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

在过去的几年中，计算全局域的伽罗瓦扩展的学科非常活跃。Gunter Malle推测了给定伽罗瓦群的大判别扩展基数的精确渐近性。最近的一个反例引起了人们对正特征的注意，在这种情况下，群的顺序可以被特征整除。这些情况目前大多被忽略，只考虑特征p有限域上的函数域及其广泛的分支扩展。反例分析表明，局部函数域的相应问题也需要进一步研究。根据Hasse的爱因斯坦定理，我们得到了给定的可被p整除的阶阿贝尔伽罗瓦群的无限多扩展，我们可以再一次要求它们在大判据下的分布。在本项目中，我们希望了解局部或全局函数域的广泛分支扩展的分布，并推导出一个关于它们的渐近行为的新猜想，以弥补Malle猜想的空白。除了理论方面，这涉及到广泛的计算机代数实验，它应该启动和认可新的猜想，并提供一个新的数据库，局部函数域。

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.