Euclidean algorithm in Galois Quartic Fields

无限多对不同四次Galois CM域具有相同判别式和调节数

An infinite family of pairs of distinct quartic Galois CM-fields with the same discriminant and regulator

本文构造了无限多个具有相同判别式和调节数的虚二元四次域和虚循环四次域的配对，并给出了具有相同判别式、调节数和类数的示例。

The shapes of Galois quartic fields

We determine the shapes of all degree $4$ number fields that are Galois. These lie in four infinite families depending on the Galois group and the tame versus wild ramification of the field. In the $V_4$ case, each family is a two-dimensional space of orthorhombic lattices and we show that the shapes are equidistributed, in a regularized sense, in these spaces as the discriminant goes to infinity (with respect to natural measures). We also show that the shape is a complete invariant in some natural families of $V_4$-quartic fields. For $C_4$-quartic fields, each family is a one-dimensional spa…

Binary quartic forms with bounded invariants and small Galois groups

In this paper, we consider integral and irreducible binary quartic forms whose Galois group is isomorphic to a subgroup of the dihedral group of order eight. We first show that the set of all such forms is a union of families indexed by integral binary quadratic forms $f(x,y)$ of non-zero discriminant. Then, we shall enumerate the $\operatorname{GL}_2(\mathbb{Z})$-equivalence classes of all such forms associated to a fixed $f(x,y)$.

非伽罗瓦四次CM域上\(\mathrm{GL}_2\)的扭曲西格尔 - 韦伊公式

Twisted Siegel-Weil formulas for $\mathrm{GL}_2$ over non-Galois quartic CM fields

研究非伽罗瓦四次CM域上\(\mathrm{GL}_2\)的扭曲西格尔 - 韦伊公式，通过联系扭曲西塔积分与赫克积分的土居 - 永野提升，证明雅克比 - 朗兰兹对应基变换是同构，还得出博切尔兹形式扭曲CM值相关结论。

有理数实四次 Galois 扩张的 Bertrand 与 Rodriguez Villegas 猜想

Bertrand's and Rodriguez Villegas' Conjecture for real quartic Galois extensions of the rationals

本文研究有理数全实四次扩张单位外平方中元素的 1-范数下界问题，并在 Galois 扩张条件下将 Costa–Friedman 的 0.802 界改进为 1.134。

单生四次多项式及其 Galois 群

Monogenic Quartic Polynomials and Their Galois Groups

本文借助 Kappe–Warren 的四次 Galois 群分类，为除循环群外的各类可能 Galois 群构造生成互异四次域的无限单参数单生四次多项式族，并讨论未解决的循环情形。

计数 $A_4$-四次域的界

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。

单源互反四次多项式及其伽罗瓦群

Monogenic Reciprocal Quartic Polynomials And Their Galois Groups

本文针对互反四次整系数多项式，在两种参数情形下，确定了所有具有给定伽罗瓦群的单源多项式。

Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic

We use explicit methods to study the 4-torsion points on the Jacobian variety of the Fermat quartic. With the aid of computer algebra systems, we explicitly give a basis of the group of 4-torsion points. We calculate the Galois action, and show that the image of the mod 4 Galois representation is isomorphic to the dihedral group of order 8. As applications, we calculate the Mordell-Weil group of the Jacobian variety of the Fermat quartic over each subfield of the 8-th cyclotomic field. We determine all of the points on the Fermat quartic defined over quadratic extensions of the 8-th cyclotomic…

例外四次域无处不在

Exceptional Quartics are Ubiquitous

本文构造性地证明了例外四次数域的普遍存在性：既有无穷多个包含给定实二次域的例外四次域，也有无穷多个不含二次子域的例外四次域，并进一步推广到所有大于4的次数。

A family of cyclic quartic fields with explicit fundamental units

We construct a family of quartic polynomials with cyclic Galois group and show that the roots of the polynomials are fundamental units or generate a subgroup of index 5.

Comparing the density of D_4 and S_4 quartic extensions of number fields

When ordered by discriminant, it is known that about 83% of quartic fields over Q have associated Galois group S_4, while the remaining 17% have Galois group D_4. We study these proportions over a general number field F. We find that asymptotically 100% of quadratic number fields have more D_4 extensions than S_4 and that the ratio between the number of D_4 and S_4 quartic extensions is biased arbitrarily in favor of D_4 extensions. Under GRH, we give a lower bound that holds for general number fields.

