Non-vanishing of quadratic twists of modular \(L\)-functions
Let \(F(z)\) be a newform of weight \(2k\) and trivial Nebentypus. For \(D\) a square-free fundamental discriminant, let \(\chi_D\) be the quadratic character attached to the field \(K={\mathbb Q}(\sqrt D)\), and let \(L(s,F,\chi_D)\) be the twisted \(L\)-function. This paper studies the proportion of \(D\) such that the central critical value \(L(k,F,\chi_D)\) is nonzero. More precisely, the authors consider \(D\) such that a finite number of primes have specified behavior (either splitting or remaining prime) in \(K\). By work of Waldspurger, this nonvanishing holds for infinitely many such \(D\). The authors show that the number of such \(D\), \(0<|D|<X\), such that \(L(k,F,\chi_D)\neq 0\) is \(\gg X/\log X\). Taking \(F\) of weight 2, a similar statement holds for the number of twists \(E(D)\) of an elliptic curve \(E\) over \({\mathbb Q}\) which are of rank \(0\). The authors also show that, under a mild hypothesis related to the Fourier coefficients of a Shimura correspondent for \(F\), \(L(k,F,\chi_D)\neq 0\) for a positive proportion of the \(D\) as above with precisely \(r\) prime factors. (The hypothesis depends on \(r\).) They use work of Basmaji to check their hypothesis for the case that \(F\) corresponds to an elliptic curve \(E\) with conductor \(\leq 100\), and conclude that for each of these curves, either \(E(p)\) or \(E(-p)\) has rank zero for a positive proportion of the primes \(p\). The proofs use a key lemma concerning the frequency of occurrence of a given prime power in the factorization of the Fourier coefficients of a half-integral weight automorphic form which is a Hecke eigenform, the passage from modular forms to Galois representations due to Shimura, Deligne, and Serre, the work of Waldspurger on the Shimura correspondence, the nonvanishing theorem of Friedberg-Hoffstein, and Kolyvagin's work on the finiteness of the Mordell-Weil group. The conjecture that a positive proportion of the \(D\), \(0<|D|<X\), have the property that \(L(k,F,\chi_D)\neq 0\) is due to Goldfeld. Previously, Hoffstein and Luo have shown that there are infinitely many \(D\) (with specified behavior of a finite number of primes, as above) with at most 4 prime factors for which the \(L\)-value is nonzero.
scientific article; zbMATH DE number 1275724