Étienne Fouvry, Peter Koymans
We study the $4$-rank of the ideal class group of $K_n := \mathbb{Q}(\sqrt{-n}, \sqrt{n})$. Our main result is that for a positive proportion of the squarefree integers $n$ we have that the $4$-rank of $\text{Cl}(K_n)$ equals $ω_3(n) - 1$, where $ω_3(n)$ is the number of prime divisors of $n$ that are $3$ modulo $4$.
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We show that for $100\%$ of the odd, squarefree integers $n > 0$ the $4$-rank of $\text{Cl}(\mathbb{Q}(i, \sqrt{n}))$ is equal to $ω_3(n) - 1$, where $ω_3$ is the number of prime divisors of $n$ that are $3$ modulo $4$.
By Dirichlet's Unit Theorem, under the log embedding the units in the ring of integers of a number field form a lattice, called the log-unit lattice. We investigate the geometry of these lattices when the number field is a biquadratic or cyclic cubic extension of $\mathbb{Q}$. In the biquadratic case, we determine when the log-unit lattice is orthogonal. In the cyclic cubic case, we show that the log-unit lattice is always equilateral triangular.
Sum of two squares in biquadratic fields
本文基于二次形式迷向性的局部-全局原理，提出一种算法，用于判定双二次域中的数是否为两平方和。
In this paper, based on the theory of genus fields of biquadartic fields, we find a new larger family of biquadratic fields having a non-principal Euclidean ideal class, which implies the main results of \cite{JNT}.
Given a pair of modular forms with complex multiplication by distinct imaginary quadratic fields, the four dimensional Galois representation associated to their Rankin--Selberg convolution is induced from a character over an imaginary biquadratic field $F$. Using the Euler system of Lei, Loeffler and Zerbes we bound a Selmer group associated to this character, over the unique $\mathbb{Z}_{p}^{3}$-extension of $F$.
The aim of this article is to study (additively) indecomposable algebraic integers $\mathcal O_K$ of biquadratic number fields $K$ and universal totally positive quadratic forms with coefficients in $\mathcal O_K$. There are given sufficient conditions for an indecomposable element of a quadratic subfield to remain indecomposable in the biquadratic number field $K$. Furthermore, estimates are proven which enable algorithmization of the method of escalation over $K$. These are used to prove, over two particular biquadratic number fields $\mathbb{Q}(\sqrt{2}, \sqrt{3})$ and $\mathbb{Q}(\sqrt{6},…
Class numbers of Imaginary bicyclic biquadratic number fields
本文提出一种利用类数整除$2n$的虚二次数域列表，计算类数为$n$的虚双循环双二次数域及其类群结构的方法，并列出了类数为4、6和7的域及其子域的类群结构。
An elementary formulation of Kubota's proof of Satz 4 about biquadratic number fields
本文旨在提供Kubota关于双二次数域Satz 4证明的初等表述，因为该命题是计算Polya群阶的基石。
Euclidean Algorithms for Ideal Classes in Biquadratic fields: A Genus-Theoretic Perspective
本文从亏元理论角度研究实数双二次域中的欧几里得理想类，证明在特定条件下存在欧几里得理想，并展示相关双二次域集合的密度为零。
Additively indecomposable quadratic forms over biquadratic and simplest cubic fields
本文研究了双二次和最简三次域上加法不可分解二次型的性质，证明了存在经典二元二次型，并给出了其数量的下界。
Lenstra introduced the notion of a Euclidean ideal class, which is a generalization of the Euclidean domain. Lenstra also proved that the Euclidean ideal in a number field $K$ implies that the class group of $K$ is cyclic. We construct a family of biquadratic fields with a Euclidean ideal whenever the class number is 2. This extends the families given by Graves, Hsu, Chattopadhyay, and Muthukrishnan.
Let $d$ be a positive square-free integer. In this paper we shall investigate the structure of the $2$-class group of the cyclotomic $\mathbb{Z}_2$-extension of the imaginary biquadratic number field $\mathbb{Q}(\sqrt{d},\sqrt{-1})$. Furthermore, we deduce the structure of the $2$-class group of cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}(\sqrt{-d})$.
