Study on Number Theoretic Algorithms and Developments of Systems for Number Theory

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As in the references below, many theoretical results were obtained on number theoretic algorithms and their applications related to Scholz's conjecture on addition chains, Iwasawa invariants, elliptic curves, prime ideal decomposition in polynomial rings, hyperplane arrangement, unramified extensions of quadratic fields, several zeta values, class groups and unit groups of number fields, and cryptology, for example.The 4th to 11th meetings were held each with about 50 participants by our main research organization JANT, and, the details are reported in http://ntw.e-one.uec.ac.jp/jant. As a result, we edited two special issues of Transactions of the JSIAM on the topics of the most advanced results and a general survey on number theoretic algorithms, and held an organized session at the yearly meeting of the JSIAM. To exchange with broader areas of algebra, we organized the 4th and 5th meetings of Algebra and Computation' (AC2001 and AC2003) each with almost 150 participants including those from abroad and published the proceedings at ftp://tnt.math.metro-u.ac.jp/pub/ac0[13]/. For wide international exchange of mathematical software research, we help the international Congress of Mathematical Softwares ICMS2002 at Beijin with about 110participants.We took over the system for number theory SIMATH, developed in Germany since 1980's with highly efficient functions for elliptic curves, and have been maintaining and supporting via http://simath.info/. We have succeeded in implementing it on 64 bit machines, have released it as Ver.4.6, and packaged and installed the CEM up to cubic fields. After that, restricting ourselves to maintain SIMATH as it is because of the problems of license and development environment, we started to develop a new system NZMATH by the script language Python, released its alpha version at http://tnt.math.metro-u.ac.jp/nzmath/, and we are going to create a system for number theory constructed together by developers and users of the system.