A Nonstandard Approach to Real Multiplication
نویسنده
چکیده
This paper looks at the use of Model Theoretic ideas to study certain non-Hausdorff spaces with a view to their application in Manin’s proposed theory of Real Multiplication, which seeks to establish a framework in which to prove Hilbert’s twelfth problem for real quadratic fields. We study morphisms between these non-Hausdorff spaces using Nonstandard Analysis, and as a consequece in certain special cases we are able to describe an action of a certain Galois group on isomorphism classes of these spaces.
منابع مشابه
A New Approach to Nonstandard Analysis
In this paper, we propose a new approach to nonstandard analysis without using the ultrafilters. This method is very simple in practice. Moreover, we construct explicitly the total order relation in the new field of the infinitesimal numbers. To illustrate the importance of this work, we suggest comparing a few applications of this approach with the former methods.
متن کاملUltraproducts and Hyperreal Numbers
Notions of infinite and infinitesimal numbers have been around since the earliest days of calculus; in particular, Leibniz found them a great inspiration in his co-invention of calculus. Later generations of analysts, including Weierstrass, frowned upon such notions, pointing out the lack of rigor in many of their arguments. Yet even Cauchy, who was largely known for his efforts to ‘clean up’ a...
متن کاملNoncommutative Tori, Real Multiplication and Line Bundles
This thesis explores an approach to Hilbert's twelfth problem for real quadratic number elds, concerning the determination of an explicit class eld theory for such elds. The basis for our approach is a paper by Manin proposing a theory of Real Multiplication realising such an explicit theory, analogous to the theory of Complex Multiplication associated to imaginary quadratic elds. Whereas ellip...
متن کاملA nonstandard finite difference scheme for solving fractional-order model of HIV-1 infection of CD4^{+} t-cells
In this paper, we introduce fractional-order into a model of HIV-1 infection of CD4^+ T--cells. We study the effect of the changing the average number of viral particles $N$ with different sets of initial conditions on the dynamics of the presented model. The nonstandard finite difference (NSFD) scheme is implemented to study the dynamic behaviors in the fractional--order HIV-1 ...
متن کاملA numerical method for discrete fractional--order chemostat model derived from nonstandard numerical scheme
In this paper, the fractional--order form of three dimensional chemostat model with variable yields is introduced. The stability analysis of this fractional system is discussed in detail. In order to study the dynamic behaviours of the mentioned fractional system, the well known nonstandard (NSFD) scheme is implemented. The proposed NSFD scheme is compared with the forward Euler and ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2006