Representation of integers by positive-definite quadratic forms in totally real algebraic number fields

1985 ◽  
Vol 29 (3) ◽  
pp. 1253-1264
Author(s):  
V. V. Golovizin
Keyword(s):  
Algebraic Number ◽  
Quadratic Forms ◽  
Number Fields ◽  
Positive Definite ◽  
Algebraic Number Fields ◽  
Representation Of Integers ◽  
Totally Real

Some Generalizations of Ramanujan's Sum

1980 ◽  
Vol 32 (5) ◽  
pp. 1250-1260 ◽  
Author(s):  
K. G. Ramanathan ◽  
M. V. Subbarao
Keyword(s):  
Algebraic Number ◽  
Number Fields ◽  
Sums Of Squares ◽  
Modular Functions ◽  
Algebraic Number Fields ◽  
Representation Of Integers ◽  
Ramanujan’S Sum ◽  
Image Position ◽  
Ramanujan's Sum

Ramanujan's well known trigonometrical sum C(m, n) denned bywhere x runs through a reduced residue system (mod n), had been shown to occur in analytic problems concerning modular functions of one variable, by Poincaré [4]. Ramanujan, independently later, used these trigonometrical sums in his remarkable work on representation of integers as sums of squares [6]. There are various generalizations of C(m, n) in the literature (some also to algebraic number fields); see, for example, [9] which gives references to some of these. Perhaps the earliest generalization to algebraic number fields is due to H. Rademacher [5]. We here consider a novel generalization involving matrices.

On Lattices in a Module Over a Matrix Algebra

1966 ◽  
Vol 9 (1) ◽  
pp. 57-61 ◽  
Author(s):  
Nobuo Nobusawa
Keyword(s):  
Algebraic Number ◽  
Quadratic Forms ◽  
Matrix Algebra ◽  
Symmetric Matrix ◽  
Algebraic Number Field ◽  
Number Fields ◽  
Algebraic Number Fields ◽  
Left Module ◽  
Bilinear Mapping ◽  
The Matrix

Let A be the matrix algebra of type n × n over a finite algebraic number field F, and V the module of matrices of type n × m over F. V is naturally an A-left module. Given a non-singular symmetric matrix S of type m × m over F, we have a bilinear mapping f of V on A such that f(x, y) = xSy' for elements x and y in V where y' is the transpose of y. In this case, corresponding to the arithmetic of A([l]), the arithmetical theory of V will be discussed to some extent as we establish the arithmetic of quadratic forms over algebraic number fields ([2]). In this note, we shall define a lattice in V with respect to a maximal order in A. and determine its structure (Theorem 1), and after giving a structure of a complement of a lattice (Theorem 2), we shall give a finiteness theorem of class numbers of lattices under some assumption (Theorem 3).

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