Malle proposed a conjecture for counting the number of $G$-extensions $L∕K$ with discriminant bounded above by $X$, denoted $N\left(K,G;X\right)$, where $G$ is a fixed transitive subgroup $G\subset S_n$ and $X$ tends towards infinity. We introduce a refinement of Malle’s conjecture, if $G$ is a group with a nontrivial Galois action then we consider the set of crossed homomorphisms in $Z^1\left(K,G\right)$ (or equivalently $1$-coclasses in $H^1\left(K,G\right)$) with bounded discriminant. This has a natural interpretation given by counting $G$-extensions $F∕L$ for some fixed $L$ and prescribed extension class $F∕L∕K$.

If $T$ is an abelian group with any Galois action, we compute the asymptotic growth rate of this refined counting function for $Z^1\left(K,T\right)$ (and equivalently for $H^1\left(K,T\right)$) and show that it is a natural generalization of Malle’s conjecture. The proof technique is in essence an application of a theorem of Wiles on generalized Selmer groups, and additionally gives the asymptotic main term when restricted to certain local behaviors. As a consequence, whenever the inverse Galois problem is solved for $G\subset S_n$ over $K$ and $G$ has an abelian normal subgroup $T⊴G$ we prove a nontrivial lower bound for $N\left(K,G;X\right)$ given by a nonzero power of $X$ times a power of $logX$. For many groups, including many solvable groups, these are the first known nontrivial lower bounds. These bounds prove Malle’s predicted lower bounds for a large family of groups, and for an infinite subfamily they generalize Klüners’ counterexample to Malle’s conjecture and verify the corrected lower bounds predicted by Türkelli.

Malle 提出了一个猜想$G$- 扩展$\mathrm{大号}∕ķ$上面的判别式为$X$, 表示$ñ\left(ķ,G;X\right)$， 在哪里$G$是一个固定传递子群$G\subset {\mathrm{小号}}_n$和$X$趋于无穷大。我们引入了对 Malle 猜想的改进，如果$G$是一个具有非平凡伽罗瓦动作的群，那么我们考虑交叉同态的集合$Z^1\left(ķ,G\right)$（或等效地$1$-coclasses$H^1\left(ķ,G\right)$) 有界判别式。这有一个通过计数给出的自然解释$G$- 扩展$F∕\mathrm{大号}$对于一些固定的$\mathrm{大号}$和规定的扩展等级$F∕\mathrm{大号}∕ķ$.

如果$吨$是具有任何伽罗瓦作用的阿贝尔群，我们计算这个精化计数函数的渐近增长率$Z^1\left(ķ,吨\right)$（等价于$H^1\left(ķ,吨\right)$) 并证明它是马勒猜想的自然推广。证明技术本质上是 Wiles 定理在广义 Selmer 群上的应用，并且在受限于某些局部行为时还给出了渐近主项。因此，每当求解逆伽罗瓦问题时$G\subset {\mathrm{小号}}_n$超过$ķ$和$G$有一个阿贝尔正规子群$吨⊴G$我们证明了一个非平凡的下界$ñ\left(ķ,G;X\right)$由非零幂给出$X$次幂$日志X$. 对于许多组，包括许多可解组，这些是第一个已知的非平凡下限。这些界限证明了 Malle 对一个大族群的预测下界，对于一个无限的亚族，他们将 Klüners 的反例推广到 Malle 猜想，并验证了 Türkelli 预测的校正下界。