Bounds on the Pythagoras number and indecomposables in biquadratic fields
本文研究实双二次域代数整数环的毕达哥拉斯数，证明在不含特定平方根时其值至少为6，并给出加法不可分解整数的范数与最小余差分迹的上界。
We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms. Interestingly, the correspondence between these ideals in the larger ring and those in the smaller ones extends to the divisibility of principal ideals in their respective rings, with some exceptions that we explicitly provide. Finally, we hint at possible applications of this correspondence.
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。
We study the capitulation of ideal classes in an infinite family of imaginary bicyclic biquadratic number fields consisting of fields $k =Q(\sqrt{2pq}, i)$, where $i=\sqrt{-1}$ and $p\equiv -q\equiv1 \pmod 4$ are different primes. For each of the three quadratic extensions $K/k$ inside the absolute genus field $k^{(*)}$ of $k$, we compute the capitulation kernel of $K/k$. Then we deduce that each strongly ambiguous class of $k/Q(i)$ capitulates already in $k^{(*)}$, which is smaller than the relative genus field $\left(k/Q(i)\right)^*$.
Pólya-Ostrowski Group and Unit Index in Real Biquadratic Fields
本文研究了实双二次域中Pólya-Ostrowski群与Hasse单位指数的关系，并改进了分歧素数个数的上界。
H. W. Lenstra \cite{lenstra} introduced the notion of an Euclidean ideal class, which is a generalization of norm-Euclidean ideals in number fields. Later, families of number fields of small degree were obtained with an Euclidean ideal class (for instance, in \cite{hester1} and \cite{cathy}). In this paper, we construct certain new families of biquadratic number fields having a non-principal Euclidean ideal class and this extends the previously known families given by H. Graves \cite{hester1} and C. Hsu \cite{cathy}.
An infinite family of imaginary biquadratic fields with a large class number
本文通过组合两个虚二次域族，利用Kuroda类数公式，构造了具有大类数的虚双二次域的无限族，旨在类群中获得高阶元素。
Denote $Θ_C$ as the Frobenius class of a curve $C$ over the finite field $\mathbb{F}_q$. In this paper we determine the expected value of Tr$(Θ_C^n)$ where $C$ runs over all biquadratic curves when $q$ is fixed and $g$ tends to infinity. This extends work done by Rudnick and Chinis who separately looked at hyperelliptic curves and Bucur, Costa, David, Guerreiro and Lowry-Duda who looked at $\ell$-cyclic curves, for $\ell$ a prime, as well as cubic non-Galois curves.
Greenberg's conjecture and Iwasawa module of Real biquadratic fields II
本文研究实双二次域的$2$-类群秩稳定性，验证了部分数域的格林伯格猜想，并确定了具有平凡$2$-Iwasawa模的实双二次域列表。
Class groups of imaginary biquadratic fields
本文构造了两族各含无穷多个成员且理想类群较大的虚双二次域，分别借助椭圆曲线及其二次扭曲，以及椭圆曲线与超椭圆曲线的组合，并利用 Soleng 与 Banerjee–Hoque 的结果。
Constructions of lattices via biquadratic and triquadratic fields
本研究利用代数数论工具，证明$D_4$-格可由无穷全虚双二次域族构造，并扩展方法证明$D_8$与$E_8$格可通过该族衍生的三二次域实现，为格构造提供了新的代数数论途径。
We examine the Pythagoras number $\mathcal{P}(\mathcal{O}_K)$ of the ring of integers $\mathcal{O}_K$ in a totally real biquadratic number field $K$. We show that the known upper bound $7$ is attained in a large and natural infinite family of such fields. In contrast, for almost all fields $\mathbb{Q}(\sqrt5, \sqrt{s})$ we prove $\mathcal{P}(\mathcal{O}_K)=5$. Further we show that $5$ is a lower bound for all but seven fields $K$ and $6$ is a lower bound in an asymptotic sense.
