Study of modular varieties and modular forms

我们研究了多元Bernoulli多项式、Arakawa-Kaneko Zeta函数和有限特征的Dedekind型和。T.Arakawa和M.Kaneko引入了一种Zeta函数，称为Arakawa-Kaneko Zeta函数，以阐明多重Zeta值与多重Bernoulli数之间的关系。A.Bayad和我引进并研究了多项式Bernoulli多项式和Hurwitz型Arakawa-Kaneko Zeta函数。我们的研究结果发表在《九州数学杂志》上。在2011年。之后，我们将Bernoulli多项式和Zeta函数推广到两个方向。这些都是适用的。我们在2010年的RIMS研讨会上报告了我们的部分成果。接下来，我们将报告有关Dedekind和的结果。Dedekind型函数的变换公式是由Dedekind型函数的Dedekind型和来描述的。这些金额有很多用途。扎吉尔给出了这些和的一个更高维度的版本。研究了有限特征下的Dedekind型和。我们引入了高维Dedekin和，建立了互易律、合理性和Knopp恒等式。此外，我们还引入了广义高维Dedekin和，并用Dedekin和刻画了L函数的值。

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.