TAILIEUCHUNG - Đề tài " Twisted Fermat curves over totally real fields "

Let p be a prime number, F a totally real field such that [F (µp ) : F ] = 2 and [F : Q] is odd. For δ ∈ F × , let [ δ ] denote its class in F × /F ×p . In this paper, we show Main Theorem. There are infinitely many classes [ δ ] ∈ F × /F ×p such that the twisted affine Fermat curves Wδ : have no F -rational points. Remark. It is clear that if [ δ ] = [ δ ], then Wδ is isomorphic to Wδ over F . For any δ ∈ F × , Wδ /F has rational points locally everywhere. | Annals of Mathematics Twisted Fermat curves over totally real fields By Adrian Diaconu and Ye Tian Annals of Mathematics 162 2005 1353 1376 Twisted Fermat curves over totally real fields By Adrian Diaconu and Ye Tian 1. Introduction Let p be a prime number F a totally real field such that F pp F 2 and F Q is odd. For Ỗ e Fx let Ỗ denote its class in Fx Fxp. In this paper we show Main Theorem. There are infinitely many classes Ỗ e Fx Fxp such that the twisted affine Fermat curves Ws Xp Yp Ỗ have no F-rational points. Remark. It is clear that if Ỗ Ỗ then Ws is isomorphic to Ws over F. For any Ỗ e Fx Ws F has rational points locally everywhere. To obtain this result consider the smooth open affine curve Cs Vp U Ỗ - U and the morphism fis Ws Cs x y I xp xy . Let Cs Js be the Jacobian embedding of Cs F defined by the point 0 0 . We will show that 1 If L 1 Js F 0 then Js F is a finite group cf. Theorem . of 2 . The proof is based on Zhang s extension of the Gross-Zagier formula to totally real fields and on Kolyvagin s technique of Euler systems. One might use techniques of congruence of modular forms to remove the restriction that the degree F Q is odd. 2 There are infinitely many classes Ỗ such that L 1 Js F 0 cf. Theorem . of 3 see also . . The proof is based on the theory of double Dirichlet series. The condition that F pp F 2 is essential for the technique we use here. 1354 ADRIAN DIACONU AND YE TIAN Combining 1 and 2 one can see that the set n ỗ G Fx Fxp I Js F is torsionỊ is infinite. . Proof of the Main Theorem assuming 1 and 2 . For any ỗ G Fx consider the twisting isomorphism defined over F -ựỗ Is Cs - Ci u v I - u ỗ v ỷ ỗ2 . Define ns Js - J1 to be the homomorphism associated to Is. Let Ss denote the set Is Cs F . It is easy to see that i Ss Ss if ỗ ỗ ii Ss n Ss 0 0 1 0 otherwise. For any ỗ G Fx with ỗ G n and ỗ 1 the diagram Ws F I Cs F - Js F is ns Ci F Vó J1 F -ựỗ commutes. Since the set u Ji F tfỗ tor c Ji F seF is finite by the Northcott .

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