A. Agboola
Let $K$ be a number field and $G$ a finite abelian group. We study the asymptotic behaviour of the number of tamely ramified $G$-extensions of $K$ with ring of integers of fixed realisable class as a Galois module.
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In this article we present a new method to obtain polynomial lower bounds for Galois orbits of torsion points of one dimensional group varieties.
We obtain strong information on the asymptotic behaviour of the counting function for nilpotent Galois extensions with bounded discriminant of arbitrary number fields. This extends previous investigations for the case of abelian groups. In particular, the result confirms a conjecture by the second author on this function for arbitrary groups in the nilpotent case. We further prove compatibility of the conjecture with taking wreath products with the cyclic group of order 2 and give examples in degree up to 8.
Counting wildly ramified quartic extensions with prescribed discriminant and Galois closure group
本文针对2进数域上具有指定判别式赋值与Galois闭包群的完全分歧四次扩张建立计数公式，并据此改进Serre质量公式，为数域算术统计提供工具。
We use Massey products and their relations to unipotent representations to parametrize and find an explicit formula for the number of Galois extensions of a given local field with the prescribed Galois group ${\mathbb U}_4({\mathbb F}_p)$ consisting of unipotent four by four matrices over ${\mathbb F}_p$. Further applications of this method involve the counting of certain Galois extensions with restricted ramifications, and counting the numbers of Galois ${\mathbb U}_4({\mathbb F}_p)$-extensions of some other fields. For each Demushkin pro-$p$-group, we find a very simple version of the condit…
Counting Polynomials via Galois Actions on Root Subsets
本文研究了高度不超过H的单项整系数多项式数量，其伽罗华群在根上的作用同构于给定的置换群(G,Ω)。针对多种群族获得了新的上界。
Let $G$ be a Frobenius group with an abelian Frobenius kernel $F$ and let $k$ be a finite extension of $\mathbb{Q}$. We obtain an upper bound for the number of degree $|F|$ algebraic extensions $K/k$ with Galois group $G$ with the norm of the discriminant $\mathcal{N}_{k/\mathbb{Q}}(d_{K/k})$ bounded above by $X$. We extend this method for any group $G$ that has an abelian normal subgroup. If $G$ has an abelian normal subgroup, then we obtain upper bounds for the number of degree $|G|$ extensions $N/k$ with Galois group $G$ with bounded norm of the discriminant. Malle made a conjecture about w…
We prove an upper bound for the asymptotics of counting functions of number fields with nilpotent Galois groups.
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。
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.
In this paper we give a survey of recent methods for the asymptotic and exact enumeration of number fields with given Galois group of the Galois closure. In particular, the case of fields of degree up to 4 is now almost completely solved, both in theory and in practice. The same methods also allow construction of the corresponding complete tables of number fields with discriminant up to a given bound.
We describe a method for counting the number of extensions of $\mathbb{Q}_p$ with a given Galois group $G$, founded upon the description of the absolute Galois group of $\mathbb{Q}_p$ due to Jannsen and Wingberg. Because this description is only known for odd $p$, our results do not apply to $\mathbb{Q}_2$. We report on the results of counting such extensions for $G$ of order up to $2000$ (except those divisible by $512$), for $p=3,5,7,11,13$. In particular, we highlight a relatively short list of minimal $G$ that do not arise as Galois groups. Motivated by this list, we prove two theorems abo…
Enumerative Galois theory for number fields
本文通过确定方法为偶数伽罗瓦群提供上界，解决了数域计数难题，拓展了算术统计领域。
For an odd prime number $p$, we consider degree $p$ extensions $L/K$ of $p$-adic fields with normal closure $\widetilde{L}$ such that the Galois group of $\widetilde{L}/K$ is the dihedral group of order $2p$. We shall prove a complete characterization of the freeness of the ring of integers $\mathcal{O}_L$ over its associated order $\mathfrak{A}_{L/K}$ in the unique Hopf-Galois structure on $L/K$, which is analogous to the one already known for cyclic degree $p$ extensions of $p$-adic fields. We shall derive positive and negative results on criteria for the freeness of $\mathcal{O}_L$ as $\mat…
Let $F$ be a number field with ring of integers $O_F$ and let $G$ be a finite group. We describe an approach to the study of the set of realisable classes in the locally free class group $Cl(O_FG)$ of $O_FG$ that involves applying the work of the second-named author in the context of relative algebraic $K$ theory. When $G$ is of odd order, we show (subject to certain conditions) that the set of realisable classes is a subgroup of $Cl(O_FG)$. This may be viewed as being a partial analogue of a classical theorem of Shafarevich on the inverse Galois problem for soluble groups in the setting of Ga…
Bhargava proved a formula for counting, with certain weights, degree n etale extensions of a local field, or equivalently, local Galois representations to S_n. This formula is motivation for his conjectures about the density of discriminants of S_n-number fields. We prove there are analogous ``mass formulas'' that count local Galois representations to any group that can be formed from symmetric groups by wreath products and cross products, corresponding to counting towers and direct sums of etale extensions. We obtain as a corollary that the above mentioned groups have rational character table…
We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by discriminant.
