L-series with Nonzero Central Critical Value

Suppose that f = ∑ n≥1 an(f)q n is a cusp form of weight 2k (k ∈ N). We denote by L(f, s) the L-function of f . For Re(s) sufficiently large, the value of L(f, s) is given by L(f, s) = ∑ n≥1 an(f) ns , and one can show that L(f, s) has analytic continuation to the entire complex plane. The value of L(f, s) at s = k will be of particular interest to us, and we will refer to this value as the central critical value of L(f, s). Let χ D denote the Dirichlet character associated to the extension Q( √ D)/Q, that is, χ D (n) = ( ∆D n ) , where ∆D denotes the discriminant of Q( √ D)/Q. Define the D quadratic twist of f to be fχ D = ∑ n≥1 an(f)χD (n)q . For any integer D, the L-function of fχ D is the D quadratic twist of L(f, s), that is, L(fχ D , s) = ∑ n≥1 an(f)χD (n) ns . We will be interested in determining how often L(fχD , s) has nonzero central critical value as D varies over all integers. Since χ Dm2 = χ D , we will restrict our attention to the square-free integers D. We expect that as we let D vary over all of the square-free integers, a positive proportion of the L-functions L(fχ D , s) will have nonzero central critical value. Indeed, Goldfeld [7] conjectures that for newforms f of weight 2, L(fχ D , 1) 6= 0 for 12 of the square-free integers. Given an elliptic curve E : y = x +Ax +Bx+C (A,B,C ∈ Z) with conductor NE and an integer D, we define the D quadratic twist of E to be the curve ED : y = x+ADx+BDx+CD. Let L(ED, s) denote the L-function associated to ED. For square-free D coprime to 2NE, L(ED, s) is simply the D quadratic twist of L(E1, s). If f ∈ S2(N) is a newform with integer coefficients, we know via the theory of Eichler and Shimura that there is an elliptic curve E over Q having conductor N so that L(E, s) = L(f, s). Thus if D is coprime to 2N , then L(ED, s) = L(fχ D , s). Also, one knows from the work of Kolyvagin [13], as supplemented by the work of Murty and Murty [17] or that of Bump, Friedberg and Hoffstein [3] (see also [10] for a shorter proof), that if E is a modular elliptic curve and if L(E, 1) 6= 0, then the rank of E is 0. Thus, if f has the property that a positive proportion of the twists of L(f, s) have nonzero central critical value, then this implies that a positive density of the quadratic twists ED have rank 0. There have been many papers which have proved results in this direction. For example, in [2], [3], [6], [10], [16], [17], [19], [28] one can find general theorems on