On The 2-part Of The Birch And Swinnerton-Dyer Conjecture For Quadratic Twists Of Elliptic Curves

Mathematicians have always been fascinated by the problem of describing all so-lutions in whole numbers a, b,c to algebraic equations like a2+b2=c2.This well known equation describes the relation between the three sides a, b, c of a right angled triangle, and is one of the simplest examples of a Diophantine equation. Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. In 1637, Fermat stated that the more general equation an+bn=cn had no solutions in positive integers, if n is an integer greater than 2. Although he claimed to have a general proof of his conjecture, Fermat left no details of his proof apart from the special case n=4. The proof of Fermat’s Last Theorem in full, for all n, was finally accomplished by Andrew Wiles in 1994. In the proof of Fermat’s Last Theorem, one of the key steps is an apparent link between the modularity theorem and Fermat’s Last Theorem, which was noticed by Gerhard Frey in 1984 and proven by Ribet in 1986, yielding that, if a solution to Fermat’s equation existed, it could be used to create a semistable elliptic curve of the form y2=x(x-ap)(x+bp), which was not modular. Then in 1994, Wiles proved the modularity theorem (Taniyama-Shimura-Weil conjecture) for semistable elliptic curves, which, together with Ribet’s theorem, provides a proof for Fermat’s Last Theorem.For any elliptic curve E, we can find an affine model in Weierstrass form with A, B ∈ Z, and which is a nonsingular plane curve of genus one. If E is an elliptic curve over Q, then for some integer r≥ 0, where E(Q)tors is a finite abelian group, which was proved by Mordell in 1922. The integer r is called the rank of E, which is a fundamental arithmetic invariant attached to it. It is zero if and only if E(Q) is finite. Let C be an positive integer, and let Γ0(C) denote the subgroup of SL2(Z) consisting of all matrices with the bottom left hand corner entry divisible by C, and we write X0(C) for the corresponding compactified modular curve. By the theorem of Wiles for E semistable, and its generalization to all E by Breuil-Conrad-Diamond-Taylor, there is a non-constant rational map defined over Q, which maps the cusp at infinity, which we denote by [∞], to the zero element O of E. Write [0] for the cusp of the zero point in the complex plane, so that φ([0]) is a torsion point in E(Q) by the theorem of Manin-Drinfeld. We call C the conductor of E. Let q be a prime. We now setNq:=#solutions of y2=x3+Ax+B mod q} and aq=q-Nq. Then we can define the complex L-series of E by where ε=0, ±1 according as the reduction type of E(Q) at q. We view this as a function of the complex variable s and this infinite Euler product is then known to converge for Re(s)> 3/2. It has been proved that L(E,s) has a holomorphic continuation as a function of s to the whole complex plane, and satisfies a functional equation relating, for any s, L(E, s) to L(E,2-s).To this day what is surprising is that one has no certifiable method to compute the rank for general values of A and B. That there should exist such a method is predicted by the conjecture of Birch and Swinnerton-Dyer, which is one of the most important open problems in contemporary number theory. Birch and Swinnerton-Dyer predicted that the rank is equal to an analytic invariant of the elliptic curve, namely the order of vanishing at s=1 of the L-function of the elliptic curve, i.e. ords=1L(E, s)=rank of E(Q). This conjecture was subsequently extended to include the prediction of the precise leading Taylor coefficient of the L-function at s=1. It is conjecturally given by Here |Ⅲ(E)| is the order of the Tate-Shafarevich group of the elliptic curve E, which is defined by a group which is not known in general to be finite although it is conjectured to be so. It counts the number of equivalence classes of homogeneous spaces of E which have points in all local fields. The term R∞(E) is an r × r determinant whose matrix entries are given by a height pairing applied to a system of generators of E(Q)/E(Q)tors. The cp’s are elementary local factors, namely Tamagawa factors, and ω(E) is a simple multiple of the least real period of E.For each square-free integer M, prime to C, with M=1 mod 4, we define where Ω∞(E(M)) is the least positive real period of E(M).It