Torsion Part of ℤ-module

Одной из актуальных современных проблем алгебры и теории чисел является проблема существования и поиска фундаментальных S-единиц в гиперэллиптических полях. Проблема существования и поиска S-единиц в гиперэллиптических полях эквивалентна разрешимости норменного уравнения - функционального уравнения Пелля - с некоторыми дополнительными условиями на вид этого уравнения и его решения. Существуетглубокая связь между точками конечного порядка в якобиевом многообразии (якобиане) гиперэллиптической кривой и нетривиальными S-единицами соответствующего гиперэллиптического поля. Эта связь легла в основу предложенного В. П. Платоновым алгебраического подхода к известной фундаментальной проблеме об ограниченности кручения в якобиевых многообразиях гиперэллиптических кривых. Для эллиптических кривых над полем рациональных чисел проблема кручения была решена Мазуром в 1970-ых годах. Для кривых рода 2 и выше над полем рациональных чисел проблема кручения оказалась значительно сложнее, и пока далека от своего полного решения. Основные результаты, полученные к настоящему времени в этом направлении, относятся к описанию подгрупп кручения якобиевых многообразий конкретных гиперэллиптических кривых, а также к описанию некоторых семейств гиперэллиптических кривых рода g >= 2.В данной статье нами найден новый метод исследования разрешимости функциональных норменных уравнений, дающий полное описание гиперэллиптических кривых над полем рациональных чисел, якобиевы многообразия которых обладают точками кручения данных порядков. Наш метод основан на аналитическом изучении представителей дивизоров конечного порядка в группе классов дивизоров степени ноль и их представлений Мамфорда. В качестве иллюстрации работы нашего метода в данной статье непосредственно найдены все параметрические семейства гиперэллиптических кривых рода два над полем рациональных чисел, якобиевы многообразия которых обладают рациональными точками кручения порядков не превосходящих пяти. Более того, наш метод позволяет определить, какому найденному параметрическому семейству принадлежит данная кривая, якобиан которой обладает точкой кручения порядка, не превосходящего пяти.

An explicit bound is obtained for the torsion of elliptic curves over the field of rational numbers. Let Γ be an elliptic curve over the field of rational numbers R, and Qm a primitive R-point of order m on it; here m is a prime or a double prime. Hence if m=2p, then p ≤ 509, whereas if m=p, then p < 6144.

The classical problem of the periodicity of continued fractions for elements of hyperelliptic fields has a long and deep history. This problem has up to now been far from completely solved. A surprising result was obtained in [1] for quadratic extensions defined by cubic polynomials with coefficients in the field of rational numbers: except for trivial cases there are only three (up to equivalence) cubic polynomials over whose square root has a periodic continued fraction expansion in the field of formal power series. In view of the results in [1], we completely solve the classification problem for polynomials with periodic continued fraction expansion of in elliptic fields with the field of rational numbers as the field of constants.

We present a new proof of a theorem of Schur's determining the least common multiple of the orders of all finite groups of complex $n \times n$-matrices whose elements have traces in the field of rational numbers. The basic method of proof goes back to Minkowski and proceeds by reduction to the case of finite fields. For the most part, we work over an arbitrary number field rather than the rationals. The first half of the article is expository and is intended to be accessible to graduate students and advanced undergraduates. It gives a self-contained treatment, following Schur, over the field of rational numbers.

Let A be a commutative Noetherian ring of dimension n (n ≥ 3). Let I be a local complete intersection ideal in A[T] of height n. Suppose I/I 2 is free A[T]/I-module of rank n and (A[T]/I) is torsion in K 0(A[T]). It is proved in this article that I is a set theoretic complete intersection ideal in A[T] if one of the following conditions holds: (1) n ≥ 5, odd; (2) n is even, and A contains the field of rational numbers; (3) n = 3, and A contains the field of rational numbers.

It is commonly accepted that the knowledge and learning of rational numbers is more complex than that of the whole number field. This complexity includes the broader range of application of rational numbers, the increased level of technical complexity in the mathematical structure and symbol systems of this field and the more complex nature of many conceptual properties of the rational number field. Research on rational number learning is divided as to whether children’s difficulties in learning rational numbers arise only from the increased complexity or also include elements of conceptual change. This article argues for a fundamental conceptual difference between whole and rational numbers. It develops the position that rational numbers are fundamentally relational in nature and that the move from absolute counts to relative comparisons leads to a further level of abstraction in our understanding of number and quantity. The argument is based on a number of qualitative, in-depth research projects with children and adults. These research projects indicated the importance of such a relational understanding in both the learning and teaching of rational numbers, as well as in adult representations of rational numbers on the number line. Acknowledgement of such a conceptual change could have important consequences for the teaching and learning of rational numbers.

