Three cubes in arithmetic progression over quadratic fields

We study the problem of the existence of arithmetic progressions of three cubes over quadratic number fields ${{\mathbb{Q}(\sqrt{D})}}We study the problem of the existence of arithmetic progressions of three cubes over quadratic number fields \mathbbQ(?D){{\mathbb{Q}(\sqrt{D})}}, where D is a squarefree integer. For this purpose, we give a characterization in terms of \mathbbQ(?D){{\mathbb{Q}(\sqrt{D})}}-rational points on the elliptic curve E : y ² = x ³ − 27. We compute the torsion subgroup of the Mordell–Weil group of this elliptic curve over \mathbbQ(?D){{\mathbb{Q}(\sqrt{D})}} and we give an explicit answer, in terms of D, to the finiteness of the free part of E(\mathbbQ(?D)){E({\mathbb{Q}(\sqrt{D})})} for some cases. We translate this task to computing whether the rank of the quadratic D-twist of the modular curve X ₀(36) is zero or not.     

Quaternionic Manin symbols, Brandt matrices, and Hilbert modular forms

In this paper, we propose a generalization of the algorithm we developed previously. Along the way, we also develop a theory of quaternionic -symbols whose definition bears some resemblance to the classical -symbols, except for their combinatorial nature. The theory gives a more efficient way to compute Hilbert modular forms over totally real number fields, especially quadratic fields, and we have illustrated it with several examples. Namely, we have computed all the newforms of prime levels of norm less than 100 over the quadratic fields and , and whose Fourier coefficients are rational or are defined over a quadratic field.

     

New identities involving sums of the tails related to real quadratic fields

In previous work, the authors discovered new examples of q-hypergeometric series related to the arithmetic of $\mathbb {Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ . Building on this work, we construct in this paper sum of the tails identities for which some which some of these functions occur as error terms. As an application, we obtain formulas for the generating function of a certain zeta functions for real quadratic fields at negative integers.     

A note on the 3-rank of quadratic fields

The difference between the 3-rank of the ideal class group of an imaginary quadratic field and that of the associated real quadratic field is equal to 0 or 1. In this note, we give an infinite family of examples in each case.Received: 9 September 2002     

A fast, rigorous technique for computing the regulator of a real quadratic field

We present a new algorithm for computing the regulator of a real quadratic field based on an algorithm for unconditionally verifying the correctness of the regulator produced by a subexponential algorithm, that runs in expected time under the Generalized Riemann Hypothesis. The correctness of our algorithm relies on no unproven hypotheses and is currently the fastest known unconditional algorithm for computing the regulator. A number of implementation issues and performance enhancements are discussed, and we present the results of computations demonstrating the efficiency of the new algorithm.

     

限制零的m进制数集之Hausdorff测度

对于任意整数表示mkz的分数部分.给出了数集Fm的Hausdorff测度是Fm的HausdorfF维数.     

Normal bases of ray class fields over imaginary quadratic fields

We develop a criterion for a normal basis (Theorem 2.4), and prove that the singular values of certain Siegel functions form normal bases of ray class fields over imaginary quadratic fields other than ${\mathbb{Q}(\sqrt{-1})}$ and ${\mathbb{Q}(\sqrt{-3})}$ (Theorem 4.2). This result would be an answer for the Lang-Schertz conjecture on a ray class field with modulus generated by an integer (≥2) (Remark 4.3).     

Class numbers of quadratic fields and

Let be a square free integer. We shall show that there exist infinitely many positive fundamental discriminants 0$"> with a positive density such that the class numbers of quadratic fields and are both not divisible by 3.

     

An eigenform of weight one expressed by meromorphic η-quotients

In this paper, we construct an eigenform of weight one by using meromorphic η-quotients. It is the modular form associated to an irreducible representation r: Gal(L/\mathbb Q)? GL₂(\mathbb C)\rho : {\rm Gal}(L/\mathbb {Q})\rightarrow {\rm GL}_{2}(\mathbb {C}) with L=\mathbb Q(?{-3},³?{3})L=\mathbb {Q}(\sqrt{-3},\sqrt[3]{3}).     

A mean value theorem for the Dedekind zeta-function of a quadratic number field

Let \(K = \mathbb{Q}(\sqrt d )\) be any quadratic number field with discriminantd. ζ _K (s) denotes the Dedekind zeta-function. The purpose of this note is to prove the following asymptotic formula: $$\int\limits_0^T {|\zeta _K ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + it)|^2 dt = ({6 \mathord{\left/ {\vphantom {6 {\pi ^2 }}} \right. \kern-\nulldelimiterspace} {\pi ^2 }})} \prod\limits_{p/d} {(1 + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p})^{ - 1} \cdot R_K^2 \cdot T \cdot \log ^2 T + O_\varepsilon \left\{ {\left| d \right|1 + \varepsilon \cdot T \cdot \log T} \right\},} $$ where the implied constant depends only on ε. HereR _K, denotes the residue of ζ _K (s) ats=1.     

