Ranks of abelian varieties in cyclotomic twist families
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866910736302735360 |
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| author | Shnidman, Ari Weiss, Ariel |
| author_facet | Shnidman, Ari Weiss, Ariel |
| contents | Let $A$ be an abelian variety over a number field $F$, and suppose that $\mathbb Z[ζ_n]$ embeds in $\mathrm{End}_{\bar F} A$, for some root of unity $ζ_n$ of order $n = 3^m$. Assuming that the Galois action on the finite group $A[1-ζ_n]$ is sufficiently reducible, we bound the average rank of the Mordell--Weil groups $A_d(F)$, as $A_d$ varies through the family of $μ_{2n}$-twists of $A$. Combining this with the recently proved uniform Mordell--Lang conjecture, we prove near-uniform bounds for the number of rational points in twist families of bicyclic trigonal curves $y^3 = f(x^2)$, as well as in twist families of theta divisors of cyclic trigonal curves $y^3 = f(x)$. Our main technical result is the determination of the average size of a $3$-isogeny Selmer group in a family of $μ_{2n}$-twists. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_06803 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Ranks of abelian varieties in cyclotomic twist families Shnidman, Ari Weiss, Ariel Number Theory Algebraic Geometry 11G10 (11E76, 11S25, 14G05) Let $A$ be an abelian variety over a number field $F$, and suppose that $\mathbb Z[ζ_n]$ embeds in $\mathrm{End}_{\bar F} A$, for some root of unity $ζ_n$ of order $n = 3^m$. Assuming that the Galois action on the finite group $A[1-ζ_n]$ is sufficiently reducible, we bound the average rank of the Mordell--Weil groups $A_d(F)$, as $A_d$ varies through the family of $μ_{2n}$-twists of $A$. Combining this with the recently proved uniform Mordell--Lang conjecture, we prove near-uniform bounds for the number of rational points in twist families of bicyclic trigonal curves $y^3 = f(x^2)$, as well as in twist families of theta divisors of cyclic trigonal curves $y^3 = f(x)$. Our main technical result is the determination of the average size of a $3$-isogeny Selmer group in a family of $μ_{2n}$-twists. |
| title | Ranks of abelian varieties in cyclotomic twist families |
| topic | Number Theory Algebraic Geometry 11G10 (11E76, 11S25, 14G05) |
| url | https://arxiv.org/abs/2107.06803 |