Number Theory and Its Applications to Discrete Mathematics

关于代数数域中整数环的结构、离散数学和编码理论的主题，我们邀请了一位外国合作研究者kim Hyun Kwang教授[浦项理工大学]，并支持了日本-韩国[2005，Kujyu]和韩国和日本[2006，KAIST]数论及其在相关领域的应用联合研讨会。我们项目的三个目标已经由调查人员完成如下：[05]有限次阿贝尔域上幂积分基的Hasse问题及类群结构的研究[j] . Hasse问题，首席研究员，上海大学。Peshawar)和Y.Motoda （Yatsushiro National College of Technology）给出了整数环具有幂积分基的阿贝尔场的新表征[JNT， 1986]。也就是说，如果一个域K是一个octic域Q(sqrt{mn}, sqrt{lm}, sqrt{l})，其中lmn是一个无平方整数，那么K除了一个5之外没有幂积分基…More ld [mn (Arch.Math.2004)]。因此，我们提出了一个开放性问题[同上]；问题。对于任何秩为3的octic 2-初等阿贝尔扩展K，如果环Z_K具有幂积分基，那么K是否与第24环切场Q（zeta_{24}）重合？最近，我们基于一种新的约简分类思想证明了上述问题[提交给九州数学. j .]。我们在2005年马赛的Journees Arithmetiques上介绍了主要部分。2003年10月，我们在名古屋大学举办了“Yokoi-Chowla猜想及相关问题”国际研讨会。最近这个猜想被匈牙利年轻的数学家A.Bir'o解决了。包括他在内，我们邀请了几位来自匈牙利、韩国和加拿大的教授，他们是数论和相关主题的高级研究者。我们发表了2003年名古屋会议论文集Yokoi-Chowla猜想及相关问题，由s.g atayama（Tokushima U.）编辑。C.Levesque (U。Laval, Qu'ebec)和T.Nakahara(Saga U.)，修订本。，其中包括15篇关于开放问题和没有精确证明的定理思想的论文[BKLN]。对于$u$ / $mathbb{Q}$中的一元不可约三次多项式$P(u)$，研究了由方程$w^3=P(u)$定义的平面曲线$E=E(P(u))$的算法。它是一条j不变量等于0的椭圆曲线。Miyake可以证明E$的简写形式是一条莫德尔曲线，y^1=x^3+k$，有一个由P(u)$的系数决定的有理数k$。还指出$E(P(u))$本质上依赖于多项式$P(u)$而不是三次域$K$，尽管$E[mathbb{Q}]$完全由三次域的子集$mathcal{W}(xi)$描述。事实上，我们证明了一般多项式$P(u)=u^3+tu+t，t在mathbb{Q}$中给出了这样的椭圆曲线的一个优势族。Katayama调查了两个主题。第一个主题是代数数字段的类数。他构造了一个p-1次的循环域族，它具有p-秩至少为2的理想类群。我们注意到$p等于1 mod 4$。（2005）数论在算术几何和代数几何中的应用。ichikawa计算了由共形场论导出的Teichm ' {u}ller群的单调表示，证明了在Schottky均化Riemann曲面上，任何0度的稳定向量束都是由Schottky群的线性表示得到的。并给出了肖特基群线性表示的模空间的Verlinde公式。田口证明了Fontaine-Mazur的有限性猜想和kare - moon的mod $p$有限性猜想的两个版本之间的逻辑关系。与此相关，我研究了伽罗瓦表示的模空间。接下来，他给出了经典θ级数的模恒等式的一个简单证明[CT（与Y.Choie联合工作）]。最后，他对某些模2伽罗瓦表示进行了分类，并推导了一些模形式的傅里叶系数的2进性质，其中涉及到一些经典算术或组合函数模2幂的同余性质的应用。[0405]数论在编码理论和离散数学中的应用苏哈拉建立了一种构造最小距离大于已知最小距离的厄密码的方法。他证明了二进制BCH码的简单列表解码是由欧几里得解码执行的。利用各种代数系统，我们开发了一种具有良好纠错能力的低密度奇偶校验码的构造方法。关于构造具有新冗余函数的RSA签名，Katayama研究了这些新RSA签名在乘法攻击下的安全性，并验证了新签名具有良好的安全性[YK]。少

On the Structure of the ring of integers in an algebraic number field, Discrete math and Coding theory of the main theme, we invited a foreign co-investigator Prof.Kim Hyun Kwang[Pohang Univ.of Science and Technology] and supported the Japan-Korea[2005, Kujyu] and Korea-Japan[2006, KAIST] Joint Seminar on Number Theory and its Application to the Related Area.The three aims of our project have been accomplished by the investigators as follows ;A0405)Investigation of Hasse's problem for the power integral bases and the structures of the class groups of abelian fields of finite degree over the rationalsOn Hasse's problem, the head investigator, S.I.A.Shah(Univ.Peshawar) and Y.Motoda (Yatsushiro National College of Technology) gave a new characterization of abelian fields whose rings of integers have power integral bases[JNT, 1986]. Namely if a field K is an octic field Q(sqrt{mn}, sqrt{lm}, sqrt{l}), where lmn is a square-free integer, then K has no power integral basis except for one fie … More ld [MN(Arch.Math.2004)]. Thus we proposed an open problem[ibid] ; Problem. For any octic 2-elementary abelian extension K of rank 3, if the ring Z_K has a power integral basis, then does K coincide with the 24-th cyclotomic field Q(zeta_{24})? Recently we proved the above problem based on a new idea for the reduced classification [submitted to Kyushu Math.J.]