Anton Deitmar, Mark Pavey
We prove an asymptotic formula for class numbers of totlally imaginary quartic number fields, ie for number fields of degree 4 over Q with only complex embeddings. After previous work for real quadratic fields (Sarnak) and complex cubic fields (Deitmar), this is the last remaining case of number fields with unit rank one.
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Some upper bounds for the number of monogenizations of quartic orders are established by considering certain classical Diophantine equations, namely index form equations in quartic number fields, and cubic and quartic Thue equations.
In this paper, we consider maximal and irreducible quartic orders which arise from integral binary quartic forms, via the construction of Birch and Merriman, and whose field of fractions is a quartic $D_4$-field. By a theorem of M. Wood, such quartic orders may be regarded as quartic $D_4$-fields whose ring of integers has a monogenic cubic resolvent. We shall give the asymptotic number of such objects when ordered by conductor, as well as estimate the asymptotic number of such objects when ordered by discriminant.
In this paper, based on techniques of Newton polygons, a result which allows the computation of a p integral basis of every quartic number field is given. For each prime integer p, this result allows to compute a p-integral basis of a quartic number field K defined by an irreducible polynomial $P(X) = X4 + aX + b \in\z[X]$ in methodical and complete generality.
We give an asymptotic formula for the number of $D_4$ quartic extensions of a function field with discriminant equal to some bound, essentially reproducing the analogous result over number fields due Cohen, Diaz y Diaz, and Olivier, but with a stronger error term. We also study the relative density of $D_4$ and $S_4$ quartic extensions of a function field and show that with mild conditions, the number of $D_4$ quartic extensions can far exceed the number of $S_4$ quartic extensions
In this paper, we investigate the 2-rank of the class group of some real cyclic quartic number fields. Precisely, we consider the case where the quadratic subfield is Q(\sqrt{l}) with l congruent to 5 modulo 8 is a prime.
Minimal Mahler Measure in Quartic Galois Number Fields
本研究聚焦四次伽罗瓦数域中最小整马勒测度与域判别式的关联，推导了有条件依赖ABC猜想的密度结论及若干无条件结论。
For every prime integer $p$, an explicit factorization of the principal ideal $p\z_K$ into prime ideals of $\z_K$ is given, where $K$ is a quartic number field defined by an irreducible polynomial $X^4+aX+b\in\z[X]$.
In this paper, we determine the 2-rank of the class group of certain classes of real cyclic quartic number fields. Precisely, we consider the case in which the quadratic subfield is Q(\sqrt{l}) with l=2 or a prime congruent to 1 mod 8.
We consider the simplest quartic number fields $\mathbb{K}_m$ defined by the irreducible quartic polynomials $$x^4-mx^3-6x^2+mx+1,$$ where $m$ runs over the positive rational integers such that the odd part of $m^2+16$ is squarefree. In this paper, we study the common index divisor $I(\mathbb K_m)$ and determine explicitly the prime ideal decomposition for any prime number in any simplest quartic number fields $\mathbb{K}_m$. On the other hand, we establish an asymptotic formula for the number of simplest quartic fields with discriminant $\leq x$ and given index.
We will give upper bounds for the number of integral solutions to quartic Thue equations. Our main tool here is a logarithmic curve $ϕ(x, y)$ that allows us to use the theory of linear forms in logarithms. This manuscript improves the results of author's earlier work with Okazaki by giving special treatments to forms with respect to their signature.
Let $E$ be an elliptic curve defined over $\mathbb{Q}$, and let $K$ be a number field of degree four that is Galois over $\mathbb{Q}$. The goal of this article is to classify the different isomorphism types of $E(K)_{\text{tors}}$.
We show that an order in a quartic field has fewer than $3000$ essentially different generators as a $\mathbb Z$-algebra (and fewer than $200$ if the discriminant of the order is sufficiently large). This significantly improves the previously best known bound of $2^{72}$. Analogously, we show that an order in a quartic field is isomorphic to the invariant order of at most $10$ classes of integral binary quartic forms (and at most $7$ if the discriminant is sufficiently large). This significantly improves the previously best known bound of $2^{80}$.
For $n=3$, 4, and 5, we prove that, when $S_n$-number fields of degree $n$ are ordered by their absolute discriminants, the lattice shapes of the rings of integers in these fields become equidistributed in the space of lattices.
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.
