Connected Components of Hurwitz Schemes and Malle's Conjecture
Connected Components of Hurwitz Schemes and Malle's Conjecture
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赫尔维茨方案和马勒猜想的连通部分
DOI：
10.1016/j.jnt.2015.03.005
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
Seyfi Turkelli
中科院分区：
文献类型：
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作者：
Seyfi Turkelli
令 Z (X) 为 F q (t) 具有某个指定伽罗瓦群且以 X 为界的判别式的 n 度扩展数。计算 Z (X) 渐近线的问题可以与计算某些 Hurwitz 空间上的 F q 有理点的问题相关。 Ellenberg 和 Venkatesh 使用这个想法开发了 Z 0 (X) 渐近行为的启发式，即几何连通扩展的数量，并表明这与 Malle 对函数域的猜想一致。我们扩展了 Ellenberg-Venkatesh 的论证，以处理 P 1 的覆盖可能不具有几何连接的更复杂的情况，并表明由此产生的启发式建议对 Malle 猜想进行自然修改，避免了 Klüners 造成的原始猜想的反例。
Let Z (X) be the number of degree-n extensions of F q (t) with some specified Galois group and with discriminant bounded by X. The problem of computing the asymptotics for Z (X) can be related to a problem of counting F q-rational points on certain Hurwitz spaces. Ellenberg and Venkatesh used this idea to develop a heuristic for the asymptotic behavior of Z 0 (X), the number of–geometrically connected–extensions, and showed that this agrees with the conjectures of Malle for function fields. We extend Ellenberg–Venkatesh's argument to handle the more complicated case of covers of P 1 which may not be geometrically connected, and show that the resulting heuristic suggests a natural modification to Malle's conjecture which avoids the counterexamples, due to Klüners, to the original conjecture.