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Masanobu Kaneko

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Number Theory and its Applications

Masanobu Kaneko (Unit Leader)
Degree: Doctor of Science (the University of Tokyo)
Research Interests: Number theory
Unit:Number & Equation

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Hardy once proudly said that his number theory research had nothing to do with applications, such as military purposes. However, in the late 1970s, a kind of cryptography employing number theory, which was considered to be remotely related to practical applications, was announced. Since then, explosive progress has occurred in information and communications technologies, which can be represented by the Internet, and applications of number theory to areas such as information security have been more widely and thoroughly researched. The RSA cryptography, the first example of number-theoretical cryptography, was within a realm of elementary number theory, including prime factorization and Euler’s theorem. However, nowadays the more advanced number theories such as elliptic curves, hyper-elliptic curves, and algebraic number theory are used.

I, myself, once conducted research into the so-called Jacobi sum, and was surprised to learn that there was a method of prime factorization using the Jacobi sum. Application research, such as that mentioned above, has been conducted not only in one-way from academic mathematics to the industry, but also in the opposite direction to provide feedback to mathematics by creating a new field, which can be called Computational Number Theory. As Math-for-Industry’s motto suggests to collaborate interactively between academic mathematicians and industry, this is one example of such a collaboration that is already productive.

I have studied number theory as a branch of pure mathematics. I initially focused on arithmetic fundamental groups and the Galois representation on those groups, and then my focus has shifted to elliptic curves, Bernoulli numbers, modular forms, and zeta functions.

I once coauthored a paper with cryptography researchers. I honestly do not know to what extent this work was appreciated, however, I think that the interpretation of the Sato-Tate conjecture (which was recently solved) using the theory of complex multiplication that I wrote in the appendix of that paper is interesting. My work on elliptic curves mainly concerns supersingular elliptic curves over finite fields, which often appears in the works of cryptography as a curve not suitable for efficient crypto-system. A paper that I published 20 years ago in this area has been quoted by a cryptography researcher.

Hiroshi Yoshida, who is a SSP researcher at the Graduate School of Mathematics of Kyushu University, discovered recently that a group of rational points of an elliptic curve over the rationals appear in the model of the development of a multicellular organism, and this may be related to embryonic stem cells, which is currently a very popular research topic. I am now collaborating with Dr. Yoshida on this topic. This may be an interesting new application of number theory to biology.

Modular forms and modular functions are also important research topics for me. Classically, these are defined as functions on the complex upper half-plane that satisfy a certain transformation law. As the Fourier coefficients of modular forms or functions, often arithmetically interesting sequences appear, and one of the most interesting ones is the so-called “Monstrous Moonshine”, which relates the Fourier coefficients of the elliptic modular j-function to the degrees of irreducible representations of the largest sporadic simple group “monster”. A physicist, Freeman Dyson, once wrote that his humble dream is that a physicist in the 21st century would find a monster incorporated in the structure of the universe in an unexpected way.

I once have discovered a formula expressing Fourier coefficients of this j-function in terms of the quantities that appear in the classical theory of complex multiplication. Does the theory of complex multiplication have something to do with the structure of the universe? Currently, I am working to find some meaning in the “values” of this j-function at real quadratic irrationalities, especially to find some arithmetic in there. Despite the long history of arithmetic of real quadratic fields, some problems like Gauss’ conjecture remain unsolved. In particular, to construct a theory which corresponds to the theory of complex multiplication for imaginary quadratic fields is a big challenge in number theory, including whether it is possible. I have discovered just a phenomenon whose meaning is not yet clear, but I, like Dyson, have a humble dream that this leads to something important.

Another topic of my research is Bernoulli numbers and their generalization as well as multiple zeta values, which are related to them. Several years ago, a graduate student of information technology in the US discovered an unexpected interpretation of poly-Bernoulli numbers, which is a generalization of Bernoulli numbers, from a combinatorial point of view. My attitude towards research is to investigate topics that interest me and my graduate students, and I would be more than happy if our activities lead to an unexpected application or are found to be connected with another topic.

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