We study the asymptotic count of dihedral quartic extensions over a fixed number field with bounded norm of the relative discriminant. The main term of this count (including a summation formula for the constant) can be found in the literature (see Cohen--Diaz y Diaz--Olivier for the statement without proof and see Klüners for a proof), but a power-saving for the error term has not been explicitly determined except in the case that the base field is $\mathbb{Q}$. In this article, we describe the argument for obtaining both the explicit main term and a power-saving error term for the number of $…
Let us consider the pure quartic fields of the form $\K=\Q(\sqrt[4]{p})$ where $0<p\equiv 7\pmod{16}$ is a prime integer. We prove that the $2$-class group of $\K$ has order $2$. As a consequence of this, if the class number of $\K$ is $2$, then the Hilbert class field of $\K$ is $\H_\K=\K(\sqrt{2})$. Finally, we find a criterion to decide if an ideal of the ring of integers or $\K$ is principal or non-principal.
We prove that all imaginary biquadratic fields and cyclic quartic fields of class number $1$ are Euclidean.
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 average number of primitive quadratic Dirichlet characters of modulus n tends to a constant as n->infty. The same is true for primitive cubic characters. It is therefore surprising that, as n->infty, the average number of primitive quartic characters of modulus n grows with ln(n), and that the average number of primitive octic characters of modulus n grows with ln(n)^2. Leading coefficients in the asymptotic expressions are also computed.
Let $p$ be a prime. The $2$-primary part of the class group of the pure quartic field $\mathbb{Q}(\sqrt[4]{p})$ has been determined by Parry and Lemmermeyer when $p \not\equiv \pm 1\bmod 16$. In this paper, we improve the known results in the case $p\equiv \pm 1\bmod 16$. In particular, we determine all primes $p$ such that $4$ does not divide the class number of $\mathbb{Q}(\sqrt[4]{p})$. We also conjecture a relation between the class numbers of $\mathbb{Q}(\sqrt[4]{p})$ and $\mathbb{Q}(\sqrt{-2p})$. We show that this conjecture implies a distribution result of the $2$-class numbers of $\mat…
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.
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。
Let $K=\mathbb{Q}(\sqrt[4]{pd^{2}})$ be a real pure quartic number field and $k=\mathbb{Q}(\sqrt{p})$ its real quadratic subfield, where $p\equiv 5\pmod 8$ is a prime integer and $d$ an odd square-free integer coprime to $p$. In this work, we calculate $r_2(K)$, the $2$-rank of the class group of $K$, in terms of the number of prime divisors of $d$ that decompose or remain inert in $\mathbb{Q}(\sqrt{p})$, then we will deduce forms of $d$ satisfying $r_2(K)=2$. In the last case, the $4$-rank of the class group of $K$ is given too.
The Thue-Siegel method is applied to derive an upper bound for the number of solutions to Thue's equation $F(x,y) = 1$ where $F$ is a quartic diagonalizable form with negative discriminant. Computation is used in this argument to handle forms whose discriminant is small in absolute value. We then apply our results to bound the number of integral points on a certain family of elliptic curves.
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。
In this paper the number of $\bar{\mathbb{F}}_q$-isomorphism classes of general Jacobi quartic curves, i.e., the number of general Jacobi quartic curves with distinct $j$-invariants, over the finite field $\mathbb{F}_q$ is enumerated.
We prove an upper bound for $\ell$-torsion in class groups of almost all fields in certain families of $D_4$-quartic fields. Our key tools are a new Chebotarev density theorem for these families of $D_4$-quartic fields and a lower bound for the number of fields in the families.
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…
We establish an asymptotic formula for the number of integral solutions of bounded height for pairs of diagonal quartic equations in $26$ or more variables. In certain cases, pairs in $25$ variables can be handled.
Let $Y$ be the complement of a plane quartic curve $D$ defined over a number field. Our main theorem confirms the Lang-Vojta conjecture for $Y$ when $D$ is a generic smooth quartic curve, by showing that its integral points are confined in a curve except for a finite number of exceptions. The required finiteness will be obtained by reducing it to the Shafarevich conjecture for K3 surfaces. Some variants of our method confirm the same conjecture when $D$ is a reducible generic quartic curve which consists of four lines, two lines and a conic, or two conics.
On $ p-$Rationality of Cubic and Quartic Number Fields
本文提出新的判据，通过三次递推序列的p-可除性判断某些复三次数字段的p-理性性，并探讨广义abc猜想与p-理性性的关系，获得满足Greenberg广义猜想的显式数域。
We determine the odd order torsion subgroup of the Brauer group of diagonal quartic surfaces over the field of rational numbers. We show that a non-constant Brauer element of odd order always obstructs weak approximation but never the Hasse principle.
