LetE=E σ :y 2 =x(x+σp)(x+σq) be elliptic curves, where σ= ±1,p andq are prime numbers withp + 2= q. (i) Selmer groups S2(E/Q), S03D5(E/Q), and $S^{\widehat{(\varphi )}} \left( {E/Q} \right)$ are explicitly de...LetE=E σ :y 2 =x(x+σp)(x+σq) be elliptic curves, where σ= ±1,p andq are prime numbers withp + 2= q. (i) Selmer groups S2(E/Q), S03D5(E/Q), and $S^{\widehat{(\varphi )}} \left( {E/Q} \right)$ are explicitly determined, e.g. S2(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) Whenp ≡ 5 (3, 5 for σ= ?1) (mod 8), it is proved that the Mordell-Weil group E(Q) ? Z/2Z ⊕ Z/2Z, rankE(Q) = 0, and Shafarevich-Tate group III (E/Q)[2]= 0. (iii) In any case, the sum of rankE(Q) and dimension of III (E/Q)[2] is given, e.g. 0, 1, 2 whenp ≡ 5, 1 (or 3), 7 (mod 8) for σ= 1. (iv) The Kodaira symbol, the torsion subgroup E(K)tors for any number fieldK, etc. are also obtained.展开更多
Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) ...Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) with the prime number p = r^2+s^2 and s is even, then C_1h^-≡B_((p-1)/_4)B_(3(p-1)/4) (mod p) for p≡1 (mod 8); and C_2h^-≡E_((p-5)/8)E_((3p-7)/8)(mod p) for p≡5 (mod 8)where B_n and E_n are the Bernoulli and the Euler numbers. If the real K = Q((v(5+2(5^(1/2))))^(1/2),then C_3h^-≡h(Q((-v)^(1/2))) h (Q((-5v)^(1/2))) (mod 5). If 3 ramifies in K = Q(θ^(1/2)), then C_4h(K)≡h(K~*) (mod 3) with K~* = Q((-3θ^(1/2))). All the above C_i are explicitly given constants.Some relations between the factors of class numbers h^- are also obtained. These results forcyclic quartic fields are an extension of the results for quadratic fields obtained by Ankeny-Artin-Chowla, Kiselev, Carlitz and Lu Hong-wen from 1948 to 1983.展开更多
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 order n.Some criteria satisfying the condition are also obtained.And eight ...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 order n.Some criteria satisfying the condition are also obtained.And eight types of such fields are proved to have this property,e.g.fields with m=(zn+t+12)+4t(with t|zn-1),which contains the well-known fields with m=4zn+1 and m=z2n+4 as special cases.展开更多
By the theory of semi-simple (generalized) continued fractions, the Diophantine equation x^2-dy^2=c (1) will be studied, and simple criteria for solvability and explicit sets of solutions will be given.These could be ...By the theory of semi-simple (generalized) continued fractions, the Diophantine equation x^2-dy^2=c (1) will be studied, and simple criteria for solvability and explicit sets of solutions will be given.These could be used to obtain structure theorems on ideal class groups of real quadratic fields and to improve the Cohen-Lenstra heuristic under certain conditions. Minimal continued fractions are discussed finally.展开更多
For real quadratic fields K, especially for fields K of ERD-type, a series of criteria of ideal class numbers h(K)=1 and h(K)】1 will be given via results of Diophantine equations in [1] and continued fraction theory....For real quadratic fields K, especially for fields K of ERD-type, a series of criteria of ideal class numbers h(K)=1 and h(K)】1 will be given via results of Diophantine equations in [1] and continued fraction theory. The problem of class numbers of real quadratic fields, after Gauss’conjecture, has been studied. For example, Lu Hong-wen展开更多
Let K be an abelian number field, and let K_G be its genus field. A simple construction of K_G was given in Ref. [1]. We here give a further description of K_G, which determines the conductor f(K) and discriminant D(K...Let K be an abelian number field, and let K_G be its genus field. A simple construction of K_G was given in Ref. [1]. We here give a further description of K_G, which determines the conductor f(K) and discriminant D(K). And, using these results. we prove that an extension L/K of type (q^s, q^s,…, q^s) has a relative integral basis under some conditions. Suppose that L is any algebraic number field containing K as a subfield. Since展开更多
基金This work ws supported by the National Natural Science Foundation of China(Grant No.10071041).
