Nonvanishing of motivic L-functions

Abstract

Let E be a modular elliptic curve over ℚ and let L(E, s) denote the associated L-function. The Birch and Swinnerton-Dyer conjecture then predicts that L(E, s) has a zero at s = 1 of order precisely equal to the rank of the Mordell–Weil group E(ℚ). According to Waldspurger's theorem [26] we know that there exists a real quadratic character χ such that the twisted L-function L(E, χ, s) does not vanish at s = 1. Recently Kolyvagin [16] has proved that E(ℚ) is finite provided L(E, 1) ≠ 0 and there exists a suitable real quadratic character χ_t such that L(E, χ_t, s) has a simple zero at s = 1. The latter condition was proved to be true for infinitely many t [4, 19]. Iwaniec [15] and Perelli and Pomykała [20] have proved quantitative results on this condition; Pomykała [21] has generalized it to the nth derivative of L(E, χ, s).

Variants and generalizations are possible. For instance, Friedberg and Hoffstein [10] established a nonvanishing theorem for quadratic twists of the L-series of an arbitrary cuspidal automorphic form on GL(2) over any number field. In some cases such a result can be applied to the construction of l-adic representations associated to modular forms over imaginary quadratic fields [25]. Further examples as well as some perspective concerning the higher rank case is discussed in an excellent survey article [5].

We should also mention that there are examples of irreducible cuspidal automorphic representations of GL(2) over a number field such that the corresponding twisted L-function vanishes at the centre of the critical strip for all quadratic characters [22].

Here is our contribution to the above picture.

Let M be a pure motive over ℚ, and let L(M, s) denote the corresponding L-function. In this paper we prove, under certain assumptions, (a quantitative version of) a nonvanishing theorem for n-th derivatives of quadratic twists of L(M, s) at the centre of the critical strip (Theorem 3). Also, assuming the Birch and Swinnerton-Dyer conjecture for abelian varieties over ℚ, we obtain a bound for the order of the (twisted) Shafarevich–Tate group (Theorems 4 and 5).

All the results of this paper except Section 5 could be stated (and proved) without using any motivic language. Almost all we need to know about motives concerns the conjectural description of their L-functions (see Section 1).