On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields
Consider the modular group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>PSL</mi></mrow><mrow><mo>(</mo><mn>2</mn><mo>,</mo><...
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MDPI AG
2025-04-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/14/5/335 |
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| author | Abdulaziz Deajim |
| author_facet | Abdulaziz Deajim |
| author_sort | Abdulaziz Deajim |
| collection | DOAJ |
| description | Consider the modular group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>PSL</mi></mrow><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">Z</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>⟨</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>y</mi><mspace width="0.166667em"></mspace><mo>|</mo><mspace width="0.166667em"></mspace><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><msup><mi>y</mi><mn>3</mn></msup><mo>=</mo><mn>1</mn><mo>⟩</mo></mrow></mrow></semantics></math></inline-formula> generated by the transformations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>:</mo><mi>z</mi><mo>↦</mo><mo>−</mo><mn>1</mn><mo>/</mo><mi>z</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>:</mo><mi>z</mi><mo>↦</mo><mo>(</mo><mi>z</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>z</mi></mrow></semantics></math></inline-formula>. Let <i>H</i> be the proper subgroup <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⟨</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>v</mi><mspace width="0.166667em"></mspace><mo>|</mo><mspace width="0.166667em"></mspace><msup><mi>y</mi><mn>3</mn></msup><mo>=</mo><msup><mi>v</mi><mn>3</mn></msup><mo>=</mo><mn>1</mn><mo>⟩</mo></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>PSL</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">Z</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>=</mo><mi>x</mi><mi>y</mi><mi>x</mi></mrow></semantics></math></inline-formula>. For a positive square-free integer <i>n</i>, this article studies the action of <i>H</i> on the subset <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">{</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>a</mi><mo>+</mo><msqrt><mrow><mo>−</mo><mi>n</mi></mrow></msqrt></mrow><mi>c</mi></mfrac></mstyle><mspace width="0.166667em"></mspace><mo stretchy="false">|</mo><mspace width="0.166667em"></mspace><mi>a</mi><mo>,</mo><mi>b</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow><mi>c</mi></mfrac></mstyle><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mo>,</mo><mi>c</mi><mo>≠</mo><mn>0</mn><mo stretchy="false">}</mo></mrow></semantics></math></inline-formula> of the imaginary quadratic number field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></semantics></math></inline-formula> where, in particular, the accurate estimate of the number of orbits arising from this action is given, correcting the estimate given in some of the relevant literature. |
| format | Article |
| id | doaj-art-fb349380360a4414b1ff172ea809d5ec |
| institution | DOAJ |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-fb349380360a4414b1ff172ea809d5ec2025-08-20T03:14:32ZengMDPI AGAxioms2075-16802025-04-0114533510.3390/axioms14050335On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number FieldsAbdulaziz Deajim0Department of Mathematics, King Khalid University, Abha 61413, Saudi ArabiaConsider the modular group <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>PSL</mi></mrow><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">Z</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>⟨</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>y</mi><mspace width="0.166667em"></mspace><mo>|</mo><mspace width="0.166667em"></mspace><msup><mi>x</mi><mn>2</mn></msup><mo>=</mo><msup><mi>y</mi><mn>3</mn></msup><mo>=</mo><mn>1</mn><mo>⟩</mo></mrow></mrow></semantics></math></inline-formula> generated by the transformations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>:</mo><mi>z</mi><mo>↦</mo><mo>−</mo><mn>1</mn><mo>/</mo><mi>z</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>y</mi><mo>:</mo><mi>z</mi><mo>↦</mo><mo>(</mo><mi>z</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>/</mo><mi>z</mi></mrow></semantics></math></inline-formula>. Let <i>H</i> be the proper subgroup <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>⟨</mo><mi>y</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>v</mi><mspace width="0.166667em"></mspace><mo>|</mo><mspace width="0.166667em"></mspace><msup><mi>y</mi><mn>3</mn></msup><mo>=</mo><msup><mi>v</mi><mn>3</mn></msup><mo>=</mo><mn>1</mn><mo>⟩</mo></mrow></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>PSL</mi></mrow><mo>(</mo><mn>2</mn><mo>,</mo><mi mathvariant="double-struck">Z</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>v</mi><mo>=</mo><mi>x</mi><mi>y</mi><mi>x</mi></mrow></semantics></math></inline-formula>. For a positive square-free integer <i>n</i>, this article studies the action of <i>H</i> on the subset <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">{</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><mi>a</mi><mo>+</mo><msqrt><mrow><mo>−</mo><mi>n</mi></mrow></msqrt></mrow><mi>c</mi></mfrac></mstyle><mspace width="0.166667em"></mspace><mo stretchy="false">|</mo><mspace width="0.166667em"></mspace><mi>a</mi><mo>,</mo><mi>b</mi><mo>=</mo><mstyle scriptlevel="0" displaystyle="true"><mfrac><mrow><msup><mi>a</mi><mn>2</mn></msup><mo>+</mo><mi>n</mi></mrow><mi>c</mi></mfrac></mstyle><mo>,</mo><mi>c</mi><mo>∈</mo><mi mathvariant="double-struck">Z</mi><mo>,</mo><mi>c</mi><mo>≠</mo><mn>0</mn><mo stretchy="false">}</mo></mrow></semantics></math></inline-formula> of the imaginary quadratic number field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="double-struck">Q</mi><mo>(</mo><msqrt><mrow><mo>−</mo><mi>n</mi></mrow></msqrt><mo>)</mo></mrow></semantics></math></inline-formula> where, in particular, the accurate estimate of the number of orbits arising from this action is given, correcting the estimate given in some of the relevant literature.https://www.mdpi.com/2075-1680/14/5/335quadratic fieldsmodular groupgroup actionorbits |
| spellingShingle | Abdulaziz Deajim On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields Axioms quadratic fields modular group group action orbits |
| title | On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields |
| title_full | On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields |
| title_fullStr | On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields |
| title_full_unstemmed | On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields |
| title_short | On the Action of a Subgroup of the Modular Group on Imaginary Quadratic Number Fields |
| title_sort | on the action of a subgroup of the modular group on imaginary quadratic number fields |
| topic | quadratic fields modular group group action orbits |
| url | https://www.mdpi.com/2075-1680/14/5/335 |
| work_keys_str_mv | AT abdulazizdeajim ontheactionofasubgroupofthemodulargrouponimaginaryquadraticnumberfields |