Let $K$ be a complete discrete valuation field of characteristic zero with residue field $k_K$ of characteristic $p>0$. Let $L/K$ be a finite Galois extension with Galois group $G=\Gal(L/K)$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. Let $\mathbb{W}_n(\cdot)$ denote the ring of $p$-typical Witt vectors of length $n$. Hesselholt conjectured that the pro-abelian group $\{H^1(G,\mathbb{W}_n(\mathcal{O}_L))\}_{n\geq 1}$ is isomorphic to zero. Hogadi and Pisolkar have recently provided a proof of this conjecture. In this paper, we provide an alternative proof o…
Counting $D_4$-field extensions by multi-invariants
本文按 Gundlach 引入的多变元计数具有固定 Galois 群 $D_4$ 的 Galois 扩张，验证了 Malle 猜想的渐近行为，并将主项常数与 Loughran 和 Santens 的近期预测进行了比较。
A Galois scaffold, in a Galois extension of local fields with perfect residue fields, is an adaptation of the normal basis to the valuation of the extension field, and thus can be applied to answer questions of Galois module structure. Here we give a sufficient condition for a Galois scaffold to exist in fully ramified Galois extensions of degree $p^2$ of characteristic $p$ local fields. This condition becomes necessary when we restrict to $p=3$. For extensions $L/K$ of degree $p^2$ that satisfy this condition, we determine the Galois module structure of the ring of integers by finding necessa…
Let $N(5,D_5,X)$ be the number of quintic number fields whose Galois closure has Galois group $D_5$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect that $N(5,D_5,X) \sim C X^{1/2}$ for some constant $C$. The best known upper bound is $N(5,D_5,X)\ll X^{3/4 + ε}$, and we show this could be improved by counting points on a certain variety defined by a norm equation; computer calculations give strong evidence that this number is $\ll X^{2/3}$. Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for $A_4$ q…
Let $S$ be a Shimura variety and let $h$ be a Weil height function on $S$. We conjecture that the heights of special points in $S$ are discriminant negligible. Assuming this conjecture to be true, we prove that the sizes of the Galois orbits of special points grow as a fixed power of their discriminant (an invariant we will define in the text). In the case of Shimura varieties of abelian type, the height bound holds by the recently proved averaged Colmez formula, and our theorem gives a new proof of Tsimerman's Galois lower bound in this case. The main novelty is that our approach avoids the u…
Galois module structure of algebraic integers of cyclic cubic fields
研究通过根式通用循环三次多项式确定三次循环域中整数环的伽罗瓦模结构，并显式给出其生成元。
Mass formulas for local Galois representations and quotient singularities I: a comparison of counting functions
本文研究局部伽罗瓦表示中 Artin 导子与野 Deligne-Mumford 栈 motivic 积分权的关系，并借此证明 McKay 对应的一种形式，将 Bhargava 质量公式与点的 Hilbert 概型联系起来。
Let L/K be an extension of number fields where L/\Q is abelian. We define such an extension to be Leopoldt if the ring of integers O_L of L is free over the associated order A_L/K. Furthermore we define an abelian number field K to be Leopoldt if every finite extension L/K with L/Q abelian is Leopoldt in the sense above. Previous results of Leopoldt, Chan & Lim, Bley, and Byott & Lettl culminate in the proof that the n-th cyclotomic field Q^(n) is Leopoldt for every n. In this paper, we generalize this result by giving more examples of Leopoldt extensions and fields, along with explicit genera…
Galois groups of random integer matrices
本文研究特征多项式 Galois 群非全对称群的整数矩阵数量，使用大筛法及 Bhargava 证明 van der Waerden 猜想的 Fourier 分析与几何筛法改进了部分界。
For positive integers $n$, the truncated binomial expansions of $(1+x)^n$ which consist of all the terms of degree $\le r$ where $1 \le r \le n-2$ appear always to be irreducible. For fixed $r$ and $n$ sufficiently large, this is known to be the case. We show here that for a fixed positive integer $r \ne 6$ and $n$ sufficiently large, the Galois group of such a polynomial over the rationals is the symmetric group $S_{r}$. For $r = 6$, we show the number of exceptional $n \le N$ for which the Galois group of this polynomial is not $S_r$ is at most $O(\log N)$.
