ABSTRACT: Let E be an elliptic curve defined over Q and without complex multiplication. For a prime p of good reduction, let E be the reduction of E modulo p. Assuming that certain Dedekind zeta functions have no zeros in Re(s) > 3/4, we determine how often E(Fp) is a cyclic group. This result was previously obtained by J. -P. Serre using the full Generalized Riemann Hypothesis for the same Ded...

In this paper, by using elementary tools of commutative algebra,we prove the persistence property for two especial classes of rings. In fact, thispaper has two main sections. In the first main section, we let R be a Dedekindring and I be a proper ideal of R. We prove that if I1, . . . , In are non-zeroproper ideals of R, then Ass1(Ik11 . . . Iknn ) = Ass1(Ik11 ) [ · · · [ Ass1(Iknn )for all k1,...

The simplest quartic field was introduced by M. Gras and studied by A. J. Lazarus. In this paper, we will evaluate the values of the Dedekind zeta functions at s = −1 of the simplest quartic fields. We first introduce Siegel’s formula for the values of the Dedekind zeta function of a totally real number field at negative odd integers, and will apply Siegel’s formula to the simplest quartic fiel...

We state several conditions under which comultiplication and weak comultiplication modulesare cyclic and study strong comultiplication modules and comultiplication rings. In particular,we will show that every faithful weak comultiplication module having a maximal submoduleover a reduced ring with a finite indecomposable decomposition is cyclic. Also we show that if M is an strong comultiplicati...

It is known that the 3-manifold SL(2,Z)\ SL(2,R) is diffeomorphic to the complement of the trefoil knot in S. E. Ghys showed that the linking number of this trefoil knot with a modular knot is given by the Rademacher symbol, which is a homogenization of the classical Dedekind symbol. The Dedekind symbol arose historically in the transformation formula of the logarithm of Dedekind’s eta function...

In this paper, new probability estimates are derived for ideal lattice codes from totally real number fields using ideal class Dedekind zeta functions. In contrast to previous work on the subject, it is not assumed that the ideal in question is principal. In particular, it is shown that the corresponding inverse norm sum depends not only on the regulator and discriminant of the number field, bu...

Richard Dedekind has had an incredible influence on modern mathematics, largely due to his methodological demands which are still valued by mathematicians today. Through an investigation of some of his works written between 1854 and 1877, I reveal a connection between these methodological demands and features of axiomatic reasoning that he employed. I discuss two foundational/philosophic works ...

An i?-module Mis a generator (of the category of modules) provided every module is a homomorphic image of a suitable direct sum of copies of M. Equivalently, some M has R as a summand. Except in the last section, all rings are assumed to be commutative, Noetherian domains, and modules are usually finitely generated. In this context generators are exactly those modules that have non-zero free su...

2 Local symplectic, complex and Kähler geometry: a quick review 10 2.1 Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Symplectic quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Complex manifolds . . . . . . . . . . . . . . ....

Bài báo này nghiên cứu tính hữu hạn địa phương của các đối đại số được biết đến như là định lý cơ bản cho trên vành Dedekind. Trước tiên, chúng tôi đưa ra một chứng minh chất xạ ảnh môđun miền iđêan chính mà không sử dụng trường. Tiếp theo, phiên phẳng Dedekind, dĩ nhiên mở rộng Cuối cùng, áp kết quả tọa độ lược đồ nhóm affine phẳng.