Study of modular varieties and modular forms

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We investigated poly-Bernoulli polynomials, Arakawa-Kaneko zeta functions, and Dedekind sums in finite characteristic. T. Arakawa and M. Kaneko introduced a certain zeta function, which is called the Arakawa-Kaneko zeta function, to clarify the relation between multiple zeta values and poly-Bernoulli numbers. A. Bayad and I introduced and studied poly-Bernoulli polynomials and the Arakawa-Kaneko zeta function of Hurwitz type. Our result was published in Kyushu J. Math. in 2011. After that, we extended our Bernoulli polynomials and the zeta function into two directions. These can be applicable. We reported a part of our results at RIMS workshop in 2010. We next report the results about the Dedekind sums. The Dedekind sums were introduced by Dedekind to describe the transformation formula of the Dedekind eta-function. These sums have many applications. Zagier gave a higher dimensional version of these sums. We have investigated the Dedekind sums in finite characteristic. We introduced higher dimensional Dedekind sums, and established the reciprocity law, rationality, and the Knopp identity. Moreover, we introduced the generalized higher dimensional Dedekind sums, and described the values of L-functions in terms of the Dedekind sums.