On the Frobenius fields of abelian varieties over number fields
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866914418055446528 |
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| author | Burungale, Ashay A. Hida, Haruzo Lai, Shilin |
| author_facet | Burungale, Ashay A. Hida, Haruzo Lai, Shilin |
| contents | Let $A$ be a non-CM simple abelian variety over a number field $K$. For a place $v$ of $K$ such that $A$ has good reduction at $v$, let $F(A,v)$ denote the Frobenius field generated by the corresponding Frobenius eigenvalues. Assuming $A$ has connected monodromy groups, we show that the set of places $v$ such that $F(A,v)$ is isomorphic to a fixed number field has upper Dirichlet density zero. Assuming the GRH, we give a power saving upper bound for the number of such places. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07935 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the Frobenius fields of abelian varieties over number fields Burungale, Ashay A. Hida, Haruzo Lai, Shilin Number Theory Let $A$ be a non-CM simple abelian variety over a number field $K$. For a place $v$ of $K$ such that $A$ has good reduction at $v$, let $F(A,v)$ denote the Frobenius field generated by the corresponding Frobenius eigenvalues. Assuming $A$ has connected monodromy groups, we show that the set of places $v$ such that $F(A,v)$ is isomorphic to a fixed number field has upper Dirichlet density zero. Assuming the GRH, we give a power saving upper bound for the number of such places. |
| title | On the Frobenius fields of abelian varieties over number fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2402.07935 |