On counting totally imaginary number fields
该研究定义了CM型与TR型全虚数域，得出四次全虚数域中两类的占比，还依据伽罗瓦群及复共轭位置完成了六次全虚数域的类型分类。
Let $p>5$ be a prime integer and $K/\mathbb{Q}_p$ a finite ramified extension with ring of integers $\mathcal{O}$ and uniformizer $π$. Let $n>1$ be a positive integer and $ρ_n:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O}/π^n)$ be a continuous Galois representation. In this article we prove that under some technical hypotheses the representation $ρ_n$ can be lifted to a representation $ρ:G_\mathbb{Q} \to \text{GL}_2(\mathcal{O})$. Furthermore, we can pick the lift restriction to inertia at any finite set of primes (at the cost of allowing some extra ramification) and get a deformation problem whose…
Let $K$ be a complete discrete valued field of characteristic zero with residue field $k_K$ of characteristic $p > 0$. Let $L/K$ be a finite Galois extension with the Galois group $G$ and suppose that the induced extension of residue fields $k_L/k_K$ is separable. In his paper, Hesselholt conjectured that $H^1(G,W(\sO_L))$ is zero, where $\sO_L$ is the ring of integers of $L$ and $W(\sO_L)$ is the Witt ring of $\sO_L$ w.r.t. the prime $p$. He partially proved this conjecture for a large class of extensions. In this paper, we prove Hesselholt's conjecture for all Galois extensions.
Let $k$ be a number field. For $\mathcal{H}\rightarrow \infty$, we give an asymptotic formula for the number of algebraic integers of absolute Weil height bounded by $\mathcal{H}$ and fixed degree over $k$.
On the influence of the Galois group structure on the Chebyshev bias in number fields
本文研究伽罗瓦数域扩张中弗罗贝尼乌斯元分布的切比雪夫偏差，证明极端偏差与素理想计数均等的二分性，引入相关群论性质，分别刻画阿贝尔扩张和$p$-群情形下的极端偏差条件并得到Linnik型有效结论。
We show that there are at most $O_{n,ε}(H^{n-2+\sqrt{2}+ε})$ monic integer polynomials of degree $n$ having height at most $H$ and Galois group different from the full symmetric group $S_n$, improving on the previous 1973 world record $O_{n}(H^{n-1/2}\log H)$.
Let $p$ be an odd prime. For field extensions $L/\mathbb{Q}_p$ with Galois group isomorphic to the dihedral group $D_{2p}$ of order $2p$, we consider the problem of computing a basis of the associated order in each Hopf Galois structure and the module structure of the ring of integers $\mathcal{O}_L$. We solve the case in which $L/\mathbb{Q}_p$ is not totally ramified and present a practical method which provides a complete answer for the cases $p=3$ and $p=5$. We see that within this family of dihedral extensions, the ring of integers is always free over the associated orders in the different…
We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $φ(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.
In this paper, we construct, for some $2$-groups $G$, explicit Galois extensions $E/\mathbb{Q}(T)$ of group $G$ with $E\cap\overline{\mathbb{Q}}=\mathbb{Q}$. We also provide explicit arithmetic progressions of integers $t_0$ such that the specialization $E_{t_0}/\mathbb{Q}$ of $E/\mathbb{Q}(T)$ at $t_0$ has Galois group $G$.
