Hiroyuki OGAWA

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Algebraic number theory
Keywords Galois groups, algebraic curves, continued fraction expansion, periodic points, iterative action

I have an interest in periodic objects. Expanding rational numbers into decimal numbers is delightful. The decimal number expansion becomes the repeat of a sequence of some integers. I have an appetite for continued fraction expansions, never get tired to calculate it, and want to find continued fractions with sufficiently long period. It is on the way to Gauss' class number one conjecture. Recently, I am studying iteration of rational functions. For a rational function g(x) with rational coefficients, a complex number z with g(g(...g(z)...))=z is called a periodic point on g(x) and is an algebraic number. I expect that number theoretical properties which such an algebraic number z has is described by the rational function g(x). This does not seem to work out anytime, but one can find many rational functions g(x) that describe the Galois group, the class number, the class group, and so on of a periodic point of g(x). I think that this should be surely useful, and calculate like these every day.