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

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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.