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

在过去的几年里，计算全局域的伽罗瓦扩张的学科一直非常活跃。Gunter Malle证明了在给定Galois群下，对于大判别式，扩张的基数的一个精确的渐近性质.最近的一个反例提请注意的情况下，积极的特点，其中集团秩序是整除的特点。这些情况到目前为止大多被忽略，并考虑特征p的有限域上的函数域及其广泛分歧的扩张。反例的分析表明，局部函数域上的一个相应问题也需要研究。由哈塞的Einseinheitenbrand我们得到无限多的扩展与给定的阿贝尔伽罗瓦群的顺序可分的p和再次我们可能会要求他们的分布大的判别式。在建议的项目中，我们希望了解分布的广泛分歧的扩展的本地或全球的功能领域，并得出一个新的猜想，其渐近行为关闭差距在Malle的猜想。除了理论方面，这涉及广泛的计算机代数实验，这应该启动和认可的新猜想，以及提供一个新的数据库的地方功能领域。

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.