On the fourth power level of $\mathfrak{p}$-adic completions of biquadratic number fields
本文研究了双二次数域的 $\mathfrak{p}$-adic 完成的四次方级，通过计算 ramification指数为4且inertial度为平凡时的四次方级，得出了此类数域的下界。
We consider the signature rank of the units in real multiquadratic fields. When the three quadratic subfields of a real biquadratic field $K$ either (a) all have signature rank 2 (that is, fundamental units of norm $-1$), or (b) all have signature rank 1 (that is, have totally positive fundamental units), we provide explicit examples to show there exist infinitely many $K$ having each of the possible unit signature ranks (namely signature rank 3 or 4 in case (a) and signature rank 1,2, or 3 in case (b)). We make some additional remarks for higher rank real multiquadratic fields, in particular …
Let $K$ be an imaginary biquadratic field and $K_1$, $K_2$ be its imaginary quadratic subfields. For integers $N>0$, $μ\geq 0$ and an odd prime $p$ with $\gcd(N,p)=1$, let $K_{(Np^μ)}$ and $(K_i)_{(Np^μ)}$ for $i=1,2$ be the ray class fields of $K$ and $K_i$, respectively, modulo $Np^μ$. We first present certain class fields $\widetilde{K_{N,p,μ}^{1,2}}$ of $K$, in the sense of Hilbert, which are generated by Siegel-Ramachandra invariants of $(K_i)_{(Np^{μ+1})}$ for $i=1,2$ over $K_{(Np^μ)}$ and show that $K_{(Np^{μ+1})}=\widetilde{K_{N,p,μ}^{1,2}}$ for almost all $μ$.
In this paper, we use the theory of genus fields to study the Euclidean ideals of certain real biquadratic fields $K.$ Comparing with the previous works, our methods yield a new larger family of real biquadratic fields $K$ having Euclidean ideals; and the conditions for our family seem to be more efficient for the computations. Moreover, the previous approaches mainly focus on the case if $h_K=2$, while the present approach can also deal with the general case when $h_K=2^t (t\geq1)$, where $h_K$ denotes the ideal class number of $K$. In particular, if $h_K\geq 4$, it shows that $H(K)$, the Hil…
We study totally positive definite quadratic forms over the ring of integers $\mathcal{O}_K$ of a totally real biquadratic field $K=\mathbb{Q}(\sqrt{m}, \sqrt{s})$. We restrict our attention to classical forms (i.e., those with all non-diagonal coefficients in $2\mathcal{O}_K$) and prove that no such forms in three variables are universal (i.e., represent all totally positive elements of $\mathcal{O}_K$). This provides further evidence towards Kitaoka's conjecture that there are only finitely many number fields over which such forms exist. One of our main tools are additively indecomposable el…
Anticyclotomic Euler system over biquadratic fields
该研究构造了虚双二次域上反分圆Hecke特征扭化的新形式伽罗瓦表示的Jetchev-Nekovar-Skinner型反分圆欧拉系，并给出其在Bloch-Kato猜想与Iwasawa-Greenberg主猜想整除性问题上的数论应用。
On biquadratic fields: when 5 squares are not enough
本文研究了完全实二次biquadratic域K的Pythagoras数P(OK)，继续Tinková的工作，证明了K满足P(OK)≥6当且仅当K中不包含√2或√5（除了有限多个例外）。我们解决了剩余三个类别中的两个，证明了除了有限多个外，所有包含√6或√7的K都满足P(OK)≥6。此外，我们还提出了支持猜想的想法和计算，也适用于包含√3的K，从而通过明确列出例外域来细化猜想。
Greenberg's conjecture and Iwasawa module of Real biquadratic fields I
本文研究实双二次域的Greenberg猜想，提出并探讨使Iwasawa模的秩与第一层类群的秩相等的问题，并构造了多族使Iwasawa模平凡或同构于特定循环群及直积的实双二次域，同时给出实三二次域类群2-秩的相关结果。
Stability of 2-class groups in the $\mathbb{Z}_2$-extension of certain real biquadratic fields
本文通过初等群论与类域论方法，验证了某无限族实双二次域分圆 $\mathbb{Z}_2$-扩张中各层 $2$-类群的 Greenberg 猜想，并关联了理想类 capitulation 与类群相对大小。
