Geoff Bailey, Stephen D. Cohen, Nicole Sutherland, Tim Trudgian
With $\Fq$ the finite field of $q$ elements, we investigate the following question. If $γ$ generates $\Fqn$ over $\Fq$ and $β$ is a non-zero element of $\Fqn$, is there always an $a \in \Fq$ such that $β(γ+ a)$ is a primitive element? We resolve this case when $n=3$, thereby proving a conjecture by Cohen. We also improve substantially on what is known when $n=4$.
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In this paper we determine the group of rational automorphisms of binary cubic and quartic forms with integer coefficients and non-zero discriminant in terms of certain quadratic covariants of cubic and quartic forms. This allows one to give precise asymptotic formulae for the number of integers in an interval representable by a binary cubic or quartic form and extends work of Hooley. Further, we give the field of definition of lines contained in certain cubic and quartic surfaces related to binary cubic and quartic forms.
Proof of a Conjecture on Primitive Quartic Polynomials over Finite Fields
针对有限域上本原四次多项式的猜想，本文在q足够大时，借助Fu和Wan的定理，通过AI辅助完成了证明。
We establish the non-singular Hasse Principle for pairs of diagonal quartic equations in 22 or more variables. Our methods involve the estimation of a certain entangled two-dimensional 21st moment of quartic smooth Weyl sums via a novel cubic moment of Fourier coefficients.
The first moment of central value of primitive quartic $L$-functions with fixed genus
在非Kummer设定下，利用双Dirichlet级数研究$\mathbb{F}_q(T)$上原始四次$L$-函数第一个矩的均值，得到误差项$q^{(\frac{3}{5}+\varepsilon)g}$。
In this article, we give a largely self-contained proof that the quartic extension $\mathbb{F}_{q^4}$ of the finite field $\mathbb{F}_q$ contains a primitive element $α$ such that the element $α+α^{-1}$ is also a primitive element of ${\mathbb{F}_{q^4}},$ and $Tr_{\mathbb{F}_{q^4}|\mathbb{F}_{q}}(α)=a$ for any prescribed $a \in \mathbb{F}_q$. The corresponding result for finite field extensions of degrees exceeding 4 has already been established by Gupta, Sharma and Cohen.
Shifted moments of cubic and quartic Dirichlet $L$-functions
本文在广义黎曼猜想下，建立了三次和四次狄利克雷L函数位移矩的上界，并应用此结果给出了三次和四次狄利克雷特征和的矩界。
We evaluate the average of cubic and quartic Dirichlet character sums with the modulus going up to a size comparable to the length of the individual sums. This generalizes a result of Conrey, Farmer and Soundararajan on quadratic Dirichlet character sums.
We study the $2k$-th moment of central values of the family of primitive cubic and quartic Dirichlet $L$-functions. We establish sharp lower bounds for all real $k \geq 1/2$ unconditionally for the cubic case and under the Lindelöf hypothesis for the quartic case. We also establish sharp lower bounds for all real $0 \leq k<1/2$ and sharp upper bounds for all real $k \geq 0$ for both the cubic and quartic cases under the generalized Riemann hypothesis (GRH). As an application of our results, we establish quantitative non-vanishing results for the corresponding $L$-values.
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.
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.
We are interested in solving the congruences $f^3+g^3+1\equiv 0\pmod{fg}$ and $f^4-4g^2+4\equiv 0\pmod{fg}$ in polynomials $f, g$ with rational coefficients. Moreover, we present results of computations of all integer points on certain one parametric curves of genus 1 and 3, related to cubic and quartic fields, respectively.
Large values of cubic and quartic Dirichlet $L$-functions of prime moduli
本文在广义黎曼假设下，应用谐振子方法研究素数模的三次与四次Dirichlet L-函数族，展示其在中心点处的大值。
We show that there are $O_\varepsilon(H^{1.5+\varepsilon})$ monic, cubic polynomials with integer coefficients bounded by $H$ in absolute value whose Galois group is $A_3$. We also show that the order of magnitude for $D_4$ quartics is $H^2 (\log H)^2$, and that the respective counts for $A_4$, $V_4$, $C_4$ are $O(H^{2.91})$, $O(H^2 \log H)$, $O(H^2 \log H)$. Our work establishes that irreducible non-$S_3$ cubic polynomials are less numerous than reducible ones, and similarly in the quartic setting: these are the first two solved cases of a 1936 conjecture made by van der Waerden.
We construct a collection of matrices defined by quadratic residue symbols, termed "quadratic residue matrices", associated to the splitting behavior of prime ideals in a composite of quadratic extensions of $\mathbb{Q}$, and prove a simple criterion characterizing such matrices. We also study the analogous classes of matrices constructed from the cubic and quartic residue symbols for a set of prime ideals of $\mathbb{Q}(\sqrt{-3})$ and $\mathbb{Q}(i)$, respectively.
