Number Theory and Its Applications to Discrete Mathematics

点击翻译按钮获取中文摘要

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