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。
Let $p_1\equiv p_2\equiv -q\equiv1 \pmod4$ be different primes such that $\displaystyle\left(\frac{2}{p_1}\right)= \displaystyle\left(\frac{2}{p_2}\right)=\displaystyle\left(\frac{p_1}{q}\right)=\displaystyle\left(\frac{p_2}{q}\right)=-1$. Put $d=p_1p_2q$ and $i=\sqrt{-1}$, then the bicyclic biquadratic field ${k}=\mathbb{Q}(\sqrt{d},i)$ has an elementary abelian $2$-class group, $\mathbf{C}l_2(k)$, of rank $3$. In this paper, we study the principalization of the $2$-classes of ${k}$ in its fourteen unramified abelian extensions $\mathbb{K}_j$ and $\mathbb{L}_j$ within ${k}_2^{(1)}$, that is t…
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}_{\…
On the maximal unramified pro-2-extension of $\mathbb{Z}_2$-extension of certain real biquadratic fields
研究实双二次域的\(\mathbb{Z}_2\)-扩张的极大非分歧 pro - 2 - 扩张，通过构造特定实数字域，证明其伽罗瓦群性质，构造出满足条件的实双二次域族，满足格林伯格猜想。
Class Number Formulas for Certain Biquadratic Fields
本文利用Shintani的工作，针对类数为1的全实二次域的虚二次扩张，提出了两个初等类数公式，分别基于有理函数幂级数展开系数的交错和与$1/p$展开式的周期长度。
In this paper, we investigate the existence of an elementary abelian closure in characteristic not $2$ for biquadratic extensions. We discover that it exists for any non-cyclic extension. We make use of it to obtain a classification for this class of extensions up to isomorphism via descent. This permits us to describe the geometry of this moduli space in group theoretic terms. We also provide two families of polynomials with two parameters that can describe any quartic extensions.
Minimal rank of universal lattices and number of indecomposable elements in real multiquadratic fields
本文通过给出实多重二次域中泛二次格最小秩及不可分解元素数量的上界，证明了相应域的密度为零，并研究了实双二次域中不可分解元素的结构与具体计算。
For a Galois extension $K/F$ with $\text{char}(K)\neq 2$ and $\text{Gal}(K/F) \simeq \mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}$, we determine the $\mathbb{F}_2[\text{Gal}(K/F)]$-module structure of $K^\times/K^{\times 2}$. Although there are an infinite number of (pairwise non-isomorphic) indecomposable $\mathbb{F}_2[\mathbb{Z}/2\mathbb{Z}\oplus\mathbb{Z}/2\mathbb{Z}]$-modules, our decomposition includes at most $9$ indecomposable types. This paper marks the first time that the Galois module structure of power classes of a field has been fully determined when the modular representatio…
Quaternion algebras and square power classes over biquadratic extensions
针对 Klein 4-群双二次扩张，本文利用 Br(F) 的特定子空间确定平方幂类 Galois 模分解中各求和项类型的重数，并证明所有非例外类型均可实现。
We prove that all imaginary biquadratic fields and cyclic quartic fields of class number $1$ are Euclidean.
Let $X$ be a proper smooth variety having an affine open subset defined by the normic equation $N_{k(\sqrt{a},\sqrt{b})/k}({x})=Q(t_{1},...,t_{m})^{2}$ over a number field $k$. We prove that : (1) the failure of the local-global principle for zero-cycles is controlled by the Brauer group of $X$; (2) the analogue for rational points is also valid assuming Schinzel's hypothesis.
Hilbert's Theorem 90 is a classical result in the theory of cyclic extensions. The quadratic case of Hilbert 90, however, generalizes in noncyclic directions as well. Informed by a poem of Richard Wilbur, the article explores several generalizations, discerning connections among multiplicative groups of fields, values of binary quadratic forms, a bit of module theory over group rings, and even Galois cohomology.