Investigations of monogenity and power integral bases were recently extended from the absolute case (over Q) to the relative case (over algebraic number fields). Formerly, in the relative case we only succeeded to calculate generators of power integral bases when the ground field is an imaginary quadratic field. This is the first case when we consider monogenity in the more difficult case, in extensions of real quadratic fields. We give efficient algorithms for calculating generators of power integral bases in cubic and quartic extensions of real quadratic fields, more exactly in composites of…
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.
Non-vanishing of primitive cubic, quartic and sextic L-functions
该研究在非库默尔情形下，通过适配Gao和Zhao的截断指数方法，改进了三次狄利克雷$L$-函数的非零比例下界，对四次与六次族得到正下界及六次一阶矩渐近公式，所有结果均无条件成立。
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。
One of the main themes in this thesis is the description of the signature of both the infinite place and the finite places in cubic function fields of any characteristic and quartic function fields of characteristic at least 5. For these purposes, we provide a new theory which can be applied to cubic and quartic function fields and to even higher dimensional function fields. One of the striking advantages of this theory to other existing methods is that is does not use the concept of p-adic completions and we can dispense of Cardano's formulae. Another key result comprises the construction of …
We show that a quartic $p$-adic form with at least $3192$ variables possesses a non-trivial zero. We also prove new results on systems of cubic, quadratic and linear forms. As an example, we show that for a system comprising two cubic forms $132$ variables are sufficient.
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。
For a given group $G$ and an elliptic curve $E$ defined over a number field $K$, I discuss the problem of finding $G$-extensions of $K$ over which $E$ gains rank. I prove the following theorem, extending a result of Fearnley, Kisilevsky, and Kuwata: Let $n = 3,4,$ or $6$. If $K$ contains its $n^{th}$-roots of unity then, for any elliptic curve $E$ over $K$, there are infinitely many $\mathbb{Z}/n\mathbb{Z}$-extensions of $K$ over which $E$ gains rank.
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。
Non-orthogonality of the cubic and quartic large sieves via Rankin-Selberg
研究三次和四次大筛法的非正交性，利用久保田狄利克雷级数二阶矩平均林德勒夫上界及亚纯θ函数兰金 - 塞尔伯格卷积傅里叶系数平均下界，采用扎吉尔正则化方法，还对特定赫克特征族提出猜想。
Let $k$ be a cubic field. We give an explicit formula for the Dirichlet series $\sum_K|\Disc(K)|^{-s}$, where the sum is over isomorphism classes of all quartic fields whose cubic resolvent field is isomorphic to $k$. Our work is a sequel to an unpublished preprint of Cohen, Diaz y Diaz, and Olivier, and we include complete proofs of their results so as not to rely on unpublished work. This is a companion to a previous paper where we compute the Dirichlet series associated to cubic fields having a given quadratic resolvent.
We show that for an elliptic divisibility sequence on a twist of the Fermat cubic, u^3+v^3=m, with m cube-free, all the terms beyond the first have a primitive divisor.
We give a bijection between binary quartic forms and quartic rings with a monogenic cubic resolvent ring, relating the rings associated to binary quartic forms with Bhargava's cubic resolvent rings. This gives a parametrization of quartic rings with monogenic cubic resolvents. We also give a geometric interpretation of this parametrization.
The families of simplest cubic, simplest quartic and simplest sextic fields and the related Thue equations are well known. The family of simplest cubic Thue equations was already studied in the relative case, over imaginary quadratic fields. In the present paper we give a similar extension of simplest quartic and simplest sextic Thue equations over imaginary quadratic fields. We explicitly give the solutions of these infinite parametric families of Thue equations over arbitrary imaginary quadratic fields.
On quadratic character sums over quartics
本文研究了多项式变元的二次特征和问题，推导出了四次和三次多项式变元的二次特征和的变换公式。
Cubic and quartic points on modular curves using generalised symmetric Chabauty
针对 Zureick-Brown 的问题，提出部分相对对称 Chabauty 方法，确定多个模曲线的三次点及 X_0(65) 的四次点，并严格计算其雅可比的有理 Mordell--Weil 群。
Twisted second moment of primitive cubic L-functions
本文研究非Kummer设定下$\mathbb{F}_q(T)$上本原三次L-函数扭曲二阶矩的均值，采用双重Dirichlet级数方法建立了相应的误差项。
One can define class invariants for a quartic primitive CM field K as special values of certain Siegel (or Hilbert) modular functions at CM points corresponding to K. We provide explicit bounds on the primes appearing in the denominators of these algebraic numbers. This allows us, in particular, to construct S-units in certain abelian extensions of K, where S is effectively determined by K. It also yields class polynomials for primitive quartic CM fields whose coefficients are S-integers.
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.