Norm-Euclidean Galois fields

Let K be a Galois number field of prime degree $\ell$. Heilbronn showed that for a given $\ell$ there are only finitely many such fields that are norm-Euclidean. In the case of $\ell=2$ all such norm-Euclidean fields have been identified, but for $\ell\neq 2$, little else is known. We give the first upper bounds on the discriminants of such fields when $\ell>2$. Our methods lead to a simple algorithm which allows one to generate a list of candidate norm-Euclidean fields up to a given discriminant, and we provide some computational results.

Norm-Euclidean Galois fields and the Generalized Riemann Hypothesis

Assuming the Generalized Riemann Hypothesis (GRH), we show that the norm-Euclidean Galois cubic fields are exactly those with discriminant $Δ=7^2,9^2,13^2,19^2,31^2,37^2,43^2,61^2,67^2,103^2,109^2,127^2,157^2$. A large part of the proof is in establishing the following more general result: Let $K$ be a Galois number field of odd prime degree $\ell$ and conductor $f$. Assume the GRH for $ζ_K(s)$. If $38(\ell-1)^2(\log f)^6\log\log f<f$, then $K$ is not norm-Euclidean.

On the largest prime factor of quartic polynomial values: the cyclic and dihedral cases

Let $P(X)\in\mathbb{Z}[X]$ be an irreducible, monic, quartic polynomial with cyclic or dihedral Galois group. We prove that there exists a constant $c_P>0$ such that for a positive proportion of integers $n$, $P(n)$ has a prime factor $\ge n^{1+c_P}$.

在对应于单基因多项式族的某些四次域中计算幂整基

Calculating power integral bases in some quartic fields corresponding to monogenic families of polynomials

本研究针对J. Harrington、L. Jones及P. Voutier提出的单基因四次多项式族，刻画其生成数域中幂整基的全部生成元，补充了相关单基因数域的幂整基生成元研究。

Plane quartic twists of X(5,3)

Given an odd representation of the absolute Galois group of Q onto PGL(2,3) and a positive integer N, there exists a twisted modular curve defined over Q whose rational points classify the quadratic Q-curves of degree N realizing the representation. The paper gives a method to provide an explicit plane quartic model for this curve in the genus-three case N=5.

Euclidean ideal classes in Galois number fields of odd prime degree

Weinberger in 1972, proved that the ring of integers of a number field with unit rank at least $1$ is a principal ideal domain if and only if it is a Euclidean domain, provided the generalised Riemann hypothesis holds. Lenstra extended the notion of Euclidean domains in order to capture Dedekind domains with finite cyclic class group and proved an analogous theorem in this setup. More precisely, he showed that the class group of the ring of integers of a number field with unit rank at least $1$ is cyclic if and only if it has a Euclidean ideal class, provided the generalised Riemann hypothesis…

计数D₄-四次域的一个改进误差项

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。

On Greenberg's generalized conjecture for imaginary quartic fields

For an algebraic number field $K$ and a prime number $p$, let $\widetilde{K}/K$ be the maximal multiple $\mathbb{Z}_p$-extension. Greenberg's generalized conjecture (GGC) predicts that the Galois group of the maximal unramified abelian pro-$p$ extension of $\widetilde{K}$ is pseudo-null over the completed group ring $\mathbb{Z}_p[\![\mathop{\mathrm{Gal}}\nolimits(\widetilde{K}/K)]\!]$. We show that GGC holds for some imaginary quartic fields containing imaginary quadratic fields and some prime numbers.

Euclid's Algorithm in quartic CM-fields

In this note we present techniques to compute inhomogeneous minima of norm forms; as an application, we determine all norm-Euclidean complex bicyclic quartic number fields.

The number of quartic $D_4$-fields ordered by conductor

We consider families of number fields of degree 4 whose normal closures over $\mathbb{Q}$ have Galois group isomorphic to $D_4$, the symmetries of a square. To any such field $L$, one can associate the Artin conductor of the corresponding 2-dimensional irreducible Galois representation with image $D_4$. We determine the asymptotic number of such quartic $D_4$-fields ordered by conductor, and compute the leading term explicitly as a mass formula, verifying heuristics of Kedlaya and Wood. Additionally, we are able to impose any local splitting conditions at any finite number of primes (sometimes…

