Splitting of almost ordinary abelian surfaces in families and the $S$-integrality conjectures
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913783767629824 |
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| author | Jiang, Ruofan |
| author_facet | Jiang, Ruofan |
| contents | Let $A$ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $Δ$ is an infinite set of positive integers, such that $\left(\frac{m}{p}\right)=1$ for $\forall m\in Δ$. If $A$ does not admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of $A$ has endomorphism ring containing $\mathbb{Z}[x]/(x^2-m)$ for some $m\in Δ$. This implies that there are infinitely many places modulo which $A$ is not simple, generalizing the main result of arXiv:1812.11679 to the non-ordinary case. As an another application, we also generalize the $S$-integrality theorem for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of abelian surfaces over global function fields. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_07715 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Splitting of almost ordinary abelian surfaces in families and the $S$-integrality conjectures Jiang, Ruofan Number Theory Algebraic Geometry Let $A$ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic $p$. Suppose $Δ$ is an infinite set of positive integers, such that $\left(\frac{m}{p}\right)=1$ for $\forall m\in Δ$. If $A$ does not admit any global real multiplication, we prove the existence of infinitely many places modulo which the reduction of $A$ has endomorphism ring containing $\mathbb{Z}[x]/(x^2-m)$ for some $m\in Δ$. This implies that there are infinitely many places modulo which $A$ is not simple, generalizing the main result of arXiv:1812.11679 to the non-ordinary case. As an another application, we also generalize the $S$-integrality theorem for elliptic curves over number fields, as proved in arXiv:math/0509485, to the setting of abelian surfaces over global function fields. |
| title | Splitting of almost ordinary abelian surfaces in families and the $S$-integrality conjectures |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2304.07715 |