N is the set of natural numbers. Q is the field of complex rational numbers, i.e. the field of numbers a+bi, where a and b are rational. A sequence σ of elements of Q (i.e. a function σ, σ o:N→Q) is recursive iff there exist recursive functions fj(j=1,2,3,4; fj: N→N) such that for all n in N $$ \sigma (n) = \frac{{{f_1}(n)}} {{{f_2}\left( n \right) + 1}} + \frac{{{f_3}\left( n \right)}}{{{f_4}\left( n \right) + 1}}i$$ Such a sequence σ is recursively convergent iff there exists a recursive function k(k: N→N) such that for all h,j,n in N $$h \geqslant k(n)\Lambda \,j \geqslant k(n) \Rightarrow \left| {\sigma (h) - \sigma (j)} \right| < \frac{1} {{n + 1}}$$ (1)

CHAPTER TEN - ROOTS OF POLYNOMIALS

We let Z denote the ring of rational integers, Q the field of rational numbers, R the field of real numbers, and C the field of complex numbers.For elements e and f of a Lie algebra, [e,f] denotes the bracket of e and f. A generalized Cartan matrix C = (cij) is a square matrix of integers satisfying cii = 2, cij ≦ 0 if i ≠ j, cij = 0 if and only if cji = 0. For any generalized Cartan matrix C = (cij) of size l × l and for any field F of characteristic zero, denotes the Lie algebra over F generated by 3l generators e1, …, el, h1, …, hl, f1, …, fl with the defining relationsfor all i, j,for distinct i, j.

Throughout these notes V will be a finite dimensional K vector space and I will denote the identity operator. For all our applications there is no loss in specializing K to be the field of rational numbers, but at some point in our development we will need to assume that K is the field of complex numbers.

We study the problem of when a periodic square matrix of order $n\times n$ over an arbitrary field $\mathbb{F}$ is decomposable into the sum of a square-zero matrix and a torsion matrix and show that this decomposition can always be obtained for matrices of rank at least $\frac{n}{2}$ when $\mathbb{F}$ is either a field of prime characteristic, or the field of rational numbers, or an algebraically closed field of zero characteristic. We also provide a counterexample to such a decomposition when $\mathbb{F}$ equals the field of the real numbers.

It is known that the Brauer--Manin obstruction to the Hasse principle is vacuous for smooth Fano hypersurfaces of dimension at least $3$ over any number field. Moreover, for such varieties it follows from a general conjecture of Colliot-Thélène that the Brauer--Manin obstruction to the Hasse principle should be the only one, so that the Hasse principle is expected to hold. Working over the field of rational numbers and ordering Fano hypersurfaces of fixed degree and dimension by height, we prove that almost every such hypersurface satisfies the Hasse principle provided that the dimension is at least $3$. This proves a conjecture of Poonen and Voloch in every case except for cubic surfaces.

Summary The notion of the characteristic of rings and its basic properties are formalized [14], [39], [20]. Classification of prime fields in terms of isomorphisms with appropriate fields (ℚ or ℤ/p) are presented. To facilitate reasonings within the field of rational numbers, values of numerators and denominators of basic operations over rationals are computed.

– A method for optimization of polynomial expressions in terms of fixed polarities for discrete functions is presented. The method is based on the principle of extended dual polarity, which provides a simple way of ordering polarities to obtain an effective way of finding the optimal polarity. The method still implies exhaustive search, but it is an optimized search, which may be expressed in very simple rules. Experimental results illustrate the effectivity of the proposed method.

The arithmetic properties of the values of the hypergeometric functions can in some cases be studied by the effective construction of a system of linear approximating forms. This construction was used for functions with several parameters in [I], where functions of the type in part (B) of the theorem below were studied for a(x) ~ 1 and b(x) ~ i. The arithmetic properties of the values of polylogarithms with several parameters were studied in [2, 3]. The estimates of linear forms obtained in this manner are more accurate than those obtained (in a more general situation) by means of the Siegel method (see Remark 1 below). The proof of the theorem given below is based on the effective construction of a system of linear approximating forms carried out in [4]. Here to prove each part of the theorem we use a special auxiliary function which plays the same role in the proof as the function ~(z) from Lemma 1 of [4]. THEOREM. Let I be an imaginary quadratic field or a field of rational numbers Q; a ( x ) = ( x + a , ) . . . (x+a~.); b ( x ) = ( x + ~ , ) . . . (x+~m); 5; a 1 . . . . . a r ; $1 . . . . . Sm; ~1 . . . . ,A t a r e r a t i o n a l number s ; a i ~ j , ~h,--~h~Z (i=1 . . . . . r; ] = t . . . . , m; ki, k ~ = t , . . . , t ; k ~ # k 2 ) ; ( x + ~ + ~ ) ( x + ~ k ) a ( x ) b ( x ) # O f o r x = t , 2, 3, . . . ; k = t . . . . . t; ~ I ~ ~r Furthermore, let ; , x -~ ~ l-[ ~ (~ + ~ ) ~ (x) (A) r (z) = ~'--,=o ~-1 b (x) '