Twists of the Burkhardt Quartic Threefold

We study twists of the Burkhardt quartic threefold over non-algebraically closed base fields of characteristic different from 2,3,5. We show they all admit quartic models in projective four-space. We identify a Galois-cohomological obstruction that measures if a given twist is birational to a moduli space of abelian varieties. This obstruction has implications for the rational points on these varieties. As a result, we see that all possible 3-level structures can be realized by abelian surfaces, whereas Kummer 3-level structures that group-theoretically may be admissible, may not be realizable…

单生成循环四次三项式

Monogenic Cyclic Quartic Trinomials

本文证明了恰好存在三个形如 $x^4+bx^2+d$ 且 Galois 群为四阶循环群的单生成三项式。

On Splitting Types, Discriminant Bounds, and Conclusive Tests for the Galois Group

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…

Two classes of number fields with a non-principal Euclidean ideal

This paper introduces two classes of totally real quartic number fields, one of biquadratic extensions and one of cyclic extensions, each of which has a non-principal Euclidean ideal. It generalizes techniques of Graves used to prove that the number field $\mathbb{Q}(\sqrt{2},\sqrt{35})$ has a non-principal Euclidean ideal.

The local-global property for bitangents of plane quartics

We study the arithmetic of bitangents of smooth quartics over global fields. With the aid of computer algebra systems and using Elsenhans--Jahnel's results on the inverse Galois problem for bitangents, we show that, over any global field of characteristic different from $2$, there exist smooth quartics which have bitangents over every local field, but do not have bitangents over the global field. We give an algorithm to find such quartics explicitly, and give an example over $\mathbb{Q}$. We also discuss a similar problem concerning symmetric determinantal representations. This paper is a summ…

The density of discriminants of quintic rings and fields

We determine, asymptotically, the number of quintic fields having bounded discriminant. Specifically, we prove that the asymptotic number of quintic fields having absolute discriminant at most X is a constant times X. In contrast with the quartic case, we also show that a density of 100% of quintic fields, when ordered by absolute discriminant, have Galois closure with full Galois group $S_5$. The analogues of these results are also proven for orders in quintic fields. Finally, we give an interpretation of the various constants appearing in these theorems in terms of local masses of quintic ri…

具有循环类群的低次扩张

Low degree extensions with Cyclic class group

本文证明了阿贝尔四次域及某些类数为2的双二次域中存在欧几里得理想类，并探讨了特定三次和二次扩张中的相关性质。

5-Class towers of cyclic quartic fields arising from quintic reflection

Let zeta5 be a primitive fifth root of unity and d<>1 be a quadratic fundamental discriminant not divisible by 5. For the 5-dual cyclic quartic field M=Q((zeta5-zeta5^-1)*d^1/2) of the quadratic fields k1=Q(d^1/2) and k2=Q((5*d)^1/2) in the sense of the quintic reflection theorem, the possibilities for the isomophism type of the Galois group G(5,2)M=Gal(M(5,2)/M) of the second Hilbert 5-class field M(5,2) of M are investigated, when the 5-class group Cl5(M) is elementary bicyclic of rank two. Usually, the maximal unramified pro-5-extension M(5,infinity) of M coincides with M(5,2) already. The …

Malle 猜想对于 $\mathbb Q$ 上的 Galois 八次数域

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。

HT90 and "simplest" number fields

A standard formula (1) leads to a proof of HT90, but requires proving the existence of $θ$ such that $α\ne 0$, so that $β=α/σ(α)$. We instead impose the condition (M), that taking $θ=1$ makes $α=0$. Taking $n=3$, we recover Shanks's simplest cubic fields. The "simplest" number fields of degrees 3 to 6, Washington's cyclic quartic fields, and a certain family of totally real cyclic extensions of $\Q(\cos(π/4m))$ all have defining polynomials whose zeroes satisfy \eqref{e:M}. Further investigation of (M) for $n=4$ leads to an elementary algebraic construction of a 2-parameter family of octic pol…

Relative Ideal Classes of Arbitrary Order

We adapt a known technique for searching for ideal classes of arbitrary order and then apply it to three families of number fields. We show that a family of cyclic sextic number fields has infinitely many fields in it that contain a relative ideal class of order $r,$ where $r$ is a positive integer relatively prime to the degree of the extension. We then show that the same holds true for a family of cyclic quartic number fields. Though the technique is traditionally applied to Galois extensions, we show how it may be adapted to handle a family of non-Galois cubic number fields and prove that t…

