On the non-vanishing of cubic twists of automorphic $L$-series

Abstract

Abstract. Let f be a normalised new form of weight 2 for Γ 0 (N ) over Q and F , its base change lift to Q( √ −3). A sufficient condition is given for the nonvanishing at the center of the critical strip of infinitely many cubic twists of the L-function of F . There is an algorithm to check the condition for any given form. The new form of level 11 is used to illustrate our method. IntroductionThere has been a tremendous amount of recent research on the nonvanishing of L-functions. On GL(2), the first results are due to Shimura [15], who proved that a given L-function can be twisted by a character of finite order so that the twisted L-function does not vanish at a certain point. Shimura's nonvanishing results were generalized by Rohrlich [14]. The study of the nonvanishing of quadratic twists rather than arbitrary finite order twists was started by Goldfeld, Hoffstein and Patterson [7] Iwaniec [10] to obtain nonvanishing results. The first results on the non-vanishing of cubic twists were obtained by Lieman [11]. He applied the theory of automorphic forms on the cubic cover of GL(3) to the L-series of the CM elliptic curve x 3 +y 3 = 1. In this way he obtained a non-vanishing result for cubic twists of the L-series of the automorphic form corresponding to the curve. However, because the curve has complex multiplication, the L-series in this instance is a GL(1) Hecke L-series with grossencharacter. It is this particular fact which made the GL(3) theory applicable.In this paper we will give the first non-vanishing results for cubic twists of autmorphic L-series on GL(2) that are not lifts from GL(1). In particular, we will show that if f is the new form of weight 2 and level 11, then infinitely many cubic twists of the L-series of f are non-zero at the center of the critical strip. It then follows as an immediate corollary that there are infinitely many cubic extensions K of Q( √ −3) such that the analytic rank of E = X 0 (11) over K is zero. Our approach gives a method of obtaining a similar result for any given GL(2) automorphic form. However, for reasons that will be described below, there is one step in the computation which can be verified for any given form, but which cannot yet be done in general.Received by the editors September 27, 1996 and, in revised form, February 14, 1997. 1991 Mathematics Subject Classification. Primary 11F66; Secondary 11F70, 11M41, 11N75. Our work is based on a technique recently introduced in [5]. This method involves the convolution of an automorphic form f with an Eisenstein series on the double cover of GL (2). We have applied this technique to a similar Eisenstein series on the cubic cover of GL(2) and have succeeded in obtaining information about the non-vanishing of cubic twists of the L-series of f .However, in order for the cubic Eisenstein series to be defined, the ground field must include the cube roots of unity. Thus we have constructed F , the base change of f to Q( √ −3). This is a vector valued function on the quaternionic upper half space H. We have developed a ...