Iwasawa Theory of Quadratic Twists of X_0(49)
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摘要
The field K=Q((-7)(1/2)) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X_0(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X_0(49)by the quadratic extension K(M1/2)/K, where M is any square free element of O with M ≡1 mod 4 and(M, 7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F_∞ = K(E_(p∞)),where E_(p∞) denotes the group of p~∞-division points on E. Moreover, writing B for the twist of X_0(49)by K((-7)(1/4))/K, our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z_2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.
The field K=Q((-7)(1/2)) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X_0(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X_0(49)by the quadratic extension K(M1/2)/K, where M is any square free element of O with M ≡1 mod 4 and(M, 7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F_∞ = K(E_(p∞)),where E_(p∞) denotes the group of p~∞-division points on E. Moreover, writing B for the twist of X_0(49)by K((-7)(1/4))/K, our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z_2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.
The field K=Q((-7)(1/2)) is the only imaginary quadratic field with class number 1, in which the prime 2 splits, and we fix one of the primes p of K lying above 2. The modular elliptic curve X_0(49) has complex multiplication by the maximal order O of K, and we let E be the twist of X_0(49)by the quadratic extension K(M1/2)/K, where M is any square free element of O with M ≡1 mod 4 and(M, 7) = 1. In the present note, we use surprisingly simple algebraic arguments to prove a sharp estimate for the rank of the Mordell-Weil group modulo torsion of E over the field F_∞ = K(E_(p∞)),where E_(p∞) denotes the group of p~∞-division points on E. Moreover, writing B for the twist of X_0(49)by K((-7)(1/4))/K, our Iwasawa-theoretic arguments also show that the weak form of the conjecture of Birch and Swinnerton-Dyer implies the non-vanishing at s = 1 of the complex L-series of B over every finite layer of the unique Z_2-extension of K unramified outside p. We hope to give a proof of this last non-vanishing assertion in a subsequent paper.
引文
[1]Coates,J.:Infinite descent on elliptic curves with complex multiplication.Progr.Math.,35,321-350(1983)
[2]Coates,J.,Li,Y.,Tian,Y.,et al.:Quadratic twists of elliptic curves.Proc.London Math.Soc.,110,357-394(2015)
[3]Greenberg,R.:On the structure of certain Galois groups.Invent.Math.,47,85-99(1978)
[4]Rubin,K.:The"main conjectures"of Iwasawa theory for imaginary quadratic fields.Invent.Math.,103,25-68(1991)
[5]Serre,J.P.,Tate,J.:Good reduction of abelian varieties.Ann.of Math.,88,492-517(1968)
[6]Tate,J.:Algorithm for determining the type of a singular fiber in an elliptic pencil.Modular Functions of One Variable IV,Lecture Notes in Math.,476,1975,33-52
[2]Coates,J.,Li,Y.,Tian,Y.,et al.:Quadratic twists of elliptic curves.Proc.London Math.Soc.,110,357-394(2015)
[3]Greenberg,R.:On the structure of certain Galois groups.Invent.Math.,47,85-99(1978)
[4]Rubin,K.:The"main conjectures"of Iwasawa theory for imaginary quadratic fields.Invent.Math.,103,25-68(1991)
[5]Serre,J.P.,Tate,J.:Good reduction of abelian varieties.Ann.of Math.,88,492-517(1968)
[6]Tate,J.:Algorithm for determining the type of a singular fiber in an elliptic pencil.Modular Functions of One Variable IV,Lecture Notes in Math.,476,1975,33-52