Stephan Baier, Matthew P. Young
We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large-sieve type result for order three (and six) Dirichlet characters.
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This is an expository paper, giving a simplified proof of the cubic case of the main conjecture for Vinogradov's mean value theorem.
We prove that, on average, the monogenicity or $n$-monogenicity of a cubic field has an altering effect on the behavior of the 2-torsion in its class group.
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 obtain an asymptotic formula for the mean value of L-functions associated to cubic characters over F_q[t]. We solve this problem in the non-Kummer setting when q=2 (mod 3) and in the Kummer case when q=1 (mod 3). The proofs rely on obtaining precise asymptotics for averages of cubic Gauss sums over function fields, which can be studied using the theory of metaplectic Eisenstein series. In the non-Kummer setting we display some explicit cancellation between the main term and the dual term coming from the approximate functional equation of the L-functions.
We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument $αx^3+βx$. The first substantial advance in this topic for over 60 years, this leads to improvements in Heath-Brown's variant of Weyl's inequality, and other applications of Diophantine type.
Given any family of cubic fields defined by local conditions at finitely many primes, we determine the mean number of 2-torsion elements in the class groups and narrow class groups of these cubic fields when ordered by their absolute discriminants. For an order $\cal O$ in a cubic field, we study the three groups: $\rm Cl_2(\cal O)$, the group of ideal classes of $\cal O$ of order 2; $\rm Cl^+_2(\cal O)$, the group of narrow ideal classes of $\cal O$ of order 2; and ${\cal I}_2(\cal O)$, the group of ideals of $\cal O$ of order 2. We prove that the mean value of the difference $|\rm Cl_2({\cal…
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。
We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree $3$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le 6$. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding $2$.
By means of the Hardy-Littlewood method, we apply a new mean value theorem for exponential sums to confirm the truth, over the rational numbers, of the Hasse principle for pairs of diagonal cubic forms in thirteen or more variables.
This paper contains an account of arbitrary cubic function fields of characteristic three. We define a standard form for an arbitrary cubic curve and consider its function field. By considering an integral basis for the maximal order of these function fields, we are able to calculate the field discriminant and the genus. We also describe the splitting behavior of any place, and give composition and reduction algorithms for arithmetic in the ideal class group.
The Quadratic and Cubic Characters of 2
本文研究了2的二次与三次特征，通过引入埃森斯坦整数、高斯和雅可比和以及三次互反律，探讨了二次同余式和三次同余式的可解性问题。
An elementary approach is shown which derives the value of the Gauss sum of a cubic character over a finite field $\mathbb F_{2^s}$ without using Davenport-Hasse's theorem (namely, if $s$ is odd the Gauss sum is -1, and if $s$ is even its value is $-(-2)^{s/2}$).
Cubic arithmetic-geometric mean and isogenies of Hesse elliptic curves
本文在特征非三的任意域上构造Hesse椭圆曲线的三次同源，对应三次算术几何平均的一步，并应用于特征三中普通椭圆曲线的Serre-Tate典范提升，同时探讨与超几何函数的联系。
We establish the two-dimensional asymptotic distributions of the logarithm and logarithmic derivative of $L$-functions associated with a family of cubic Hecke characters. A crucial ingredient in the proof of our main result is an exponential decay estimate for the characteristic functions of the distributions.
An elementary approach is shown which derives the values of the Gauss sums over $\mathbb F_{p^r}$, $p$ odd, of a cubic character without using Davenport-Hasse's theorem. New links between Gauss sums over different field extensions are shown in terms of factorizations of the Gauss sums themselves, which are then rivisited in terms of prime ideal decompositions. Interestingly, one of these results gives a representation of primes $p$ of the form $6k+1$ by a binary quadratic form in integers of a subfield of the cyclotomic field of the $p$-th roots of unity.
L-values of elliptic curves twisted by cubic characters
本文研究解析秩为零的有理椭圆曲线在三次狄利克雷特征扭曲下的中心代数L值，利用整性导出的同余关系，给出了分母的q进赋值、BSD不变量表示及模q渐近密度等算术结果。
Non-vanishing of Cubic Dirichlet L-Functions over the Eisenstein Field
本文针对艾森斯坦域上的本原三次狄利克雷特征族，建立了L函数第一矩的渐近公式和第二矩的上界，无条件证明了该族中无穷多个L函数在中心点不消没。
We study the one-level density for families of L-functions associated with cubic Dirichlet characters defined over the Eisenstein field. We show that the family of $L$-functions associated with the cubic residue symbols $χ_n$ with $n$ square-free and congruent to 1 modulo 9 satisfies the Katz-Sarnak conjecture for all test functions whose Fourier transforms are supported in $(-13/11, 13/11)$, under GRH. This is the first result extending the support outside the \emph{trivial range} $(-1, 1)$ for a family of cubic L-functions. This implies that a positive density of the L-functions associated w…
Coset-refined trace statistics, nodal characters, and affine branches in cubic norm tori
本文研究了立方范数托里的余类细化迹统计、节点特征以及立方范数托里的affine分支，证明了余类细化迹定理，并探讨了迹和范数曲线上的几何输入及其对局部分支理论的影响。
The mean number of $2$-torsion elements in the class groups of cubic orders
本文确定了三次序类群中2-挠元的平均个数，并将其推广到任意奇数维情形，给出了可约SL_N(Z)-轨道的渐近计数公式。
Murmurations of quadratic and cubic characters over function fields
该研究计算函数域上二次特征族及 Kummer 框架下原始三次特征薄族的 murmuration 密度，关联其期望与超椭圆曲线模空间稳定同调，获二次 L 函数中心点非零结果。
The value-distribution of Artin $L$-functions associated with cubic fields in conductor aspect
本文研究了导子方面非Galois三次域族Artin $L$-函数的值分布，证明其均值可由可显式构造的密度函数积分表示，并借助类数公式应用于三次域类数分布的研究。
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。
Spacing statistics for a point scatterer on the cubic three-torus
研究立方三维环面上点散射体的新特征值间距统计，证明其收敛于独立于参数的分布，并刻画了小间隙渐近形式。
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