Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants
Mass Formulae for Extensions of Local Fields, and Conjectures on the Density of Number Field Discriminants
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局部域扩张的质量公式以及数域判别式密度的猜想
DOI：
10.1093/imrn/rnm052
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发表时间：
2007
影响因子：
1
通讯作者：
M. Bhargava
中科院分区：
文献类型：
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作者：
M. Bhargava
利用Serre的质量公式[15]，我们推导了一个质量公式，它计算了局部域F具有给定度n的所有（同构类）的<s:1>代数扩展。在此过程中，我们还证明了一系列用于计算具有某些性质的局部域F的<s:1>代数扩展的质量公式，例如选择的素数分裂或分支行为。然后，我们利用这些质量公式，构造了一个启发性的方法，预测了具有伽罗瓦群和全对称群Snover - π的定阶n有界判别数域的渐近数目。对那些在有限多处具有特定局部行为的数场的渐近密度作了类似的预测。所有这些预测都与已知的结果完全一致，最高可达5度。
Using Serre's mass formula [15] for totally ramified extensions, we derive a mass formula that counts all (isomorphism classes of) étale algebra extensions of a local field F having a given degree n. Along the way, we also prove a series of mass formulae for counting étale extensions of a local field F having certain properties, such as a chosen prime splitting or ramification behavior. We then use these mass formulae to formulate a heuristic that predicts the asymptotic number of number fields, of fixed degree n and bounded discriminant, whose Galois closures have Galois group the full symmetric group Snover ℚ. Analogous predictions are made for the asymptotic density of those number fields having specified local behaviors at finitely many places. All these predictions are in full agreement with the known results in degrees up to five.