Research on determinantal formulae and Bernoulli-Hurwitz numbers in the theory of Abelian functions

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Thanks to the Grant-in-Aid for Scientific Research (C), academic years 2000-2002, with theme "Research on product formula for special values of Abelian functions", the head investigator (Y.O) made two progresses :(1)Nicely generalized Bernoulli-Hurwitz numbers to algebraic functions of cyclotomic type, and(2)Gave determinantal expressions of addition formulae for Abelian functions associated to higher genus algebraic curves,Hence, we aimed to investigate these results deeply as follows :(1)should be improved by connecting the result to application on L-series as in the case of Bernoulli, and Hurwitz numbers;(2)should be generalized to non-hyperelliptic curves.The results :The problem (1)is so difficult. We only preparated a manuscript to submit a academic journal. We, however, made a big progress on (2), and published a lot of papers as included in this report.(1)is needed to investigate further ;(2)is only proved by a dedious way, and should be proved more directly.We will continue to research on these problem.Finally, we hope our results be good hints for researchers who are interesting on theory of Abelian functions.