Picard groups, pull back and class groups
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866929493100199936 |
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| author | Banerjee, Kalyan Hoque, Azizul |
| author_facet | Banerjee, Kalyan Hoque, Azizul |
| contents | Let $S$ be a certain affine algebraic surface over $\mathbb{Q}$ such that it admits a regular map to $\mathbb{A}^2/\mathbb{Q}$. We show that any non-trivial torsion line bundle in the relative Picard group $Pic^0\left(S/\mathbb{A}^2\right)$ can be pulled back to ideal classes of quadratic fields whose order can be made sufficiently large. This gives an affirmative answer to a question raised by Agboola and Pappas, in case of affine algebraic surfaces. For a closed point $P\in \mathbb{A}^2/\mathbb{Q}$, we show that the cardinality of a subgroup of the Picard group of the fiber $S_P$ remains unchanged when $P$ varies over a Zarisky open subset in $\mathbb{A}^2$. We also show by constructing an element of odd order $n\geq 3$ in the class group of certain imaginary quadratic fields that the Picard group of $S_P$ has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1903_04210 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Picard groups, pull back and class groups Banerjee, Kalyan Hoque, Azizul Algebraic Geometry Number Theory 11R29, 11R65, 14C20 Let $S$ be a certain affine algebraic surface over $\mathbb{Q}$ such that it admits a regular map to $\mathbb{A}^2/\mathbb{Q}$. We show that any non-trivial torsion line bundle in the relative Picard group $Pic^0\left(S/\mathbb{A}^2\right)$ can be pulled back to ideal classes of quadratic fields whose order can be made sufficiently large. This gives an affirmative answer to a question raised by Agboola and Pappas, in case of affine algebraic surfaces. For a closed point $P\in \mathbb{A}^2/\mathbb{Q}$, we show that the cardinality of a subgroup of the Picard group of the fiber $S_P$ remains unchanged when $P$ varies over a Zarisky open subset in $\mathbb{A}^2$. We also show by constructing an element of odd order $n\geq 3$ in the class group of certain imaginary quadratic fields that the Picard group of $S_P$ has a subgroup isomorphic to $\mathbb{Z}/n\mathbb{Z}$. |
| title | Picard groups, pull back and class groups |
| topic | Algebraic Geometry Number Theory 11R29, 11R65, 14C20 |
| url | https://arxiv.org/abs/1903.04210 |