Extremal elasticity of quadratic orders
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913728703758336 |
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| author | Fan, Steve Pollack, Paul |
| author_facet | Fan, Steve Pollack, Paul |
| contents | We study how large and small elasticity can be for orders belonging to a fixed quadratic field, in terms of the corresponding conductors. For example, we show that if $K$ is an imaginary quadratic field, then the order of conductor $f$ in $K$ has elasticity exceeding $(\log{f})^{c_1 \log\log\log{f}}$ for all $f$ that are sufficiently large. On the other hand, this elasticity is smaller than $(\log{f})^{c_2\log\log\log{f}}$ for infinitely many $f$. Here $c_1, c_2$ are universal positive constants. The proofs borrow methods from analytic number theory previously employed to study statistics of the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^{\times}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_07801 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal elasticity of quadratic orders Fan, Steve Pollack, Paul Number Theory Primary 11R27, Secondary 11N37, 11R11, 11R65, 13A05 We study how large and small elasticity can be for orders belonging to a fixed quadratic field, in terms of the corresponding conductors. For example, we show that if $K$ is an imaginary quadratic field, then the order of conductor $f$ in $K$ has elasticity exceeding $(\log{f})^{c_1 \log\log\log{f}}$ for all $f$ that are sufficiently large. On the other hand, this elasticity is smaller than $(\log{f})^{c_2\log\log\log{f}}$ for infinitely many $f$. Here $c_1, c_2$ are universal positive constants. The proofs borrow methods from analytic number theory previously employed to study statistics of the multiplicative groups $(\mathbb{Z}/m\mathbb{Z})^{\times}$. |
| title | Extremal elasticity of quadratic orders |
| topic | Number Theory Primary 11R27, Secondary 11N37, 11R11, 11R65, 13A05 |
| url | https://arxiv.org/abs/2503.07801 |