Let $S= \{ a_{1}, a_{2}, \dots, a_{n} \}$ be a finite set of non-zero integers. In \cite{KBAM21}, Karthick Babu and Anirban Mukhopadhyay calculated the explicit structure of the Galois group of multi-quadratic field $\mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots, \sqrt{a_{n}})$ over $\mathbb{Q}$. For a positive integer $d \geq 3$, $ζ_{d}$ denotes the primitive $d$-th root of unity. In this paper, we calculate the explicit structure of the Galois group of $\mathbb{Q}(\sqrt{a_{1}}, \sqrt{a_{2}}, \dots, \sqrt{a_{n}}, ζ_{d})$ over $\mathbb{Q}$ in terms of its action on $ζ_{d}$ and $\sqrt{a_{i}}$ fo…
Counting quaternion extensions by multi-heights
该研究证明了Gundlach关于Q₈群的伽罗瓦扩张数量猜想的一个情形，得到狂分歧位加局部条件的结果，并应用得出有界分歧素数乘积的Q₈扩张数量的渐近性。
Galois groups of random integer polynomials and van der Waerden's Conjecture
本文证明了范·德·瓦尔登猜想对所有次数 n 成立，即伽罗瓦群非全对称群的随机整数多项式数量上界为 O(H^(n-1))。
Let $F \in \mathbf{Z}[\boldsymbol{x}]$ be a diagonal, non-singular quadratic form in $4$ variables. Let $λ(n)$ be the normalised Fourier coefficients of a holomorphic Hecke form of full level. We give an upper bound for the problem of counting integer zeros of $F$ with $|\boldsymbol{x}| \leq X$, weighted by $λ(x_1)$.
Counting $2\times2$ integer matrices with fixed trace and determinant
该研究利用二次根的Weyl和的和的界，对带固定特征多项式的2×2整数矩阵数目做带光滑权的计数，给出主项与误差项。
Let $G$ be a subgroup of the symmetric group $S_n$, and let $δ_G=|S_n/G|^{-1}$ where $|S_n/G|$ is the index of $G$ in $S_n$. Then there are at most $O_{n, ε}(H^{n-1+δ_G+ε})$ monic integer polynomials of degree $n$ having Galois group $G$ and height not exceeding $H$, so there are only `few' polynomials having `small' Galois group.
We count various classes of algebraic integers of fixed degree by their largest absolute value. The classes of integers considered include all algebraic integers, Perron numbers, totally real integers, and totally complex integers. We give qualitative and quantitative results concerning the distribution of Perron numbers, answering in part a question of W. Thurston.
We compare two approaches to the study of Galois module structures: on the one hand factor equivalence, a technique that has been used by Fröhlich and others to investigate the Galois module structure of rings of integers of number fields and of their unit groups, and on the other hand regulator constants, a set of invariants attached to integral group representations by Dokchitser and Dokchitser, and used by the author, among others, to study Galois module structures. We show that the two approaches are in fact closely related, and interpret results arising from these two approaches in terms …
Local Galois representations associated to additive polynomials
本文针对加法多项式构造了非阿基米德局部域Weil群的不可约光滑表示，研究其不变量并给出原始性的充要条件。
Tame torsion and the tame inverse Galois problem
本文研究了 tame torsion 和 tame 逆 Galois 问题，通过分析双曲 Mumford 曲线的周期矩阵，证明了存在特定亏格的曲线满足 tame 条件，并应用于有限域上的 symplectic 矩阵群。
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…
An improved error term for counting $D_4$-quartic fields
本文改进了Cohen等人关于D₄-四次域计数公式的误差项，将其从原有结果提升至O(X^{5/8+ε})，并将类似结论推广到任意基域上的四次二面体扩张。
Using the action of the Galois group of a normal extension of number fields, we generalize and symmetrize various fundamental statements in algebra and algebraic number theory concerning splitting types of prime ideals, factorization types of polynomials modulo primes, and cycle types of the Galois groups of polynomials. One remarkable example is the removal of all artificial constraints from the Kummer-Dedekind Theorem that relates splitting and factorization patterns. Finally, we present an elementary proof that the discriminant of the splitting field of a monic irreducible polynomial with i…
Counting Algebraic Integers of Bounded Height in Cyclotomic Fields
本文针对素数幂阶分圆域，给出绝对高度不超过B的代数整数和单位个数的渐近上界。
A polynomial over a ring is called decomposable if it is a composition of two nonlinear polynomials. In this paper, we obtain sharp lower and upper bounds for the number of decomposable polynomials with integer coefficients of fixed degree and bounded height. Moreover, we obtain asymptotic formulas for the number of decomposable monic polynomials of even degree. For example, the number of monic sextic integer polynomials which are decomposable and of height at most $H$ is asymptotic to $(16ζ(3)-5/4)H^3$ as $H \to \infty$.
