橢圓曲線的二次扭變

Abstract

關於橢圓曲線的二次扭變，可以追溯到古希臘的一個數論問題，現在我們稱為congruent number problem。而在Ono的論文中提及一個猜想，給定一個橢圓曲線 E 會有無窮多個質數p，使得其對p的二次扭變的秩為零，同時也會有無窮多個質數q，使得其對q的二次扭變的秩為正。在這篇論文中，我們證明了，對於某類橢圓曲線，這個猜想是正確的，並且給出一個方法找出滿足其條件的質數。

Let E be an elliptic curve defined over Q, and for each square-free rational integer d, let E_d denote the the quadratic twist of E by d (in brief, the d-twist). The question concerning the rank rk(Ed(Q)) of the Mordell-Weil group E_d(Q) (the rank of E_d over Q) can be traced back to the ancient Greek congruent number problem for which the involved elliptic curve is nowadays called the congruent curve defined by y^2 = x^3-x: A square free integer d is a congruent number if and only if the d-twist of the congruent curve has positive rank over Q. There is a conjecture given in the Ono's paper. If E/Q is an elliptic curve, then there are infinitely many primes p for which E_p has rank 0 over Q, and there are infinitely many primes l for which E_l has positive rank over Q. The main aim of this thesis is to verify this conjecture for a large family of elliptic curves by giving an algorithm to find the suitable primes. For the technical reason, we need to assume that the subgroup of 2-torsion points, E[2], is contained in E(Q), or equivalently, the defining equation of E can be written as y2 = x(x - a)(x - b); a, b \in Z; 0< a < b; and (a,b) square free: Most of the previous works compute the rank of E_d over Q by computing Sel_2(E_d/Q). Our approach is slightly different, we consider L =Q(\sqrt{d}) and determine the rank of E_d over Q by computing the rank of E over L and then using the equality: rk(E(L)) = rk(E(Q)) + rk(E_d(Q)): The advantage of doing so is that the Selmer group Sel_2(E/L) becomes controllable if the extension L/Q satisfies certain condition that can be easily formulated via local data.