. On the main part we presented at Journees Arithmetiques 2005 in Marseille. In October 2003, We held an international symposium on "Yokoi-Chowla Conjecture and Related Problems" at Nagoya Univ.. Recently this conjecture was solved by Hungarian young mathematician A.Bir'o. Including him we invited several professors from Hungary, Korea and Canada who are higher investigators in number theory and related topics. We published the Proceedings of the 2003 Nagoya Conference Yokoi-Chowla Conjecture and Related Problems, edited by S.Katayama(Tokushima U.) C.Levesque(U.Laval, Qu'ebec) and T.Nakahara(Saga U.), revised edit., which includes 15 papers within open problems and the ideas of the theorems without precise proof[BKLN]. For a monic irreducible cubic polynomial $P(u)$ in $u$ over $mathbb{Q}$, arithmetic of the plane curve $E=E(P(u))$ defined by the equation $w^3=P(u)$ was studied. It is an elliptic curve whose $j$-invariant is equal to $0$. Miyake could show that the short form of $E$ is a Mordell curve, $y^1=x^3+k$, with a certain rational number $k$ determined by the coefficients of $P(u)$. It is also pointed out that $E(P(u))$ is essentially dependent on the polynomial $P(u)$ rather than the cubic field $K$ even though $E[mathbb{Q}]$ is completely described by the subset $mathcal{W}(xi)$ of the cubic field. Indeed, we showed that the generic polynomial $P(u)=u^3+tu+t,t in mathbb{Q}$, gives a dominant family of such elliptic curves. Katayama have investigated two topics. First topic is the class number of algebraic number fields. He has constructed a family of cyclic fields of degree p-1 which have the ideal class groups with p-rank at least 2. We note $p equiv 1 mod 4$.B0405)Applications of number theory to arithmetic geometry and algebraic geometryIchikawa calculated the monodromy representation of the Teichm"{u}ller groupoids induced from conformal field theory, proved that on a Schottky uniformized Riemann surface, any stable vector bundle of degree 0 is obtained from a linear representation of the Schottky group, and showed Verlinde's formula for the moduli space of linear representations of the Schottky group Taguchi proved a logical relation between the two versions of the finiteness conjecture of Fontaine-Mazur and the mod $p$ finiteness conjecture of Khare-Moon. Related to this, I studied the moduli space of Galois representations. Next he gave a simple proof of the modular identity for classical theta series[CT(joint work with Y.Choie)]. Finally he classified certain mod 2 Galois representations and deduced 2-adic properties of the Fourier coefficients of some modular forms which involves applications of congruence properties modulo 2-powers of some classical arithmetic or combinatorial functions[OT(joint work with K.Ono)].C0405)Applications of number theory to coding theory and discrete mathematicsUehara established a method of construction of Hermitian codes whose minimum distances are larger than those of known ones. He showed that a simple list decoding of binary BCH codes is executed by Euclid's decoding. Using various algebraic systems, we developed a method of construction of LDPC (low density parity check) codes having excellent ability of error-correcting.On the construction of the RSA signatures with new redundancy functions Katayama have studied the security of these new RSA signatures from the multiplicative attack and verified the new signature has good security[YK]. Less