Juergen Klueners
In this note we give a counter example to a conjecture of Malle which predicts the asymptotic behaviour of the counting functions for field extensions with given Galois group and bounded discriminant.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We prove Malle's conjecture for nonic Heisenberg extensions over $\mathbb{Q}$. Our main algebraic result shows that the number of nonic Heisenberg extensions over $\mathbb{Q}$ with discriminant bounded by $X$ is given by a character sum. We then extract the main term from this sum by exploiting oscillation of characters.
We show that for a large class of finite groups G, the number of Galois extensions E/Q of group G and discriminant $|d_E|\leq y$ grows like a power of $y$ (for some specified exponent). The groups G are the regular Galois groups over Q and the extensions E/Q that we count are obtained by specialization from a given regular Galois extension F/Q(T). The extensions E/Q can further be prescribed any unramified local behavior at each suitably large prime $p\leq \log (y)/δ$ for some $δ\geq 1$. This result is a step toward the Malle conjecture on the number of Galois extensions of given group and bou…
Counterexamples for Türkelli's Modification on Malle's Conjecture
本文给出了Türkelli对Malle猜想修改的反例，证明函数域与数域的$b$常数可不同，并基于Klüners反例扩展了Malle猜想，提出了数域常数$b$的新猜想值。
A power-saving error term in counting $C_2 \wr H$ extensions of an arbitrary base field parametrized by discriminants
本文提出了一种方法，用于计算任意基域由判别式参数化的 $C_2 \wr H$ 扩张的显式主项和节能误差项，并放宽了对置换群 $H$ 的假设。
This paper was motivated by a recent paper by Krumm and Pollack investigating modulo-$p$ behaviour of quadratic twists with rational points of a given hyperelliptic curve, conditional on the abc-conjecture. We extend those results to twisted Galois covers with arbitrary Galois groups. The main point of this generalization is to interpret those results as statements about the sets of specializations of a given Galois cover under restrictions on the discriminant. In particular, we make a connection with existing heuristics about the distribution of discriminants of Galois extensions such as the …
Malle's Conjecture for Galois octic fields over $\mathbb Q$
本文计算了按绝对判别式排序时 Galois 群同构于 $D_4$ 的八次数域的渐近数量，验证了 Malle 强猜想，并证明其比例常数满足 Malle--Bhargava 原理。
A refined Malle conjecture for Heisenberg groups
研究关于\(\mathbb{Q}\)的伽罗瓦\(\mathcal{H}\)-扩张的马勒猜想，基于特定猜想给出首项常数的明确预测，其为两个欧拉乘积之和，还给出计算相关首项常数的有效算法。
On rational points on classifying stacks and Malle's conjecture
本文为综述性研究，对比了计数有界判别式数域的Malle猜想与两类关于分类栈有界高度点计数的最新猜想，并借助特定分类栈示例对比较结果进行阐释。
Malle proposed a conjecture for counting the number of $G$-extensions $L/K$ with discriminant bounded above by $X$, denoted $N(K,G;X)$, 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 $Z^1(K,G)$ (or equivalently $1$-coclasses in $H^1(K,G)$) 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 $T$ i…
Malle's conjecture and Brauer groups of stacks
本文提出Malle猜想领头常数的栈版本猜想，并引入代数栈的非分歧Brauer群新概念，为研究数域判别式分布提供新框架。
The leading constant in Malle's conjecture
本文概述了作者关于有界判别式的数域Malle猜想中主常数的最新猜想，通过将Manin猜想的思想应用于分类叠，并提出了关于多高度和Bhargava启发式的新猜想。
Let Z(X) be the number of degree-d extensions of F_q(t) with bounded discriminant and some specified Galois group. The problem of computing 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'(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, a…
For a fixed finite solvable group $G$ and number field $K$, we prove an upper bound for the number of $G$-extensions $L/K$ with restricted local behavior (at infinitely many places) and ${\rm inv}(L/K)<X$ for a general invariant $"{\rm inv}"$. When the invariant is given by the discriminant for a transitive embedding of a nilpotent group $G\subset S_n$, this realizes the upper bound given in the weak form of Malle's conjecture. For other solvable groups, the upper bound depends on the size of torsion of the class group of number fields with fixed degree. In particular, the bounds we prove real…
Let $G$ be a wreath product of the form $C_2 \wr H$, where $C_2$ is the cyclic group of order 2. Under mild conditions for $H$ we determine the asymptotic behavior of the counting functions for number fields $K/k$ with Galois group $G$ and bounded discriminant. Those counting functions grow linearly with the norm of the discriminant and this result coincides with a conjecture of Malle. Up to a constant factor these groups have the same asymptotic behavior as the conjectured one for symmetric groups.
We propose a framework to prove Malle's conjecture for the compositum of two number fields based on proven results of Malle's conjecture and good uniformity estimates. Using this method we can prove Malle's conjecture for $S_n\times A$ over any number field $k$ for $n=3$ with $A$ an abelian group of order relatively prime to 2, for $n= 4$ with $A$ an abelian group of order relatively prime to 6 and for $n=5$ with $A$ an abelian group of order relatively prime to 30. As a consequence, we prove that Malle's conjecture is true for $C_3\wr C_2$ in its $S_9$ representation, whereas its $S_6$ repres…
Lower bounds for counting $A_4$-quartic fields
针对 Malle 猜想，本文给出了伽罗瓦群为 $A_4$ 的四次域计数下界，与猜想匹配。
We contribute to the Malle conjecture on the number N (K, G, y) of finite Galois extensions E of some number field K of finite group G and of discriminant of norm |N K/Q (d E)| $\le$ y. We prove the lower bound part of the conjecture for every group G and every number field K containing a certain number field K 0 depending on G : N (K, G, y) $\ge$ y $α$(G) for y 1 and some specific exponent $α$(G) depending on G. To achieve this goal, we start from a regular Galois extension F/K(T) that we specialize. We prove a strong version of the Hilbert Irreducibility Theorem which counts the number of sp…
By a heuristic argument, we relate two conjectures. One is a version of Manin's conjecture about the distribution of rational points on a Fano variety. We concern specific singular Fano varieties, namely quotients of projective spaces by finite group actions, and their singularities play a key role. The other conjecture is a generalization of Malle's conjecture about the distribution of extensions of a number field. Main tools are several Dirichlet series and previously obtained techniques, especially the untwisting, for the counterpart over a local field.
Inductive methods for counting number fields
提出按判别式渐近计数数域扩张的新方法，利用正规子群扩张的渐近计数与界，证明了 Malle's Conjecture 的多种新情形及反例。
We give a product expansion for the Drinfeld discriminant function in arbitrary rank $r$, which generalizes the formula obtained by Gekeler for the rank 2 Drinfeld discriminant function. This enables one to compute the Fourier expansion of this function much more efficiently. The formula in this article uses an $r-1$-dimensional parameter and as such provides a nice counterpoint to the formula previously obtained by Hamahata, which is written in terms of several 1-dimensional parameters.
We show how to construct counter-examples to the Hasse principle over the field of rational numbers on Atkin-Lehner quotients of Shimura curves and on twisted forms of Shimura curves by Atkin-Lehner involutions. A particular example is the quotient of the Shimura curve X attached to the indefinite rational quaternion algebra of discriminant 23*107 by the Atkin-Lehner involution w107. The quadratic twist of X by Q(sqrt{-23}) with respect to this involution is also a counter-example to the Hasse principle over Q.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。