scholarly journals A Note on Units of Algebraic Number Fields

1955 ◽  
Vol 9 ◽  
pp. 115-118 ◽  
Author(s):  
Tomio Kubota
Keyword(s):  
Existence Theorem ◽  
Algebraic Number ◽  
Present Note ◽  
Number Fields ◽  
Algebraic Number Fields

We shall prove in the present note a theorem on units of algebraic number fields, applying one of the strongest formulations, be Hasse [3], of Grunwald’s existence theorem.

scholarly journals Note on an Ordering Theorem for Subfields

1952 ◽  
Vol 4 ◽  
pp. 125-129
Author(s):  
Tadasi Nakayama
Keyword(s):  
Number Field ◽  
Algebraic Number ◽  
Present Note ◽  
Number Fields ◽  
Class Groups ◽  
Algebraic Number Fields ◽  
Original Field

In a recent paper [3] Tannaka gave an interesting ordering theorem for subfields of a p-adic number field, The purpose of the present note is firstly to observe, on modifying Tannaka’s argument a little, that his restriction to those subfields over which the original field is abelian may be removed and in fact the theorem holds for arbitrary fields which are not p-adic number fields, indeed in a much refined form, and secondly to formulate a similar ordering theorem for algebraic number fields in terms of idèle-class groups in place of element groups.

scholarly journals K-RATIONAL D-BRANE CRYSTALS

2012 ◽  
Vol 27 (22) ◽  
pp. 1250112
Author(s):  
ROLF SCHIMMRIGK
Keyword(s):  
String Theory ◽  
Algebraic Number ◽  
Special Class ◽  
Building Blocks ◽  
Number Fields ◽  
Central Point ◽  
Algebraic Number Fields ◽  
Special Values ◽  
Toroidal Compactifications ◽  
Base Points

In this paper the problem of constructing space–time from string theory is addressed in the context of D-brane physics. It is suggested that the knowledge of discrete configurations of D-branes is sufficient to reconstruct the motivic building blocks of certain Calabi–Yau varieties. The collections of D-branes involved have algebraic base points, leading to the notion of K-arithmetic D-crystals for algebraic number fields K. This idea can be tested for D0-branes in the framework of toroidal compactifications via the conjectures of Birch and Swinnerton-Dyer. For the special class of D0-crystals of Heegner type these conjectures can be interpreted as formulae that relate the canonical Néron–Tate height of the base points of the D-crystals to special values of the motivic L-function at the central point. In simple cases the knowledge of the D-crystals of Heegner type suffices to uniquely determine the geometry.

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