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.

计算全局域的伽罗瓦扩展的学科在过去几年中非常活跃，最近的一个反例引起了人们对大判别式的扩展基数的精确渐近行为的关注。其中群阶可被特征整除。到目前为止，这些情况大多被忽略，并考虑特征 p 的有限域上的函数域及其广泛的扩展。反例表明，关于局部函数域的相应问题也需要研究，通过给定的可被 p 整除的阿贝尔伽罗瓦群，我们可以得到无限多个扩展，并且我们可以再次要求它们对于大判别式的分布。希望了解局部或全局函数域的广泛扩展的分布，并对其渐近行为得出新的猜想，以缩小 Malle 的差距除了理论方面之外，这还涉及广泛的计算机代数实验，这些实验应该发起并认可新的猜想，并提供关于局部函数域的新数据库。