Fix a positive integer $g$ and rational prime $p$. We prove the existence of a genus $g$ curve $C/\mathbb{Q}$ such that the mod $p$ representation of its Jacobian is tame by imposing conditions on the endomorphism ring. As an application, we consider the tame inverse Galois problem and are able to realise general symplectic groups as Galois groups of tame extensions of $\mathbb{Q}$.
We create plots of algebraic integers in the complex plane, exploring the effect of sizing the integers according to various arithmetic invariants. We focus on Galois theoretic invariants, in particular creating plots which emphasize algebraic integers whose Galois group is not the full symmetric group--these integers we call rigid. We then give some analysis of the resulting images, suggesting avenues for future research about the geometry of so-called rigid algebraic integers.
We prove many new cases of the Inverse Galois Problem for those simple groups arising from orthogonal groups over finite fields. For example, we show that the finite simple groups Omega_{2n+1}(p) and POmega_{4n}^+(p) both occur as the Galois group of a Galois extension of the rationals for all integers n>1 and all primes p>3. We obtain our representations by studying families of twists of elliptic curves and using some known cases of the Birch and Swinnerton-Dyer conjecture along with a big monodromy result of Hall.
Counting biquadratic number fields with quaternionic and dihedral extensions
本文研究了可嵌入四元数或二面体扩张的双二次数域的计数问题，通过用 Hilbert 符号表达逆 Galois 问题的可解性，并应用 Heath-Brown 的方法界定关联二次特征标的和，建立了有界判别式下此类数域数量的渐近公式。
Given a number field $k$ and a quadratic extension $K_2$, we give an explicit asymptotic formula for the number of isomorphism classes of cubic extensions of $k$ whose Galois closure contains $K_2$ as quadratic subextension, ordered by the norm of their relative discriminant ideal. The main tool is Kummer theory. We also study in detail the error term of the asymptotics and show that it is $O(X^α)$, for an explicit $α<1$.
Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to a given $w>0$ is determined. The counting functions developed are framed in terms of convolution exponentials, and are closely related to the Riemann zeta function. At the conclusion, several estimates are derived, with special emphasis on the case of sums of square roots, i.e. $j=1$, $k=2$.
Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which a…
Let $r\ge k\ge 2$ be fixed positive integers. Let $\varrho_{r,k}$ denote the characteristic function of the set of $r$-tuples of positive integers with $k$-wise relatively prime components, that is any $k$ of them are relatively prime. We use the convolution method to establish an asymptotic formula for the sum $\sum_{n_1,\ldots,n_r\le x} \varrho_{r,k}(n_1,\ldots,n_r)$ by elementary arguments. Our result improves the error term obtained by J. Hu (2013).
Let $K$ be a fixed number field, assumed to be Galois over $\mathbb Q$. Let $r$ and $f$ be fixed integers with $f$ positive. Given an elliptic curve $E$, defined over $K$, we consider the problem of counting the number of degree $f$ prime ideals of $K$ with trace of Frobenius equal to $r$. Except in the case $f=2$, we show that "on average," the number of such prime ideals with norm less than or equal to $x$ satisfies an asymptotic identity that is in accordance with standard heuristics. This work is related to the classical Lang-Trotter conjecture and extends the work of several authors.
In this article, we prove that for a convergent sequence of residually absolutely irreducible representations of the absolute Galois group of a number field $F$ with coefficients in a domain finite over a power series ring over a $p$-adic integer ring, the set of places of $F$ where some of the representations ramifies has density zero. Using this, we extend a result of Das--Rajan to such convergent sequences. We also establish a strong multiplicity one theorem for big Galois representations.
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