org/abs/2111.01565 |