is well known that L(alg)(E(M),1) is a rational number. We write ord2 for the order valuation of Q at the prime 2, with the normalization ord2(2)=1. Also we define ord,2(0)=00. Let f(x) be the 2-division polynomial of E. When f(x) is irreducible over Q, we define F to be the field obtained by adjoining to Q one fixed root of f(x). Let q be any prime of good reduction for E, and let aq be the trace of Frobenius at q on E and denote Nq:=1+q-aq. For each integer m> 1, let E[m] denote the group of m-division points on E. Also, we define a rational prime q to be inert in the field F if it is unramified and there is a unique prime of F above q. By applying some results by Manin [10] and Cremona [5] on modular symbols, we prove the following general results.Theorem 0.0.1. Let E be an optimal elliptic curve over Q, with negative discriminant, with E[2](Q)=0, and satisfying ord2(L(alg)(E,1))=0. Let M be any integer of the form M=εq1q2…qr, satisfying (M, C)=1, where C is the conductor of E, r≥1, q1,..., qr are arbitrary distinct odd primes which are inert in the field F, and the sign ε=±1 is chosen so that M=1 mod 4. Then L(E(M),1)≠0, and we have ord2 (L(alg)(E(M),1))＝0. Hence, E(M)(Q) and Ⅲ(E(M)(Q)) are finite.Theorem 0.0.2. Assume the hypotheses of Theorem 0.0.1. We also suppose that the bad primes of E all split in Q((?)M), and that the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E. Then the 2-part of the Birch and Swinnerton-Dyer con-jecture holds for all the twists E(M).Theorem 0.0.3. Let E be an optimal elliptic curve over Q, with positive discriminant, with E[2](Q)=0, and satisfying ord2(L(alg)(E,1))=1. Let M be any positive integer of the form M=q1q2…qr, satisfying (M, C)=1, where C is the conductor of E, r≥1,q1,…,qr re arbitrary distinct odd primes which are inert in the the field F, and M≡1 mod 4. Then L(E)(M)1)≡0, and we have ord2(L(alg)(E(M),1))＝1. Hence,E(M)(Q) and Ⅲ(E(M)(Q)) are finite.Theorem 0.0.4. Assume the hypotheses of Theorem 0.0.3. We also suppose that the bad primes of E all split in Q((?)M), and that the 2-part of the Birch and Swinnerton-Dyer conjecture holds for E. Then the 2-part of the Birch and Swinnerton-Dyer con-jecture holds for all the twists E(M).Theorem 0.0.5. Let E be an optimal elliptic curve over Q, with negative discriminant, and with E[2](Q)≠0. Let M be any integer of the form M=εq, where q is an arbitrary odd prime with (q, C)=1, where C is the conductor of E, and the sign ε=±1 is chosen so that M=1 mod 4. Assume L(E,1)≠0. If ord2(Nq)=-ord2(L(alg)(E,1))≠0, then L(E(M),1)≠0, and we have ord2(L(alg)(E(M),1))=0. Hence, E(M)(Q) and Ⅲ (E(M)(Q)) are finite.Theorem 0.0.6. Let E be an optimal elliptic curve over Q, with positive discriminant, and with E[2](Q)≠0. Let q be any odd prime with q=1 mod 4, and (q, C)=1, where C is the conductor of E. Assume L(E,1)≠0. Iford2(Nq)=1-ord2(L(alg)(E,1))≠0, then L(E(q,1)≠0, and we have ord2(L(alg)(E(q),1))＝1. Hence, E(q)(Q) and Ⅲ(E(q)(Q)) are finite.Theorem 0.0.7. Let E be an optimal elliptic curve over Q, with negative discriminant, with E[2](Q)≠0, and satisfying L(E,1)≠0. Let M be any integer of the form M=εq1q2…qr and satisfying (M,C)=1, where C is the conductor of E,r≥1, q1,...,qr are arbitrary distinct odd primes, and the sign ε=±1 is chosen so that M=1 mod 4. If ord2(Nqi)> -ord2(L(alg)(E,1)) holds for at least one prime factor qi(q1≤i≤r) of M, then we have ord2(L(alg)(E(M),1))≥1.Theorem 0.0.8. Let E be an optimal elliptic curve over Q, with positive discriminant, with E[2](Q)≠0, and satisfying L(E,1)≠0. Let M≠1 be any integer with M≡ 1 mod 4 and (M, C)=1, where C is the conductor of E. Then we have ord2(L(alg)(E(M),1))≥1.Finally, we investigate the Neumann-Setzer elliptic curves, which have conductor p, where p is a prime of the form u2+64 for some integer u=1 mod 4, which have a minimal Weierstrass equation given by We prove the following theorem.Theorem 0.0.9. Let q be any prime congruent to 3 modulo 4 and inert in Q((?)p). When u=5 mod 8, then L(A(-q),1)≠0, and we have ord2(L(alg)(A(-1),1))＝0. Hence, A(-q)(Q) is finite, the Tate-Shafarevich group Ⅲ(A(-q)(Q)) is finite of odd cardinality. Moreover, the 2-part of Birch and Swinnerton-Dyer conjecture is valid for A(-q).We give a detailed discussion for the quadratic twists of Neumann-Setzer elliptic curves, and conjecture this would hold for a wider class.