关于一类实二次域的类数与基本单位的上界

设素数P≡1(mod4),k,ε分别表示实二次域Q(p~(1/2))类数和基本单位．本文改进了类数h和基本单位ε的上界,证明了:hlogeε<1/4(p~(1/2) 6)log(2ep~(1/2)),并得到了几个重要的推论．     

Brauer groups of genus zero extensions of number fields

We determine the isomorphism class of the Brauer groups of certain nonrational genus zero extensions of number fields. In particular, for all genus zero extensions of the rational numbers that are split by , .

     

A functional equation related to the product in a quadratic number field

The functional equation $$f(x_{1},y_{1})f(x_{2},y_{2})=f(x_{1}x_{2}+\alpha y_{1}y_{2},x_{1}y_{2}+x_{2}y_{1}),\ (x_{1},y_{1}),\,(x_{2},y_{2})\in \mathbb{ R}^{2}$$ arises from the formula for the product of two numbers in the quadratic field ${\mathbb{Q}(\sqrt{\alpha})}$ . The general solution ${f:\mathbb{R}\rightarrow \mathbb{R}}$ to this equation is determined. Moreover, it is shown that no more general equations arise from a change of basis in the field.     

在${\mathbb{R}}^m$中重对数广义律的精确速率

令\{$X$, $X_n$, $n\ge 1$\}是期望为${\mathbb{E}}X=(0,\ldots,0)_{m\times 1}$和协方差阵为${\rm Cov}(X,X)=\sigma^2I_m$的独立同分布的随机向量列, 记$S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. 对任意$d>0$和$a_n=o((\log\log n)^{-d})$, 本文研究了${{\mathbb{P}}(|S_n|\ge (\varepsilon+a_n)\sigma \sqrt{n}(\log\log n)^d)$的一类加权无穷级数的重对数广义律的精确速率.     

单圈图的最大特征值的上界的改进

Let G（V, E） be a unicyclic graph, Cm be a cycle of length m and Cm G, and ui ∈ V（Cm）. The G - E（Cm） are m trees, denoted by Ti, i = 1, 2,..., m. For i = 1, 2,..., m, let eui be the excentricity of ui in Ti and ec = max{eui ： i = 1, 2 , m}. Let κ = ec＋1. Forj = 1,2,...,k- 1, let δij = max{dv ： dist（v, ui） = j,v ∈ Ti}, δj = max{δij ： i = 1, 2,..., m}, δ0 = max{dui ： ui ∈ V（Cm）}. Then λ1（G）≤max{max 2≤j≤k-2 （√δj-1-1＋√δj-1）,2＋√δ0-2,√δ0-2＋√δ1-1}. If G ≌ Cn, then the equality holds, where λ1 （G） is the largest eigenvalue of the adjacency matrix of G.     

Quadratic equations over finite fields and class numbers of real quadratic fields

Letp be an odd prime and the finite field withp elements. In the present paper we shall investigate the number of points of certain quadratic hypersurfaces in the vector space and derive explicit formulas for them. In addition, we shall show that the class number of the real quadratic field (wherep1 (mod 4)) over the field of rational numbers can be expressed by means of these formulas.     

Convergence of Solutions of General Dispersive Equations Along Curve

In this paper, the authors give the local L~2 estimate of the maximal operator S_(φ,γ)~* of the operator family {S_(t,φ,γ)} defined initially by ■which is the solution(when n = 1) of the following dispersive equations(~*) along a curve γ:■where φ : R~+→R satisfies some suitable conditions and φ((-?)~(1/2)) is a pseudo-differential operator with symbol φ(|ξ|). As a consequence of the above result, the authors give the pointwise convergence of the solution(when n = 1) of the equation(~*) along curve γ.Moreover, a global L~2 estimate of the maximal operator S_(φ,γ)~* is also given in this paper.     

Heuristics for class numbers and lambda invariants

Let be an imaginary quadratic field and let be the associated real quadratic field. Starting from the Cohen-Lenstra heuristics and Scholz's theorem, we make predictions for the behaviors of the 3-parts of the class groups of these two fields as varies. We deduce heuristic predictions for the behavior of the Iwasawa -invariant for the cyclotomic -extension of and test them computationally.

     

Motives with exceptional Galois groups and the inverse Galois problem

We construct motivic ?-adic representations of $\textup {Gal}(\overline {\mathbb{Q}}/\mathbb{Q})$ into exceptional groups of type E ₇,E ₈ and G ₂ whose image is Zariski dense. This answers a question of Serre. The construction is uniform for these groups and is inspired by the Langlands correspondence for function fields. As an application, we solve new cases of the inverse Galois problem: the finite simple groups $E_{8}(\mathbb{F}_{\ell})$ are Galois groups over $\mathbb{Q}$ for large enough primes ?.