$6$-torsion and integral points on quartic threefolds
该研究证明了二次域类群6-挠点平均值的匹配上下界，并完成了仿射四次三维簇上整数解的数量计数，为数论相关领域提供了新的结论。
We prove that every elliptic curve defined over a totally real number field of degree 4 not containing $\sqrt{5}$ is modular. To this end, we study the quartic points on four modular curves.
Sum of two squares in cyclic quartic fields
本文提出一种基于二次型迷向性局部-整体原则的算法，用于判定循环四次域中的数是否为两平方数之和。
To any quartic $D_4$ extension of $\mathbb{Q}$, one can associate the Artin conductor of a 2-dimensional irreducible representation of the group. Altŭg, Shankar, Varma, and Wilson determined the asymptotic number of such fields when ordered by conductor. We refine this, realizing a secondary term and power saving error term.
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。
Pythagoras number of quartic orders containing $\sqrt{2}$
本文证明了任何包含 $\sqrt{2}$ 的四次数域中相应序的毕达哥拉斯数至多为 5，从而证实了 Krásenský、Raška 和 Sgallová 的猜想，主要工具是 Beli 的范数生成元基理论。
The Eichler--Selberg trace formula for Hilbert cusp forms, the class numbers of quartic CM fields, and their distributions
利用广义Hurwitz类数（由四次CM域类数定义），建立了窄类数为一的实二次域上Hilbert尖形式的Eichler-Selberg迹公式，并研究了类数分布与关系，进行了Hecke算子迹的数值计算。
Let $K$ be a quartic CM field, that is, a totally imaginary quadratic extension of a real quadratic number field. In a 1962 article titled On the classfields obtained by complex multiplication of abelian varieties, Shimura considered a particular family $\{F_K(m) : m \in \mathbb{Z} >0 \}$ of abelian extensions of $K$, and showed that the Hilbert class field $H_K$ of $K$ is contained in $F_K(m)$ for some positive integer m. We make this m explicit. We then give an algorithm that computes a set of defining polynomials for the Hilbert class field using the field $F_K(m)$. Our proof-of-concept imp…
Well-Rounded ideal lattices of cyclic cubic and quartic fields
本文研究了循环三次和四次域具有良圆理想格的判据，证明了循环三次域必含良圆理想，构造了循环四次域的良圆理想族，并给出了特定素理想为良圆的充要条件及奇素数分解。
It is a classical problem in algebraic number theory to decide if a number field is monogeneous, that is if it admits power integral bases. It is especially interesting to consider this question in an infinite parametric familiy of number fields. In this paper we consider the infinite parametric family of simplest quartic fields $K$ generated by a root $ξ$ of the polynomial $P_t(x)=x^4-tx^3-6x^2+tx+1$, assuming that $t>0$, $t\neq 3$ and $t^2+16$ has no odd square factors. In addition to generators of power integral bases we also calculate the minimal index and all elements of minimal index in …
On indices and monogenity of quartic number fields defined by quadrinomials
本文研究由不可约四项式生成的四次数域的指数，给出其被2或3整除的充分条件并确定指数中2、3的精确幂次，同时构造出新的无限族单基因四次数域。
Let $t$ be any imaginary quadratic integer with $|t|\geq 100$. We prove that the inequality \[ |F_t(X,Y)| = | X^4 - t X^3 Y - 6 X^2 Y^2 + t X Y^3 + Y^4 | \leq 1 \] has only trivial solutions $(x,y)$ in integers of the same imaginary quadratic number field as $t$. Moreover, we prove results on the inequalities $|F_t(X,Y)| \leq C|t|$ and $|F_t(X,Y)| \leq |t|^{2 -\varepsilon}$. These results follow from an approximation result that is based on the hypergeometric method. The proofs in this paper require a fair amount of computations, for which the code (in Sage) is provided.
