Artin's Conjecture for Abelian Varieties with Frobenius Condition
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866911623735672832 |
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| author | Hess, Florian Tomczak, Leonard |
| author_facet | Hess, Florian Tomczak, Leonard |
| contents | $A$ be an abelian variety over a number field $K$ of dimension $r$, $a_1, \dots, a_g \in A(K)$ and $F/K$ a finite Galois extension. We consider the density of primes $\frak p$ of $K$ such that the quotient $\bar{A}(k({\frak p}))/\langle \bar{a}_1,\dots,\bar{a}_g\rangle$ has at most $2r-1$ cyclic components and $\frak p$ satisfies a Frobenius condition with respect to $F/K$, where $\bar{A}$ is the reduction of $A$ modulo $\frak p$, $k(\frak p)$ is the residue class field of $\frak p$ and $\langle \bar{a}_1,\dots,\bar{a}_g\rangle$ is the subgroup generated by the reductions $\bar{a}_1,\dots,\bar{a}_g$. We develop a general framework to prove the existence of the density under the Generalized Riemann Hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_03386 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Artin's Conjecture for Abelian Varieties with Frobenius Condition Hess, Florian Tomczak, Leonard Number Theory 11R45, 11G05, 11G10 $A$ be an abelian variety over a number field $K$ of dimension $r$, $a_1, \dots, a_g \in A(K)$ and $F/K$ a finite Galois extension. We consider the density of primes $\frak p$ of $K$ such that the quotient $\bar{A}(k({\frak p}))/\langle \bar{a}_1,\dots,\bar{a}_g\rangle$ has at most $2r-1$ cyclic components and $\frak p$ satisfies a Frobenius condition with respect to $F/K$, where $\bar{A}$ is the reduction of $A$ modulo $\frak p$, $k(\frak p)$ is the residue class field of $\frak p$ and $\langle \bar{a}_1,\dots,\bar{a}_g\rangle$ is the subgroup generated by the reductions $\bar{a}_1,\dots,\bar{a}_g$. We develop a general framework to prove the existence of the density under the Generalized Riemann Hypothesis. |
| title | Artin's Conjecture for Abelian Varieties with Frobenius Condition |
| topic | Number Theory 11R45, 11G05, 11G10 |
| url | https://arxiv.org/abs/2212.03386 |