Let $p_1 \equiv p_2 \equiv5\pmod8$ be different primes. Put $i=\sqrt{-1}$ and $d=2p_1p_2$, then the bicyclic biquadratic field $k=Q(\sqrt{d}, \sqrt{-1})$ has an elementary abelian 2-class group of rank $3$. In this paper we determine the nilpotency class, the coclass, the generators and the structure of the non-abelian Galois group $\mathrm{Gal}(k_2^{(2)}/k)$ of the second Hilbert 2-class field $k_2^{(2)}$ of $k$. We study the capitulation problem of the 2-classes of $k$ in its seven unramified quadratic extensions $K_i$ and in its seven unramified bicyclic biquadratic extensions $L_i$.
On the class number of real multiquadratic fields
本文完整分类了2-类数为2的实双二次域及奇类数的实多重二次域，并给出基本单位平方根的显式公式。
Kuroda's Class Number Formula
本文给出数域双二次扩域类数公式的非解析证明
Let $k$ be a number field and $\tilde k$ a fixed quadratic extension of $k$. In this paper and its companion, we find the mean value of the product of class numbers and regulators of two quadratic extensions $F,F^*\not=\tilde k$ contained in the biquadratic extensions of $k$ containing $\tilde k$.
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.
Fix a Galois extension E/F of totally real number fields such that the Galois group G has exponent 2. Let S be a finite set of primes of F containing the infinite primes and all those which ramify in E, let S_E denote the primes of E lying above those in S, and let O_E^S denote the ring of S_E-integers of E. We then compare the Fitting ideal of K_2(O_E^S) as a Z[G]-module with a higher Stickelberger ideal. The two extend to the same ideal in the maximal order of Q[G], and hence in Z[1/2][G]. Results in Z[G] are obtained under the assumption of the Birch-Tate conjecture, especially for biquadra…
We prove that two global fields are isomorphic if and only if there is an isomorphism of groups of Dirichlet characters that preserves L-series.
We study the moments of the Dirichlet L-function when defined over the polynomial ring over finite fields. We find an asymptotic formula to the fourth moment of the central value of Dirichlet L functions in this context. We also find a lower bound to the 2kth moment of these L-functions.
Kuroda's formula relates the class number of a multi-quadratic number field $K$ to the class numbers of its quadratic subfields $k_i$. A key component in this formula is the unit group index $Q(K) = [\mathcal{O}_{K}^{\times}: \prod_i\mathcal{O}_{k_i}^{\times}]$. We study how $Q(K)$ behaves on average in certain natural families of totally real biquadratic fields $K$ parametrized by prime numbers.
Subconvexity of a double Dirichlet series over the Gaussian field
该研究针对高斯域上与二次赫克L-函数相关的双重狄利克雷级数，建立了其次凸性界，推进了数论领域相关问题的研究。
Upper bounds for shifted moments of quadratic Dirichlet $L$-functions over function fields
本文在函数域上建立了二次Dirichlet $L$-函数平移矩的锐利上界，并将其应用于证明二次Dirichlet特征和矩的若干界。
Upper bounds for shifted moments of Dirichlet $L$-functions to a fixed modulus over function fields
本文建立了函数域上固定模 Dirichlet $L$-函数族平移矩的精确上界，并以此推导出 Dirichlet 特征和矩的上界。
We consider a function-field analogue of Dirichlet series associated with the Goldbach counting function, and prove that it can, or cannot, be continued meromorphically to the whole plane. When it cannot, we further prove the existence of the natural boundary of it.
In this rather computational paper, we determine certain representation numbers of ideals in real quadratic number fields explicitly in order to obtain a representation of the associated Dirichlet series in terms of Dirichlet L-functions and a generalized divisor sum. A direct and important consequence is that the Dirichlet series has a meromorphic continuation to the whole complex plane and a simple pole at $s=2$ whose residue can be made explicit in terms of the Dirichlet L-functions and the generalized divisor sum.
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。
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