$K_2$ of families of elliptic curves over non-Abelian cubic and quartic fields
本文给出了非阿贝尔三次和四次域上椭圆曲线族的两种构造，在驯顺符号核中分别获得三个和四个线性无关的“积分”元素，证明了不忽略挠元的新积分性判据，并对90余条曲线数值验证了 Beilinson 猜想。
Let $\mathbb{F}_q$ denote the finite field with $q$ elements. In this work, we use characters to give the number of rational points on suitable curves of low degree over $\mathbb{F}_q$ in terms of the number of rational points on elliptic curves. In the case where $q$ is a prime number, we give a way to calculate these numbers. As a consequence of these results, we characterize maximal and minimal curves given by equations of the forms $ax^3+by^3+cz^3=0$ and $ax^4+by^4+cz^4=0$.
We use the method of Shintani, as developed by Taniguchi and Thorne, to prove the joint cuspidal equidistribution of the shape of quartic fields paired with the shape of its cubic resolvent, when the fields are ordered by discriminant. Our estimate saves a small power in the corresponding Weyl sums.
We study the first moments of central values of Hecke $L$-functions associated with quadratic, cubic and quartic symbols to prime moduli. This also enables us to obtain results on first moments of central values of certain families of cubic and quartic Dirichlet $L$-functions of prime moduli.
Trinomial Planar Functions on Cubic and Quartic Extensions of Finite Fields
本文研究有限域三次和四次扩域上的平面三项式，利用二次型联系与经典代数工具取得成果，并提出可推导更多扩域平面函数的通用方法。
In this paper we prove an explicit formula for the arithmetic intersection number (CM(K).G1)_{\ell} on the Siegel moduli space of abelian surfaces, generalizing the work of Bruinier-Yang and Yang. These intersection numbers allow one to compute the denominators of Igusa class polynomials, which has important applications to the construction of genus 2 curves for use in cryptography. Bruinier and Yang conjectured a formula for intersection numbers on an arithmetic Hilbert modular surface, and as a consequence obtained a conjectural formula for the intersection number (CM(K).G1)_{\ell} under str…
Solvable points on intersections of quadrics, cubics, and quartics
本文建立多项式下界，使得在二次、三次和四次超曲面交线X上存在稠密的可解点集，方法是将极超曲面经典理论与X上j维线性子空间的Fano簇联系起来。
Mean value of cubic $L$-funcitons with fixed genus
本文在非Kummer设定下，利用双重Dirichlet级数研究了 $\mathbb{F}_q(T)$ 上固定亏格本原三次 $L$-函数一阶矩的均值，并导出了大小为 $q^{(\frac{7}{8}+\varepsilon)g}$ 的误差项。
Three-Torsion Subgroups and Wild Conductor Exponents of Plane Quartics
本文提出算法，通过三次曲线与方程组描述并计算平面四次曲线雅可比簇的3-挠子群，并利用其伽罗瓦结构计算局部野导子指数。
We prove that for any prime power $q\notin\{3,4,5\}$, the cubic extension $\mathbb{F}_{q^3}$ of the finite field $\mathbb{F}_q$ contains a primitive element $ξ$ such that $ξ+ξ^{-1}$ is also primitive, and $\textrm{Tr}_{\mathbb{F}_{q^3}/\mathbb{F}_q}(ξ)=a$ for any prescribed $a\in\mathbb{F}_q$. This completes the proof of a conjecture of Gupta, Sharma, and Cohen concerning the analogous problem over an extension of arbitrary degree $n\ge3$.
Secondary terms in the counting functions of quartic fields II
本文确定了满足局部条件的判别式有界 $S_4$-四次域的平滑计数，表示为两个主项之和并带有幂次节省误差项，并证明了四次环的类似结果及 Shintani zeta 函数的极点性质。
We show that a positive proportion of quartic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof builds on the corresponding result for cubic fields that we obtained in a previous work. Along the way, we also prove that a positive proportion of quartic rings of integers do not arise as the invariant order of an integral binary quartic form despite having no local obstruction.
It is shown that the sum of class numbers of orders in totally complex quartic fields with no real quadratic subfield obeys an asymptotic law similar to the prime numbers, as the bound on the regulators tends to infinity. Here only orders which are maximal at a given set of primes containing an even number of elements are considered. The proof is accomplished by means of a Prime Geodesic Theorem for symmetric spaces formed as compact quotients of SL(4,R). This result extends work of Sarnak in the real quadratic case and of Deitmar and Hoffmann in the complex cubic case.
We study the number of elliptic curves, up to isomorphism, over a fixed quartic field $K$ having a prescribed torsion group $T$ as a subgroup. Let $T=\Z/m\Z \oplus \Z/n\Z$, where $m|n$, be a torsion group such that the modular curve $X_1(m,n)$ is an elliptic curve. Let $K$ be a number field such that there is a positive and finite number of elliptic curves $E_T$ over $K$ having $T$ as a subgroup. We call such pairs $(E_T, K)$ \emph{exceptional}. It is known that there are only finitely many exceptional pairs when $K$ varies through all quadratic or cubic fields. We prove that when $K$ varies t…
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