As we indicated in our paper [9], the standard arithmetic Chow groups introduced by Gillet-Soulé [3] are rather restricted to consider arithmetic analogues of geometric problems. In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem still holds as in papers [1], [7] and [14]. Let X → Spec(Z) be a regular ari...

Here, we initiate a program to study relationships between finite groups and arithmetic–geometric invariants in systematic way. To do this, first introduce notion of optimal module for group the setting holomorphic mock Jacobi forms. Then, classify modules cyclic prime order, special case weight 2 index 1, where class numbers imaginary quadratic fields play an important role. Finally, exhibit c...

In this paper, we prove index theorems for integrable metrized line bundles on projective varieties over complete fields and number fields respectively. As applications, we prove a non-archimedean analogue of the Calabi theorem and a rigidity theorem about the preperiodic points of algebraic dynamical systems.

For example, K. Künnemann [Ku] proved that if X is a projective space, then the conjecture is true. Here we fix a notation. We say a Hermitian line bundle (H, k) on X is arithmetically ample if (1) H is f -ample, (2) the Chern form c1(H∞, k∞) is positive definite on the infinite fiber X∞, and (3) there is a positive integer m0 such that, for any integer m ≥ m0, H(X, H) is generated by the set {...

Based on grey language multi-attribute group decision making, a kernel and grey scale scoring function is put forward according to the definition of grey language and the meaning of the kernel and grey scale. The function introduces grey scale into the decision-making method to avoid information distortion. This method is applied to the grey language hesitant fuzzy group decision making, and th...

As we indicated in our paper [10], the standard arithmetic Chow groups introduced by Gillet-Soulé [4] are rather restricted to consider arithmetic analogues of geometric problems. In this note, we would like to propose a suitable extension of the arithmetic Chow group of codimension one, in which the Hodge index theorem still holds as in papers [2], [8] and [15]. Let X → Spec(Z) be a regular ar...

Three algorithms providing rigourous bounds for the eigenvalues of a real matrix are presented. The first is an implementation of the bisection algorithm for a symmetric tridiagonal matrix using IEEE floating-point arithmetic. The two others use interval arithmetic with directed rounding and are deduced from the Jacobi method for a symmetric matrix and the Jacobi-like method of Eberlein for an ...

By Grothedieck's Anabelian conjectures, Galois representations landing in outer automorphism group of the algebraic fundamental group which are associated to hyperbolic smooth curves defined over number fields encode all arithmetic information of these curves. The goal of this paper is to develope and arithmetic teichmuller theory, by which we mean, introducing arithmetic objects summarizing th...

We answer a question asked by Hajdu and Tengely: The only arithmetic progression in coprime integers of the form (a, b, c, d) is (1, 1, 1, 1). For the proof, we first reduce the problem to that of determining the sets of rational points on three specific hyperelliptic curves of genus 4. A 2-cover descent computation shows that there are no rational points on two of these curves. We find generat...

We present an empirical study of the accuracy{cost tradeoos of Anderson's method. The various parameters that control the degree of approximation of the computational elements and the separateness of interacting computational elements govern both the arithmetic complexity and the accuracy of the method. Our experiment shows that for a given error requirement, using a near{{eld containing only n...