Generalized Jacobi forms and abelian schemes over arithmetic varieties.
We generalize Jacobi forms of an arbitrary degree and construct torus bundles over abelian schemes whose sections can be identified with such generalized Jacobi forms.
We generalize Jacobi forms of an arbitrary degree and construct torus bundles over abelian schemes whose sections can be identified with such generalized Jacobi forms.
Let , , , be integers with . The classical and the homogeneous Dedekind sums are defined by respectively, where The Knopp identities for the classical and the homogeneous Dedekind sum were the following: where . In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.
In this report we study the arithmetic of Rikuna’s generic polynomial for the cyclic group of order and obtain a generalized Kummer theory. It is useful under the condition that and where is a primitive -th root of unity and . In particular, this result with implies the classical Kummer theory. We also present a method for calculating not only the conductor but also the Artin symbols of the cyclic extension which is defined by the Rikuna polynomial.
We give a generalization of poly-Cauchy polynomials and investigate their arithmetical and combinatorial properties. We also study the zeta functions which interpolate the generalized poly-Cauchy polynomials.