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1.
Suppose that E:y2=x(x+M)(x+N) is an elliptic curve, where M相似文献
2.
V. A. Demyanenko 《Mathematical Notes》1998,63(4):444-448
The best possible estimates of torsion of the curvesy
2=x
3+s andy
2=x
3+rx over an arbitrary algebraic number field are given.Translated fromMatematicheskie Zametki, Vol. 63, No. 4, pp. 503–508, April, 1998. 相似文献
3.
Bin Yong SUN 《数学学报(英文版)》2008,24(2):305-310
The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G(k). Then the order of Г is bounded by a constant C'(k, d) which depends only on k and the dimension d of G. 相似文献
4.
E. Girondo G. González-Diez E. González-Jiménez R. Steuding J. Steuding 《Mathematica Slovaca》2009,59(3):299-306
Given any positive integer n, we prove the existence of infinitely many right triangles with area n and side lengths in certain number fields. This generalizes the famous congruent number problem. The proof allows the explicit
construction of these triangles; for this purpose we find for any positive integer n an explicit cubic number field ℚ(λ) (depending on n) and an explicit point P
λ
of infinite order in the Mordell-Weil group of the elliptic curve Y
2 = X
3 − n
2
X over ℚ(λ).
Research of the rest of authors was supported in part by grant MTM 2006-01859 (Ministerio de Educación y Ciencia, Spain). 相似文献
5.
Burcu Baran 《Journal of Number Theory》2010,130(12):2753-2772
Let be the open non-cuspidal locus of the modular curve associated to the normalizer of a non-split Cartan subgroup of level n. As Serre pointed out, an imaginary quadratic field of class number one gives rise to an integral point on for suitably chosen n. In this note, we give a genus formula for the modular curves and we give three new solutions to the class number one problem using the modular curves for n=16,20,21. These are the only such modular curves of genus ?2 that had not yet been exploited. 相似文献
6.
A necessary and sufficient condition is given for the ideal class group H(m) of a real quadratic field Q (√m) to contain a cyclic subgroup of ordern. Some criteria satisfying the condition are also obtained. And eight types of such fields are proved to have this property,
e.g. fields withm=(z
n
+t−1)2+4t(witht|z
n
−1), which contains the well-known fields withm=4z
n
+1 andm=4z
2n
+4 as special cases.
Project supported by the National Natural Science Foundation of China. 相似文献
7.
For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field. 相似文献
8.
In this paper, we present several methods for the construction of elliptic curves with large torsion group and positive rank over number fields of small degree. We also discuss potential applications of such curves in the elliptic curve factorization method (ECM). 相似文献
9.
10.
Yoshiyuki Kitaoka 《Proceedings Mathematical Sciences》1994,104(1):201-206
LetG be an algebraic group inGL
n
(C) defined over Q, andK an algebraic number field with the maximal orderO
k
. If the groupG(O
k
) of rational points ofG inM
n
(O
k
) is a finite group and if it satisfies a certain condition, which is satisfied, for example, whenK is a nilpotent extension of Q and 2 is unramified, thenG(O
k
) is generated by roots of unity inK andG(Z).
Dedicated to the memory of Professor K G Ramanathan 相似文献
11.
WU HongFeng & FENG RongQuan College of Science North China University of technology Beijing China LMAM School of Mathematical Sciences Peking University Beijing 《中国科学 数学(英文版)》2011,(9)
In this paper,the number of isomorphism classes of Legendre elliptic curves over finite field is enumerated. 相似文献
12.
13.
We investigate the rank gain of elliptic curves, and more generally, Jacobian varieties, over non-Galois extensions whose Galois closure has a Galois group permutation-isomorphic to a prescribed group G (in short, “G-extensions”). In particular, for alternating groups and (an infinite family of) projective linear groups G, we show that most elliptic curves over (for example) gain rank over infinitely many G-extensions, conditional only on the parity conjecture. More generally, we provide a theoretical criterion, which allows to deduce that “many” elliptic curves gain rank over infinitely many G-extensions, conditional on the parity conjecture and on the existence of geometric Galois realizations with group G and certain local properties. 相似文献
14.
Amador Martin-Pizarro 《Journal of Pure and Applied Algebra》2006,204(2):368-379
In this paper, we extend a previous result of A. Pillay and the author regarding existence of rational points over elliptic and hyperelliptic curves with generic moduli defined over supersimple fields to the even characteristic case. We give a detailed exposition of the affine models of these families of curves in characteristic 2 and the transformations between members in the same rational isomorphism class. 相似文献
15.
Allison M. Pacelli 《Journal of Number Theory》2004,106(1):26-49
Let be a finite field with q elements, and T a transcendental element over . In this paper, we construct infinitely many real function fields of any fixed degree over with ideal class numbers divisible by any given positive integer greater than 1. For imaginary function fields, we obtain a stronger result which shows that for any relatively prime integers m and n with m,n>1 and relatively prime to the characteristic of , there are infinitely many imaginary fields of fixed degree m such that the class group contains a subgroup isomorphic to . 相似文献
16.
Jianxing Bi 《数学学报(英文版)》1995,11(3):300-306
LetG be a finite group. If
for every primer, whereR
1 Syl
r
G andR
2 Syl
r
(L
n
(q)), thenG L
n
(q). 相似文献
17.
18.
窦志红 《纯粹数学与应用数学》2011,27(2):210-212,235
设p是奇素数,N(p)是椭圆曲线E:y2=2px(x2+1)的正整数点(x,y)的个数.主要讨论了N(p)的性质,运用初等方法及四次Diophantine方程的性质,对某些特殊素数p,给出了N(p)的上界.证明了当p≡1(mod 8)且p=s2+32t,其中s,t是正整数时,N(p)≤3;当p≡1(mod 8)且p+s... 相似文献
19.
Suppose that G is a finite group and H is a subgroup of G . H is said to be s -permutably embedded in G if for each prime p dividing |H |, a Sylow p -subgroup of H is also a Sylow p -subgroup of some s -permutable subgroup of G ; H is called weakly s -permutably embedded in G if there are a subnormal subgroup T of G and an s -permutably embedded subgroup Hse of G contained in H such that G = HT and H ∩T ≤H se . In this paper, we continue the work of [Comm. Algebra, 2009, 37: 1086-1097] to study the influence of the weakly s -permutably embedded subgroups on the structure of finite groups, and we extend some recent results. 相似文献
20.
Rob de Jeu 《Inventiones Mathematicae》1996,125(3):523-556
In this paper we consider the group K
4
(3)
(F) of a field, in case F is the function field of a smooth geometrically irreducible curve over a number field. We do this using the complexes constructed
in [4], together with an auxiliary complex. On the image in K
4
(3)
(F) of those complexes, we derive a formula for the Beilinson regulator, and compute an approximation of the boundary map at
the closed points of the curve in the localization sequence. We give a way of finding examples of elliptic curves E with elements in K
4
(3)
(E), and in some cases use computer calculations to check numerically the relation between the regulator and the L-function, as conjectured by Beilinson.
Oblatum 10-IX-1995 & 8-I-1996 相似文献