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1.
We study the classification of elliptic curves E over the rationals ℚ according to the torsion sugroups E
tors(ℚ). More precisely, we classify those elliptic curves with E
tors(ℚ) being cyclic with even orders. We also give explicit formulas for generators of E
tors(ℚ). These results, together with the recent results of K. Ono for the non-cyclic E
tors(ℚ), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.
Received July 29, 1999, Revised March 9, 2001, Accepted July 20, 2001 相似文献
2.
Suppose that E:y2=x(x+M)(x+N) is an elliptic curve, where M相似文献
3.
Let E = Eσ : y2 = x(x + σp)(x + σq) be elliptic curves, where σ = ±1, p and q are primenumbers with p+2 = q. (i) Selmer groups S(2)(E/Q), S(φ)(E/Q), and S(φ)(E/Q) are explicitly determined,e.g. S(2)(E+1/Q)= (Z/2Z)2, (Z/2Z)3, and (Z/2Z)4 when p ≡ 5, 1 (or 3), and 7(mod 8), respectively. (ii)When p ≡ 5 (3, 5 for σ = -1) (mod 8), it is proved that the Mordell-Weil group E(Q) ≌ Z/2Z Z/2Z,symbol, the torsion subgroup E(K)tors for any number field K, etc. are also obtained. 相似文献
4.
In this paper we introduce an elliptic analog of the Bloch-Suslin complex and prove that it (essentially) computes the weight
two parts of the groups K
2(E) and K
1(E) for an elliptic curve E over an arbitrary field k. Combining this with the results of Bloch and Beilinson we proved Zagier's conjecture on L(E,2) for modular elliptic curves over ℚ.
Oblatum 3-VI-1996 & 16-V-1997 相似文献
5.
Yasutsugu Fujita 《manuscripta mathematica》2005,118(3):339-360
Let E be an elliptic curve over Q and p a prime number. Denote by Qp,∞ the Zp-extension of Q. In this paper, we show that if p≠3, then where E(Qp,∞)(2) is the 2-primary part of the group E(Qp,∞) of Qp,∞-rational points on E. More precisely, in case p=2, we completely classify E(Q2,∞)(2) in terms of E(Q)(2); in case p≥5 (or in case p=3 and E(Q)(2)≠{O}), we show that E(Qp,∞)(2)=E(Q)(2). 相似文献
6.
Let E/F be a Galois extension of number fields with Galois group G=Gal(E/F), and let p be a prime not dividing #G. In this paper, using character theory of finite groups, we obtain the upper bound of #K2OE if the group K2OE is cyclic, and prove some results on the divisibility of the p-rank of the tame kernel K2OE, where E/F is not necessarily abelian. In particular, in the case of G=Cn, Dn, A4, we easily get some results on the divisibility of the p-rank of the tame kernel K2OE by the character table. Let E/Q be a normal extension with Galois group Dl, where l is an odd prime, and F/Q a non-normal subextension with degree l. As an application, we show that f|p-rank K2OF, where f is the smallest positive integer such that pf≡±1(mod l). 相似文献
7.
Let p be a large prime number, K, L, M, λ be integers with 1 ≤ M ≤ p and gcd(λ, p) = 1. The aim of our paper is to obtain sharp upper bound estimates for the number I
2(M; K, L) of solutions of the congruence
xy o l (mod p), K+ 1 £ x £ K +M, L+ 1 £ y £ L +M,xy \equiv \lambda \quad ({\rm mod} p), \quad K+ 1 \leq x \leq K +M,\quad L+ 1 \leq y \leq L +M, 相似文献
8.
Adam Osękowski 《Journal of Theoretical Probability》2011,24(3):849-874
In the paper we determine, for any K>0 and α∈[0,1], the optimal constant L(K,α)∈(0,∞] for which the following holds: If X is a nonnegative submartingale and Y is α-strongly differentially subordinate to X, then
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