Let be a Hecke eigenform, newform of weight 2 for . Its L-series is also the L-function of an elliptic curve E over . A condition for the finiteness of the group of rational points of E is for a certain quadratic character
In § 2 of [D. Bump, S. Friedberg and J. Hoffstein, Bull. Am. Math. Soc., New Ser. 21, 89-93 (1989; Zbl 0699.10038)] the theorem is announced that holds for infinitely many associated to imaginary quadratic number fields. Their proof goes along the same lines as in their earlier paper [Ann. Mat., II. Ser. 131, 53-127 (1990; Zbl 0699.10039)]. The same result follows from the main theorem in V. K. Murty [Proc. Conf. on Automorphic Forms and Analytic Number Theory, Montréal, June 1989, 89-113 (1990)].
In the paper under review a more quantitative statement is proved: for at least primitive quadratic characters with , for Y large enough. This follows from the estimates
with ; the test function F is smooth with compact support, and d runs over a set of squarefree numbers.
The proof uses an integral representation for the L-series, the symmetric square L-series associated to f, the large sieve inequality, and other techniques of analytic number theory. It is quite different from the proof of Bump, Friedberg and Hoffstein, which uses automorphic forms more heavily.