On PGL2(F7) and PSL2(F7) number fields ramified at a single prime

IF 0.6 3区数学 Q3 MATHEMATICS

Journal of Number Theory Pub Date : 2024-09-23 DOI:10.1016/j.jnt.2024.08.006

Takeshi Ogasawara , George J. Schaeffer

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引用次数: 0

Abstract

We present new examples of

{PGL}_{2} (F_{7})

and

{PSL}_{2} (F_{7})

number fields ramified at a single prime. To find these number fields we employ the following methods: (i) Specializing a modification of Malle's

{PGL}_{2} (F_{7})

polynomial, (ii) Modular method: computation of Katz modular forms of weight one over

{\overline{F}}_{7}

with prime level, and (iii) Searching for polynomials with prescribed ramification.

Method (i) quickly generates many

{PGL}_{2} (F_{7})

number fields unramified at 7 including those fields ramified at only a single prime. Method (ii) can be used to show the existence of

{PGL}_{2} (F_{7})

{PSL}_{2} (F_{7})

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.

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关于在单素数处夯实的 PGL2(F7) 和 PSL2(F7) 数域

我们提出了在一个素数上夯实的 PGL2(F7) 和 PSL2(F7) 数域的新例子。为了找到这些数域，我们采用了以下方法：(i) 马勒的 PGL2(F7) 多项式的特殊化修正；(ii) 模块法：计算 F‾7 上权重为一的卡茨模块形式的素级；(iii) 寻找具有规定斜率的多项式。方法(ii)可以用来证明只在平分的素数上有斜线的 PGL2(F7) 或 PSL2(F7) 数域的存在；然后我们可以用方法(iii)在许多情况下为这些域找到多项式。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Number Theory 数学-数学

CiteScore

1.30

自引率

14.30%

发文量

122

审稿时长

16 weeks

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.