文摘LetE=E σ :y 2 =x(x+σp)(x+σq) be elliptic curves, where σ= ±1,p andq are prime numbers withp + 2= q. (i) Selmer groups S2(E/Q), S03D5(E/Q), and $S^{\widehat{(\varphi )}} \left( {E/Q} \right)$ are explicitly determined, e.g. S2(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) Whenp ≡ 5 (3, 5 for σ= ?1) (mod 8), it is proved that the Mordell-Weil group E(Q) ? Z/2Z ⊕ Z/2Z, rankE(Q) = 0, and Shafarevich-Tate group III (E/Q)[2]= 0. (iii) In any case, the sum of rankE(Q) and dimension of III (E/Q)[2] is given, e.g. 0, 1, 2 whenp ≡ 5, 1 (or 3), 7 (mod 8) for σ= 1. (iv) The Kodaira symbol, the torsion subgroup E(K)tors for any number fieldK, etc. are also obtained.
基金Project supported by the National Natural Science Foundation of China.
文摘Let K be a cyclic quartic number field, and k its quadratic subfield. Let h(L) denote theideal class number of field L. Ten congruenees for h^- = h(K)/h(k) are obtained. In par-ticular, if K = Q((p+s(p^(1/2))))^(1/2) with the prime number p = r^2+s^2 and s is even, then C_1h^-≡B_((p-1)/_4)B_(3(p-1)/4) (mod p) for p≡1 (mod 8); and C_2h^-≡E_((p-5)/8)E_((3p-7)/8)(mod p) for p≡5 (mod 8)where B_n and E_n are the Bernoulli and the Euler numbers. If the real K = Q((v(5+2(5^(1/2))))^(1/2),then C_3h^-≡h(Q((-v)^(1/2))) h (Q((-5v)^(1/2))) (mod 5). If 3 ramifies in K = Q(θ^(1/2)), then C_4h(K)≡h(K~*) (mod 3) with K~* = Q((-3θ^(1/2))). All the above C_i are explicitly given constants.Some relations between the factors of class numbers h^- are also obtained. These results forcyclic quartic fields are an extension of the results for quadratic fields obtained by Ankeny-Artin-Chowla, Kiselev, Carlitz and Lu Hong-wen from 1948 to 1983.
基金Project supported by the National Natural Science Foundation of China.
文摘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 order n.Some criteria satisfying the condition are also obtained.And eight types of such fields are proved to have this property,e.g.fields with m=(zn+t+12)+4t(with t|zn-1),which contains the well-known fields with m=4zn+1 and m=z2n+4 as special cases.
基金the National Natural Science Foundation of China.
文摘By the theory of semi-simple (generalized) continued fractions, the Diophantine equation x^2-dy^2=c (1) will be studied, and simple criteria for solvability and explicit sets of solutions will be given.These could be used to obtain structure theorems on ideal class groups of real quadratic fields and to improve the Cohen-Lenstra heuristic under certain conditions. Minimal continued fractions are discussed finally.
基金Project supported partially by the National Natural Science Foundation of China.
文摘For real quadratic fields K, especially for fields K of ERD-type, a series of criteria of ideal class numbers h(K)=1 and h(K)】1 will be given via results of Diophantine equations in [1] and continued fraction theory. The problem of class numbers of real quadratic fields, after Gauss’conjecture, has been studied. For example, Lu Hong-wen
文摘Let K be an abelian number field, and let K_G be its genus field. A simple construction of K_G was given in Ref. [1]. We here give a further description of K_G, which determines the conductor f(K) and discriminant D(K). And, using these results. we prove that an extension L/K of type (q^s, q^s,…, q^s) has a relative integral basis under some conditions. Suppose that L is any algebraic number field containing K as a subfield. Since
