{"title":"关于在单素数处夯实的 PGL2(F7) 和 PSL2(F7) 数域","authors":"Takeshi Ogasawara , George J. Schaeffer","doi":"10.1016/j.jnt.2024.08.006","DOIUrl":null,"url":null,"abstract":"<div><div>We present new examples of <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> and <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> number fields ramified at a single prime. To find these number fields we employ the following methods: (i) Specializing a modification of Malle's <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> polynomial, (ii) Modular method: computation of Katz modular forms of weight one over <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>7</mn></mrow></msub></math></span> with prime level, and (iii) Searching for polynomials with prescribed ramification.</div><div>Method (i) quickly generates many <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> number fields unramified at 7 including those fields ramified at only a single prime. Method (ii) can be used to show the existence of <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> or <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> number fields ramified only at primes that divide the level; we can then use method (iii) to find polynomials for those fields in many cases.</div></div>","PeriodicalId":50110,"journal":{"name":"Journal of Number Theory","volume":"267 ","pages":"Pages 202-220"},"PeriodicalIF":0.6000,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On PGL2(F7) and PSL2(F7) number fields ramified at a single prime\",\"authors\":\"Takeshi Ogasawara , George J. Schaeffer\",\"doi\":\"10.1016/j.jnt.2024.08.006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We present new examples of <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> and <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> number fields ramified at a single prime. To find these number fields we employ the following methods: (i) Specializing a modification of Malle's <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> polynomial, (ii) Modular method: computation of Katz modular forms of weight one over <span><math><msub><mrow><mover><mrow><mi>F</mi></mrow><mo>‾</mo></mover></mrow><mrow><mn>7</mn></mrow></msub></math></span> with prime level, and (iii) Searching for polynomials with prescribed ramification.</div><div>Method (i) quickly generates many <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> number fields unramified at 7 including those fields ramified at only a single prime. Method (ii) can be used to show the existence of <span><math><msub><mrow><mi>PGL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> or <span><math><msub><mrow><mi>PSL</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>7</mn></mrow></msub><mo>)</mo></math></span> number fields ramified only at primes that divide the level; we can then use method (iii) to find polynomials for those fields in many cases.</div></div>\",\"PeriodicalId\":50110,\"journal\":{\"name\":\"Journal of Number Theory\",\"volume\":\"267 \",\"pages\":\"Pages 202-220\"},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2024-09-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Number Theory\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022314X24001951\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Number Theory","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X24001951","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
On PGL2(F7) and PSL2(F7) number fields ramified at a single prime
We present new examples of and number fields ramified at a single prime. To find these number fields we employ the following methods: (i) Specializing a modification of Malle's polynomial, (ii) Modular method: computation of Katz modular forms of weight one over with prime level, and (iii) Searching for polynomials with prescribed ramification.
Method (i) quickly generates many number fields unramified at 7 including those fields ramified at only a single prime. Method (ii) can be used to show the existence of or number fields ramified only at primes that divide the level; we can then use method (iii) to find polynomials for those fields in many cases.
期刊介绍:
The Journal of Number Theory (JNT) features selected research articles that represent the broad spectrum of interest in contemporary number theory and allied areas. A valuable resource for mathematicians, the journal provides an international forum for the publication of original research in this field.
The Journal of Number Theory is encouraging submissions of quality, long articles where most or all of the technical details are included. The journal now considers and welcomes also papers in Computational Number Theory.
Starting in May 2019, JNT will have a new format with 3 sections:
JNT Prime targets (possibly very long with complete proofs) high impact papers. Articles published in this section will be granted 1 year promotional open access.
JNT General Section is for shorter papers. We particularly encourage submission from junior researchers. Every attempt will be made to expedite the review process for such submissions.
Computational JNT . This section aims to provide a forum to disseminate contributions which make significant use of computer calculations to derive novel number theoretic results. There will be an online repository where supplementary codes and data can be stored.