It is a classical problem in algebraic number theory to decide if a number field admits power integral bases and further to calculate all generators of power integral bases. This problem is especially delicate to consider in an infinite parametric family of number fields. In the present paper we investigate power integral bases in the infinite parametric family of pure quartic fields $K=Q(\sqrt[4]{m})$. We often pointed out close connection of various types of Thue equations with calculating power integral bases. In this paper we determine power integral bases in pure quartic fields by reducin…
Quartic surface, its bitangents and rational points
研究四次曲面的双切线和有理点的有限性，利用四次双固有的几何方法，证明了相关点集的 Zariski 密度性质。
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…
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。
We give more evidence for Patterson's conjecture on sums of exponential sums, by getting an asymptotic for a sum of quartic exponential sums over $\Q[i].$ Previously, the strongest evidence of Patterson's conjecture over a number field is the paper of Livné and Patterson \cite{LP} on sums of cubic exponential sums over $\Q[ω], ω^3=1.$ The key ideas in getting such an asymptotic are a Kuznetsov-like trace formula for metaplectic forms over a quartic cover of $GL_2,$ and an identity on exponential sums relating Kloosterman sums and quartic exponential sums. To synthesize the spectral theory and …
It is shown that the quartic Fermat equation $x^4 +y^4=1$ has nontrivial integral solutions in the Hilbert class field $Σ$ of any quadratic field $K=\mathbb{Q}(\sqrt{-d})$ whose discriminant satisfies $-d \equiv 1$ (mod 8). A corollary is that the quartic Fermat equation has no nontrivial solution in $K=\mathbb{Q}(\sqrt{-p})$, for $p$ $( > 7)$ a prime congruent to $7$ (mod 8), but does have a nontrivial solution in the odd degree extension $Σ$ of $K$. These solutions arise from explicit formulas for the points of order 4 on elliptic curves in Tate normal form. The solutions are studied in deta…
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。
On the monogenity of quartic number fields defined by $x^4+ax^2+b$
本文研究了由不可约三项式 $x^4+ax^2+b$ 的根生成的四次数域的单生性，刻画了 $Z[α]$ 整闭的条件，给出了 $p=2,3$ 时整除公因子的最高幂次，并证明了此类域中仅存在唯一的幂整基生成元。
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。
A Classification of Multiply Monogenic Quartic Orders
本文针对所有四次数域，证明了不属于Bérczes等人定义的两类特定类型的二重单生$\u2124$-序仅有有限多个，完善了此前需附加伽罗瓦群条件的相关结论。
We present equations of $800$ conics on a Mukai's smooth quartic which is an explicit model of a K3 surface recently considered by Alex Degtyarev. The number $800$ is a current record for the number of irreducible conics on a smooth quartic in ordinary projective space.
Let $\mathbb{F}_q$ be the finite field of $q=p^m\equiv 1\pmod 4$ elements with $p$ being an odd prime and $m$ being a positive integer. For $c, y \in\mathbb{F}_q$ with $y\in\mathbb{F}_q^*$ non-quartic, let $N_n(c)$ and $M_n(y)$ be the numbers of zeros of $x_1^4+...+x_n^4=c$ and $x_1^4+...+x_{n-1}^4+yx_n^4=0$, respectively. In 1979, Myerson used Gauss sum and exponential sum to show that the generating function $\sum_{n=1}^{\infty}N_n(0)x^n$ is a rational function in $x$ and presented its explicit expression. In this paper, we make use of the cyclotomic theory and exponential sums to show that …
Given a number field $k$ and a finitely generated subgroup $\mathcal{A} \subseteq k^*$, we study the distribution of $S_4$-quartic extensions of $k$ such that the elements of $\mathcal{A}$ are norms. We show that the density of such extensions is the product of so-called "local masses" at every place of $k$. We give these local masses explicitly in almost all cases and give an algorithm for computing the remaining cases.
Squares in arithmetic progression over certain non-primitive quartic number fields
本文研究了特定非原始四次数字段上等差数列中的平方问题，通过特征化二次扩张Q(√D)上的非常数平方等差数列来探讨相关数论性质。
We give an explicit necessary condition for pairs of orders in a quartic CM-field to have the same polarised class group. This generalises a simpler result for imaginary quadratic fields. We give an application of our results to computing endomorphism rings of abelian surfaces over finite fields, and we use our results to extend a completeness result of Murabayashi and Umegaki to a list of abelian surfaces over the rationals with complex multiplication by arbitrary orders.
Let $p$ be a prime number such that $p=2$ or $p\equiv 1\pmod 4$. Let $\varepsilon_p$ denote the fundamental unit of $\mathbb{Q}(\sqrt{p})$ and let $a$ be a positive square-free integer. The main aim of this paper is to determine explicitly the Hilbert genus field of the imaginary cyclic quartic fields of the form $\mathbb{Q}(\sqrt{-a\varepsilon_p\sqrt{p}})$.
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