Asymptotics of wildly ramified Galois extensions of local or global function fields

局部或全局函数域的疯狂分支伽罗瓦扩展的渐近

• 批准号: 171354361
• 负责人: Professor Dr. Jürgen Klüners
• 金额: --
• 依托单位: Fachgebiet Computeralgebra und Zahlentheorie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/171354361?language=en
• 关键词: Asymptotics; wildly; ramified; Galois; extensions

The discipline of counting Galois extensions of global fields has been very active in the past years. Gunter Malle conjectured a precise asymptotic behavior of the cardinality of extensions with given Galois group for large discriminants. A recent counterexample draws attention to the case in positive characteristic, in which the group order is divisible by the characteristic. These cases were mostly ignored so far and regard function fields over finite fields of characteristic p and their wildly ramified extensions. The analysis of the counterexamples shows that a corresponding question on local function fields require investigation as well. By Hasse’s Einseinheitensatz we get infinitely many extensions with given Abelian Galois group of order divisible by p and again we may ask for their distribution for large discriminants. In the proposed project we wish to understand the distribution of wildly ramified extensions of local or global function fields and derive a new conjecture on their asymptotic behavior to close the gap in Malle’s conjecture. Beside theoretical aspects this involves extensive computer algebraic experiments, which should initiate and endorse the new conjecture as well as provide a new database on local function fields.

在过去的几年中，计算全球领域的Galois扩展的纪律一直非常活跃。冈特·马尔（Gunter Malle）猜想，具有给定的galois组的扩展基因基础性的精确不对称行为，用于大型判别因子。最近的反例以积极的特征提请注意该案例，其中小组顺序被特征排除。到目前为止，这些情况大多被忽略，并在特征P的有限领域及其大受损扩展上考虑了功能字段。对反见的分析表明，关于本地功能字段的相应问题也需要研究。通过Hasse的Einseinheitensatz，我们获得了无限的许多扩展，而给定的Abelian Galois组可通过P区分，我们可能会再次要求其分布大量的判别物。在拟议的项目中，我们希望了解当地或全球功能领域的大受损扩展的分布，并就其不对称行为提出新的猜想，以缩小Malle协议中的差距。除了理论方面，这还涉及广泛的计算机代数实验，该实验应启动和认可新的猜想，并在本地函数字段上提供新的数据库。