Progress Towards Counting D_5 Quintic Fields

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…

关于射线类的欧几里得系统

On Euclidean systems of ray classes

本文定义了射线类的欧几里得系统，证明其生成射线类群，并在GRH假设下证明对全实伽罗瓦数域和特定素数幂模，射线类群的生成集是欧几里得系统。

关于全虚数域的计数

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。

Genus 2 Curves with Complex Multiplication

Genus 2 curves are useful in cryptography for both discrete-log based and pairing-based systems, but a method is required to compute genus 2 curves such that the Jacobian has a given number of points. Currently, all known methods involve constructing genus 2 curves with complex multiplication via computing their three Igusa class polynomials. These polynomials have rational coefficients and require extensive computation and precision to compute. Both the computation and the complexity analysis of these algorithms can be improved by a more precise understanding of the denominators of the coeffi…

Étale代数与有限Galois模的Kummer理论

Étale algebras and the Kummer theory of finite Galois modules

本文填补了$H^1$与域扩张对应中未被记录的组合与Galois理论细节空白，以étale代数视角阐明了$H^1$在三次、四次代数参数化及局部域Tate配对计算中的应用。

Computing class polynomials for abelian surfaces

We describe a quasi-linear algorithm for computing Igusa class polynomials of Jacobians of genus 2 curves via complex floating-point approximations of their roots. After providing an explicit treatment of the computations in quartic CM fields and their Galois closures, we pursue an approach due to Dupont for evaluating $θ$- constants in quasi-linear time using Newton iterations on the Borchardt mean. We report on experiments with our implementation and present an example with class number 17608.

Ranks of elliptic curves with prescribed torsion over number fields

We study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is empty, or it contains curves of rank~0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group $T$ and a quartic field $K$ such that among the elliptic curves over $K$ with torsion subgroup $T$, there are curves of positive rank, but none of rank~0. We find examples of …

Towards a generalization of the van der Waerden's conjecture for Sn-polynomials with integral coefficients over a fixed number field extension

The van der Waerden's Conjecture states that the set $\mathscr{P}_{n,N}^0(\mathbb{Q})$ of monic integer polynomials $f(X)$ of degree $n$, with height $\le N$ such that the Galois group $G_{K_f/\mathbb{Q}}$ of the splitting field $K_f/\mathbb{Q}$ is the full symmetric group, has order $|\mathscr{P}_{n,N}^0(\mathbb{Q})|=(2N)^n+O_n(N^{n-1})$ as $N\rightarrow+\infty$. The conjecture has been shown previously for cubic and quartics polynomials by van der Waerden, Chow and Dietmann. Subsequently, Bhargava proved it for $n\ge6$. In this paper, we generalize the result for polynomials with coefficient…

Statistics for biquadratic covers of the projective line over finite fields

We study the distribution of the traces of the Frobenius endomorphism of genus $g$ curves which are quartic non-cyclic covers of $\mathbb{P}^{1}_{\mathbb{F}_{q}}$, as the curve varies in an irreducible component of the moduli space. We show that for $q$ fixed, the limiting distribution of the trace of Frobenius equals the sum of $q + 1$ independent random discrete variables. We also show that when both $g$ and $q$ go to infinity, the normalized trace has a standard complex Gaussian distribution. Finally, we extend these computations to the general case of arbitrary covers of $\mathbb{P}^{1}_{\…

27级非分裂Cartan模曲线上的有理点与数域上的二次Chabauty方法

Rational points on the non-split Cartan modular curve of level 27 and quadratic Chabauty over number fields

本文针对27级非分裂Cartan子群正规化子这一3进伽罗瓦像分类的唯一遗留问题，发展数域上模曲线的二次Chabauty方法，结合局部高度分析确定相关模曲线有理点，完成非复乘椭圆曲线3进伽罗瓦像分类。

具有循环 $p$-类群的数域的顺分歧 pro-$p$ 扩张的有限性

The finitude of tamely ramified pro-$p$ extensions of number fields with cyclic $p$-class groups

本文研究了 $p$-类群循环的数域在单个非 $p$-进素理想外不分歧的极大 pro-$p$ 扩张的 Galois 群的有限性，并证明了在生成元秩为 2 时该群对大多数素理想有限。

好整数：(T,k)-子类及其在编码理论伽罗瓦对偶中的应用

Good Integers: (T,k)-Subclasses and Applications to Galois Duality in Coding Theory

本文引入并研究了基于序列 (a^{ks+T}+b^{ks+T}) 的 (T,k)-好整数子类，发展了其算术理论，并应用于编码理论中伽罗瓦自互反不可约因子、LCD循环码和自对偶循环码的表征与枚举。