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"Rank Zero Quadratic Twists of Modular Elliptic Curves" by Ken

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RANK ZERO QUADRATIC TWISTS OF MODULAR ELLIPTIC CURVES Ken Ono Abstract. In [11] L. Mai and M. R. Murty proved that if E is a modular elliptic curve with conductor N, then there exists infinitely many square-free integers D ≡ 1 mod 4N such that ED, the D−quadratic twist of E, has rank 0. Moreover assuming the Birch and Swinnerton-Dyer Conjecture, they obtain analytic estimates on the lower bounds for the orders of their Tate-Shafarevich groups. However regarding ranks, simply by the sign of functional equations, it is not expected that there will be infinitely many square-free D in every arithmetic progression r (mod t) where gcd(r, t) is square-free such that ED has rank zero. Given a square-free positive integer r, under mild conditions we show that there ×2 exists an integer tr and a positive integer N where tr ≡ r mod Qp for all p | N, such that 0 there are infinitely many positive square-free integers D ≡ tr mod N where ED has rank 0 0 Q zero. The modulus N is defined by N := 8 p|N p where the product is over odd primes p dividing N. As an example of this theorem, let E be defined by y2 = x3 + 15x2 + 36x. If r = 1, 3, 5, 9, 11, 13, 17, 19, 21, then there exists infinitely many positive square-free in- tegers D ≡ r mod 24 such that ED has rank 0. As another example, let E be defined by y2 = x3 − 1. If r = 1, 2, 5, 10, 13, 14, 17, or 22, then there exists infinitely many positive square-free integers D ≡ r mod 24 such that ED has rank 0. More examples are given at the end of the paper. To obtain this result we use the theory of modular forms as developed by Shimura, Vign´eras, and Waldspurger. 1. Main Theorem Let E denote the Q−rational points of the elliptic curve given by the equation y2 = x3 + Ax2 + Bx + C Key words and phrases. elliptic curves, modular forms. The author is supported by NSF grants DMS-9304580 and DMS-9508976. Typeset by AMS-TEX 1 2 KEN ONO where A, B, C ∈ Z. By Mordell’s Theorem, which was subsequently generalized by Weil, E forms a finitely generated abelian group which therefore satisfies r E ˜=Etor × Z where Etor, the torsion subgroup of E, is a finite abelian group and the rank r is a non-negative integer. Moreover by Mazur’s theorem Etor satisfies Zm | where 1 ≤ m ≤ 10, or m = 12 Etor ∈ Z2 × Z2m | where 1 ≤ m ≤ 4. It is conjectured that the rank r of E is uniquely determined by the analytic behavior of the Hasse-Weil L−function L(E, s) at s = 1. To define L(E, s), for every prime p let N(p) denote the number of points (including the point at infinity) on Ep, the reduction of E modulo p and then define A(p) by A(p) := p + 1 − N(p). Then the L−function L(E, s) is defined by the Euler product ∞ X A(n) Y 1 Y 1 L(E, s) := = . ns 1 − A(p)p−s 1 − A(p)p−s + p1−2s n=1 p|∆ p-∆ The conjectured connection between L(E, 1) and the rank of E is illustrated by the following weak version of the Birch and Swinnerton-Dyer Conjecture: P∞ A(n) Conjecture. (B-SD) Let E be an elliptic curve over Q and let L(E, s) = n=1 ns be its Hasse-Weil L−function. Then L(E, s) has an analytic continuation to the entire complex plane and the rank of E is positive if and only if L(E, 1) = 0. Significant progress has been made towards a proof of this conjecture. In particular by the works of Kolyvagin, Murty, Murty, Bump, Friedberg, and Hoffstein [1,9,12] we now know a lot about modular elliptic curves, those curves for which L(E, s) is the Mellin transform of a weight 2 newform. The main theorem we use in this paper is: Theorem 1. If E is a modular elliptic curve where L(E, 1) 6= 0, then E has rank 0. Let ψ be a Dirichlet character mod M and let f(z) be a cusp form in Sk(N, χ) P∞ n 2πiz with Fourier expansion f(z) = n=1 a(n)q (q := e throughout). Then the function fψ(z), the ψ−twist of f(z), defined by ∞ X n (1) fψ(z) = ψ(n)a(n)q n=1 2 2 is a modular form in Sk(NM , χψ ). Now we relate these twists when ψ is a quadratic character to the twists of an elliptic curve. If E is given by the equation y2 = x3 + Ax2 + Bx + C and D is a square-free integer, then the equation of its D−quadratic twist is Dy2 = x3 + Ax2 + Bx + C RANK ZERO QUADRATIC TWISTS 3 P∞ A(n) and its group of Q−rational points is denoted by ED. If L(E, s) = n=1 ns is the P∞ AD (n) L−function for E, then the L−function for its quadratic twist L(ED, s) = n=1 ns satisfies ∞ X χD(n)A(n) L(E , s) = D ns n=1 √ where χD(n) is the quadratic character for Q( D)/Q. Specifically this means that for D D primes p that AD(p) = p A(p) where p is the usual Kronecker-Legendre symbol. Hence if E is a modular elliptic curve, then by (1) ED is also modular. Now we briefly describe the theory of half-integral weight modular forms as developed c by Shimura [15]. Let N be a positive integer that is divisible by 4 and define d and d by ( c c − |d| if c, d < 0 := d c |d| otherwise. 1 d ≡ 1 mod 4 d := i d ≡ 3 mod 4. 1 Also let (cz + d) 2 be the principal square root of (cz + d) (i.e. with positive imaginary part). Let χ be a Dirichlet character mod N. Then a meromorphic function f(z) on H is called a half integer weight modular form if 2λ+1 az + b c −1−2λ λ+ 1 f = χ(d) (cz + d) 2 f(z) cz + d d d a b for all ∈ Γ (N). Such a form is called a form with weight λ + 1 and character c d 0 2 χ. The set of all such forms that are holomorphic on H as well as at the cusps is denoted by M 1 (N, χ) and is a finite dimensional −vector space. The set of those f(z) in λ+ 2 C M 1 (N, χ) that also vanish at the cusps, the cusp forms, is denoted by S 1 (N, χ). λ+ 2 λ+ 2 As in the integer weight case, there are Hecke operators that preserve M 1 (N, χ) λ+ 2 and S 1 (N, χ). However for these forms the Hecke operators act on Fourier expansions λ+ 2 in square towers; specifically if p is a prime, then the Hecke operator Tp2 acts on f(z) ∈ M 1 (N, χ) by λ+ 2 ∞ λ X 2 (−1) n λ−1 2 2λ−1 2 n (2) f(z) | T 2 := (a(p n) + χ(p) p a(n) + χ(p )p a(n/p ))q . p p n=0 As in the integer weight case, a form f(z) is called an eigenform if for every prime p there exists a complex number λp such that f(z) | Tp2 = λpf(z). The critical link between half integer weight forms and the integer weight modular forms are the Shimura lifts, a family of maps which essentially takes the L−function of 4 KEN ONO a half integer weight cusp form and returns the L-function of an integer weight modular P∞ n form. More precisely let f(z) = a(n)q ∈ S 1 (N, χ) where λ ≥ 1. Let t be a pos- n=1 λ+ 2 −1λ t itive square-free integer and define the Dirichlet character ψt by ψt(n) = χ(n) n n . Now define At(n) by the formal product of L−functions ∞ ∞ 2 X At(n) X a(tn ) := L(s − λ + 1, ψ ) . ns t ns n=1 n=1 In particular, if p is prime, then At(p) is defined by 2 λ−1 (3) At(p) := a(tp ) + ψt(p)p a(t). Then Shimura proved that the Mellin transform of this product, which we denote by P∞ n N 2 St(f(z)) = n=1 At(n)q is a weight 2λ modular form in M2λ( 2 , χ ). Furthermore if λ ≥ 2, then St(f(z)) happens to be a cusp form. If f(z) is an eigenform of all the Hecke operators Tp2 and t1 and t2 are two square-free positive integers, then (4) a(t1)St2 (f(z)) = a(t2)St1 (f(z)). P∞ n If F (z) = n=1 A(n)q is a modular form, then its L−function L(F, s) is defined by ∞ X A(n) L(F, s) = . ns n=1 P∞ n If ψ is a Dirichlet character and Fψ(z) = n=1 ψ(n)A(n)q is the ψ−twist of F (z), then we denote the modular L-function for Fψ(z) by L(F, ψ, s). In [19] Waldspurger proved beautiful formulas which connect certain special values of modular L−functions to the Fourier coefficients of half-integer weight cusp forms.
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